Age of Star according to SR Question
Peri of Pera
Dirk van der Mortel is unable to explain a question about the age of a
star.
Who can help? A star is measured by astronomers to be 10 billion years
old.
It is speeding away at .9c. What is the age of the star using the SR
time
dilation theory?
Peter Riedt
dda1
No one can answer the idiotic question you asked. Cease and desist, you
are making a fool of yourself.
Peri of Pera
>> Peter Riedt
dda1, if SR claims to be a credible science about time and velocities,
it should attempt to answer questions other than
just the twin paradox. Your answer confirms that SR cannot deal with
itself and none of its adherents can explain its theory consistently.
Peter Riedt
Spoonfed
Most GR people are fond of the closed or flat Friedmann models with a
cosmological constant, which, if accurate render your question not
quite meaningless, but incredibly complex and virtually unanswerable.
However, for an answer using the Open Friedmann model and simple
Special Relativity, yes, there is no way that a star speeding away from
us at constant speed at .9c could be 10 billion years old when we are
only 13.7 billion years old. First of all, time dilation would cause
it to be less than half of 13.7 billion years old, then, on top of
that, you have the time delay caused by the propagation of light. So
the explanation commonly given is. . . well. . . no real answer, just a
lot of tensors.
The Special Relativity answer would be that the star has not been
speeding away from us at a constant speed for those 13.7 billion years.
Not because it's velocity has been changing but because ours did...
once or many times, I don't know, and it doesn't matter. During the
early universe, the incredible heat involved accelerated our section of
the universe repeatedly in random directions.
The effect of acceleration is well known and described in the Twin
Paradox. The twin that experiences acceleration ages none at all while
millions or billions of years pass for the non-accelerating twin. This
10 billion year old star moving away from us at 0.9c experienced fewer
random accelerations in the earliest moments of the universe, i.e. it
remained more "inertial."
This also explains how they can find stars further than 7 billion light
years away even though the universe most likely started as a
singularity (the common explanation here is a vague analogy with a
balloon)
It also becomes confusing because sometimes you'll hear about the
oldest galaxies in the universe, and look at what they are talking
about, and you'll find they are actually talking about stars made of
almost pure hydrogen, which means they have not yet formed heavier
elements. This actually describes a very young galaxy, but these are
often inexplicably described as the oldest galaxies. This is in the
tradition of describing things in terms opposite or unrelated to their
meaning.
Martin Hogbin
"Peri of Pera" <rie...@yahoo.co.uk> wrote in message news:1147664233....@u72g2000cwu.googlegroups.com...
What is the point in asking someone to apply a particular theory
to something that is outside the scope of that theory?
Martin Hogbin
Hexenmeister
"Martin Hogbin" <goatREMO...@hogbin.org> wrote in message
news:36ednQTFDto...@bt.com...
What is the point of you asking everyone you don't like how to
measure the speed of train, you fucking moron?
Doppler radar. Calibrate?
Androcles.
Hexenmeister
"Spoonfed" <good4...@yahoo.com> wrote in message
news:1147666288....@i40g2000cwc.googlegroups.com...
. . . well. . . no real answer,
Typical bullshit. Relativists are fucking lying cunts.
Androcles.
Hexenmeister
"Peri of Pera" <rie...@yahoo.co.uk> wrote in message
news:1147664233....@u72g2000cwu.googlegroups.com...
|
| star.
Dork is a total moron, so who is surprised?
| Who can help? A star is measured by astronomers to be 10 billion years
| old.
| It is speeding away at .9c. What is the age of the star using the SR
| time
| dilation theory?
Anything you want it to be, nobody can check your result.
Androcles.
|
| Peter Riedt
|
Dirk Van de moortel
"Peri of Pera" <rie...@yahoo.co.uk> wrote in message news:1147664233....@u72g2000cwu.googlegroups.com...
>
See
http://groups.google.com/group/sci.physics.relativity/msg/55e87007fb4c54fb
| > Dirk,
| > I don't know how much a star
| > measured by astronomers as 10billion years old
| > and speeding away at .9c
| > is older or younger according to
| > SR's time dilation theory
|
| Indeed you don't know. Otherwise you would have made the
| exercise 4 days ago. It's a good thing that you already admit
| that you have difficulty making the exercise.
| Yet you do seem to know that "SR stands in the way of scientific
| progress" and that the "fundamentals of SR defy logic". I hope
| that you also admit that something doesn't add up here.
|
| > but if you explain it to me
| > I would appreciate it.
|
| Elsewhere on this thread I already have explained and pointed
| to everything you need to make the exercise. It is all there.You
| just have to put it together into a coherent statement.
| Don't be shy - give it a try. Show how the consequences of the
| fundamentals of SR stand in the way of scientific progress.
| Show your work.
So this is how you prove to the world what kind of piece
of disgusting little troll you are?
http://users.telenet.be/vdmoortel/dirk/Physics/Fumbles/RiedtTroll.html
Well done - and well documented.
Nice show.
Dirk Vdm
Dirk Van de moortel
A very simple answer, well inside the scope of SR, could be
given to the question. That is what I asked *him* to do. See
http://users.telenet.be/vdmoortel/dirk/Physics/Fumbles/RiedtTroll.html
Dirk Vdm
Dirk Van de moortel
"Spoonfed" <good4...@yahoo.com> wrote in message news:1147666288....@i40g2000cwc.googlegroups.com...
>
> > Dirk van der Mortel is unable to explain a question about the age of a
> > star.
> > Who can help? A star is measured by astronomers to be 10 billion years
> > old.
> > It is speeding away at .9c. What is the age of the star using the SR
> > time
> > dilation theory?
> >
> > Peter Riedt
>
> Most GR people are fond of the closed or flat Friedmann models with a
> cosmological constant, which, if accurate render your question not
> quite meaningless, but incredibly complex and virtually unanswerable.
>
[snip]
Don't try to teach a pig to sing. It is a waste of time and you
will annoy the pig.
Dirk Vdm
Spoonfed
You are making assumptions about the scope of Special Relativity. I
can only guess what they are, and I'm not sure you're aware of your
assumptions or other possibilities. You may be assuming a flat
Friedmann model where redshift is not due to recession velocity and/or
you may be assuming a steady state universe, where everything was
always here, and in which local density fluctuations caused one galaxy
to form several billion years before ours.
IF you make such assumptions, then you could say that the age of such a
star is outside the scope of Special Relativity. But you should make
your assumptions clear, because Friedmann's Model has three options;
closed, flat, and open. Only one of them is correct, and unfortunately
the vast majority of work, education, and research is going to the flat
model, which is "the conservative bet" according to some, but is
actually quite misguided.
Spoonfed
Dirk Van de moortel wrote:
> "Spoonfed" <good4...@yahoo.com> wrote in message news:1147666288....@i40g2000cwc.googlegroups.com...
> >
> > Peri of Pera wrote:
> > > Dirk van der Mortel is unable to explain a question about the age of a
> > > star.
> > > Who can help? A star is measured by astronomers to be 10 billion years
> > > old.
> > > It is speeding away at .9c. What is the age of the star using the SR
> > > time
> > > dilation theory?
> > >
> > > Peter Riedt
> >
> > Most GR people are fond of the closed or flat Friedmann models with a
> > cosmological constant, which, if accurate render your question not
> > quite meaningless, but incredibly complex and virtually unanswerable.
> >
>
> will annoy the pig.
>
> Dirk Vdm
If the closed Friedmann model with cosmological constant is a song,
then I am the pig. I gather that it is pursued not because it is
likely to be true, but because it is a rich source of topics for PhD
theses.
Dirk Van de moortel
"Spoonfed" <good4...@yahoo.com> wrote in message news:1147700868....@j55g2000cwa.googlegroups.com...
Perhaps you didn't realize whom (or better what) you were talking to:
http://users.telenet.be/vdmoortel/dirk/Physics/Fumbles/RiedtTroll.html
:-)
Dirk Vdm
rot...@gmail.com
>old. It is speeding away at .9c. What is the age of the star using the SR
>time dilation theory?
Thats easy!... Its... 10 billion years old!
Unless you/we have different conceptions about "astronomers"
measurements, "age" , astronomers do/have used SR to get the 10 billion
years old etc...
Spoonfed
Actually, I hope you'll look deeper into this. I just happened to
think of the universe in a certain way from the very beginning, and
I've never been able to successfully visualize the other two.
Notwithstanding that one of the properties of the other two is that the
universe is not visualizable. I've finally found an explanation for
our difference of opinion on the meaningfulness of distances on the
highest scales, and it is that we are operating with different Friemann
solutions.
It is as though we were asked to build a fence enclosing a yard of
thirty square meters, with one side one meter longer than the other.
We plug into the equations xy=30 and y=x+1. One of us gets the
solution x=5, y=6, while the other gets -6,-5. Then book after book is
written explaining why a -6 by -5 meter fence is a legitimate solution
of the fencing problem, and anyone who expresses an interest in
pursuing the 5 X 6 solution is obviously poorly educated because they
don't even know about the -6 X -5 solution.
This is similar to the Friedmann solutions.
H^2= (8 Pi G rho) / (3 c^2) + LAMBDA/3 - k/R^2
There are three solutions k=1 for a closed universe, k=0 for flat, and
k=-1 for open. Book after book has been written favoring flat and
closed geometries. And in these geometries, special relativity does
not apply over cosmological distances--which is echoed unconditionally
and without qualification by students of General Relativity.
But the open solution where k= -1, and LAMBDA=0, and rho is small, the
equation simplifies to R=1/H. The scale of the universe is equal to
the age of the universe. There may be a little bit of modification to
this here and there due to gravitational fluctuations, but overall, the
geometry is very simple space-time.
In all discussions of cosmology where I have made blanket statements
about space-time, I have been assuming (without even realizing there
was a question) that the universe was open, whereas many of the people
I have been talking to have been assuming (also without realizing there
was a question) that the universe was flat or closed.
Now that I know there is a question, It is pretty easy to identify what
model people are using from the statements they make or the questions
they ask. For instance "The universe is like the surface of a
balloon--it expands but has no center" describes the closed model.
"Redshift is caused by the scale of the universe and gravity" describes
the flat model. Questions about how the huge region of space
encompassed by the cosmic background radiation all managed to come to
the same temperature at the same time is a question implicitly assuming
the flat or closed model. Questions involving the acoustic signature
of the cosmic background use a flat or closed model. Questions of how
a star traveling at 0.9c can be observed 10 billion years old
implicitly assumes the open model. The question of how far away the
particles producing the CBR currently are implicitly assumes the open
model. Each model has its own set of questions, and proponents of each
will see the questions asked by the other as silly.
However, if all three models are treated separately either as equal or
unequal possibilities, there might be a little less accusation of
crackpottery.
Tim
> In all discussions of cosmology where I have made blanket statements
> about space-time, I have been assuming (without even realizing there
> was a question) that the universe was open, whereas many of the people
> I have been talking to have been assuming (also without realizing there
> was a question) that the universe was flat or closed.
Another paradigm forming question:
Why spacetime?
The empirical R'xRxRxR is de facto.
Somehow it has been beyond question as to why even as string theorists
add more dimensions. But some
http://insti.physics.sunysb.edu/~siegel/research.shtml
http://arxiv.org/abs/hep-th/0506053
are working on it.
A simplistic theoretical foundation for spacetime can be found at:
http://bandtechnology.com/PolySigned/PolySigned.html
The time correspondence in one-signed numbers is strong.
I do not know which of the three forms cosmology takes on but if the
polysigned approach is right then they will probably have something to
say about it.
-Tim
Peri of Pera
Hexenmeister wrote:
Hxm, I want Dirk to tell me what he wants it to be.
Peter Riedt
Spoonfed
> Spoonfed wrote:
> > In all discussions of cosmology where I have made blanket statements
> > about space-time, I have been assuming (without even realizing there
> > was a question) that the universe was open, whereas many of the people
> > I have been talking to have been assuming (also without realizing there
> > was a question) that the universe was flat or closed.
>
> Another paradigm forming question:
>
> Why spacetime?
>
But now I read into your paper, you say "Physicists generally accept
space-time because it is observed. Few trouble over why." You aren't
questioning whether spacetime exists; you are questioning why it
exists. This is a great question, but pretty far astray from what I was
talking about. Your summary reminds me very vaguely of Feynman's
symmetries, where every physical quantity has some kind of symmetry
associated with it.
> The empirical R'xRxRxR is de facto.
> Somehow it has been beyond question as to why even as string theorists
> add more dimensions. But some
> http://insti.physics.sunysb.edu/~siegel/research.shtml
> http://arxiv.org/abs/hep-th/0506053
> are working on it.
> A simplistic theoretical foundation for spacetime can be found at:
> http://bandtechnology.com/PolySigned/PolySigned.html
> The time correspondence in one-signed numbers is strong.
> I do not know which of the three forms cosmology takes on but if the
> polysigned approach is right then they will probably have something to
> say about it.
>
> -Tim
If your approach actually derives space and time into existence,
perhaps it would settle the question of which geometry to use. I will
probably just take the approach of trying to showing that an open
spacetime geometry is consistent with observation.
Hexenmeister
Why, are you trying to teach him or something?
It won't work, he's a pee puppy. He'll piss on your leg.
|
| Peter Riedt
|
Tim
> But now I read into your paper, you say "Physicists generally accept
> space-time because it is observed. Few trouble over why." You aren't
> questioning whether spacetime exists; you are questioning why it
> exists. This is a great question, but pretty far astray from what I was
> talking about. Your summary reminds me very vaguely of Feynman's
> symmetries, where every physical quantity has some kind of symmetry
> associated with it.
> perhaps it would settle the question of which geometry to use. I will
> probably just take the approach of trying to showing that an open
> spacetime geometry is consistent with observation.
Thanks for spending some time on it.
The discrepancies with the accepted construction are suggestive.
Traditionally a zero-dimensional quantity is considered to be a point.
That is still what one-signed numbers render out to, but there is a
little bit more there.
Just enough to get time with it's arrow and without any graphical
measure.
Also multidimensional spaces are developed without a Cartesian product.
These concepts are fundamental mathematics.
I am starting to believe that we exist not in a Cartesian product space
but in a particle product space. This would allow for a polysigned
substrate whose product yields spacetime. This view may rely solely on
the superposition principle i.e. we need only look at particles two at
a time and allow that more are satisfied combinatorically.
In this context a sole distance exists between two particles, not four
independent distances. Nicely this perspective generates binary charge
with axis out of generic point particles when taken relatively through
T3(P1 + P2 + P3). I can even argue spin 1/2 and spin 1 behavior is
exhibited in P4 when this invariance is invoked, though my claim is
shaky because I don't have enough understanding of spin.
To your original point...
If this product space is accurate it needs to be implemented in
arithmetical terms. Classical force equations look promising since they
provide a product relationship. However, the /r/r term is in conflict.
Consider the following transformation:
Y = 1 / ( 1 + X )
where X is traditional Cartesian distance and Y is now a measure
ranging from unity at what was the origin to zero at far away places.
In effect we take a tape meaure, chop off the first inch, and flip it
upside down. It's the same basic tape measure slightly modified so that
now the force equation becomes:
F = q1 q2.
Here q is discrete via its sign and holds a magnitude which is the new
tape measure distance. All infinities are abolished, even at near
distance. The transform is intuitively satisfying in that unity is
local and zero (far away) things are meaningless. For large X the two
systems are equivalent. If we are discussing point particles then this
transform may be legitimated via limit theory. It is just a matter of
choosing unity in X and allowing this choice to exist on a continuum.
Strangely the resultant F is back in X (e.g. F = ma ) but this is a
good thing since it will need vector summation (superposition) which
doesn't occur easily in Y. Having taken the product we have established
spacetime according to the polysigned construction. It's still a bit
vague to me how directional vectors come about, but there are some nice
fits in the overall picture.
So, if spacetime is built upon a substrate (Y) the whole picture looks
different and the context of a lot of questions may change. It seems
fairly pertinent to your context.
To quibble over a zero at far away places versus an infinity lightens
the problem.
The worry may become about a photon from far away.
What direction can a photon from zero(Y) take?
-Tim
Spoonfed
> Spoonfed wrote:
> > But now I read into your paper, you say "Physicists generally accept
> > space-time because it is observed. Few trouble over why." You aren't
> > questioning whether spacetime exists; you are questioning why it
> > exists. This is a great question, but pretty far astray from what I was
> > talking about. Your summary reminds me very vaguely of Feynman's
> > symmetries, where every physical quantity has some kind of symmetry
> > associated with it.
> > If your approach actually derives space and time into existence,
> > perhaps it would settle the question of which geometry to use. I will
> > probably just take the approach of trying to showing that an open
> > spacetime geometry is consistent with observation.
>
> Thanks for spending some time on it.
I looked at it a little more... You say "the special properties of + do
not extend beyond the two-signed domain." However, then you mention
that (*1)(anything) = anything. In this case, at least in
multiplcation, *1 takes the role of +1.
Also, in the three-signed case, you can use complex numbers to mimic
the properties.
"-1" = e^(i 2 Pi /3) = cos(120)+i sin(120)
"+1" = e^ (i 4 Pi / 3) = cos (240) + i sin (240)
"*1" = e^(i 6 Pi / 3) = 1 = cos (360) + i sin (360)
You probably already know this, but e^(i x), sin (x) and cos(x) as well
as less familiar functions... bessel functions, legendre polynomials
can be found with Taylor Expansion series.
e^x = x^0/0! + x^1/1! + x^2/2! + x^3/3! +...
Plug an imaginary number in for x and you'll see a real and imaginary
part. The real part is the definition of cosine and the imaginary part
is the definition of sine.
Anyway, I just wondered if you were aware that there is something
similar in traditional mathematics to your multi-signed numbers in
traditional mathematics. Four signed numbers would have angles 90,
180, 270, and 360. Five signed would have angles 72, 144, 216, 288,
360. It came as some surprise to me when I realized that any root of
-1 could be described as a complex number.
But this isn't similar to yours for four-signed numbers, because in
your example you show four-signed numbers in three dimensions in a
tetrahedral pattern.
Now it occurs to me that your three signed numbers and four-signed
numbers are linearly dependent. That is, one of your unit vectors can
be described as a linear combination of the other two. I don't know
whether that's important to you or not.
> The discrepancies with the accepted construction are suggestive.
> Traditionally a zero-dimensional quantity is considered to be a point.
That could use some clarification. Quantities have units, and that
entails dimension. A location might be zero-dimensional. Also, a
quantity might have zero uncertainty in its value, so in that way,
maybe could be considered zero dimensional.
> That is still what one-signed numbers render out to, but there is a
> little bit more there.
Can you add these one-signed numbers together? Or is the only
one-signed number zero?
> Just enough to get time with it's arrow and without any graphical
> measure.
*poof* what? how'd time get there? Because it is observed?
> Also multidimensional spaces are developed without a Cartesian product.
> These concepts are fundamental mathematics.
> I am starting to believe that we exist not in a Cartesian product space
> but in a particle product space.
Is it possible there might be more than one valid representation of the
same thing? This vaguely reminds me of the Lagrangian Method, which
greatly simplifies a lot of mechanics problems by replacing the
cartesian coordinate system with a system of generalized coordinates
based on the interaction of objects.
> This would allow for a polysigned
> substrate whose product yields spacetime. This view may rely solely on
> the superposition principle i.e. we need only look at particles two at
> a time and allow that more are satisfied combinatorically.
> In this context a sole distance exists between two particles, not four
> independent distances.
Now you've lost me. Where do you get the idea anybody thinks there are
four independent distances between particles?
> Nicely this perspective generates binary charge
> with axis out of generic point particles when taken relatively through
> T3(P1 + P2 + P3). I can even argue spin 1/2 and spin 1 behavior is
> exhibited in P4 when this invariance is invoked, though my claim is
> shaky because I don't have enough understanding of spin.
>
> To your original point...
> If this product space is accurate it needs to be implemented in
> arithmetical terms. Classical force equations look promising since they
> provide a product relationship. However, the /r/r term is in conflict.
In conflict with what? The surface area of a sphere is proportional to
r^2. The part you are exposed to is 1 part of the surface area. As
you back off, the part you are exposed to decreases by 1/r^2
Did you know that the force for an infinite cylinder is proportional to
1/r, and for an infinite plane is constant?
> Consider the following transformation:
> Y = 1 / ( 1 + X )
> where X is traditional Cartesian distance and Y is now a measure
> ranging from unity at what was the origin to zero at far away places.
Kind of like taking a train track that extends infinitely to the
horizon, and marking off the positions where you see it through a glass
window?
> In effect we take a tape meaure, chop off the first inch, and flip it
> upside down. It's the same basic tape measure slightly modified so that
> now the force equation becomes:
> F = q1 q2.
I'd have to ask what you started with. If F=(q1 q2)/x^2, this yields
(q1 q2 Y^2)/(1-Y)^2
Oops, gotta shut down the computer... thunderstorm.
Peri of Pera
Spoonfed, I annoyed Dirk because I must have said something he didn't
like. Could he be the hijacker? What is his role in this NG?
Peter Riedt
Dirk Van de moortel
"Peri of Pera" <rie...@yahoo.co.uk> wrote in message news:1147839962....@38g2000cwa.googlegroups.com...
>
> Peter Riedt
Ah, don't be afraid, you didn't annoy me. Quite on the contrary.
You clearly showed what a little prat you are:
http://users.telenet.be/vdmoortel/dirk/Physics/Fumbles/RiedtTroll.html
As you can see, you did it in a very convenient and self-documenting
way. Excellent job.
Dirk Vdm
Androfizzx
"Dirk Van de moortel" <dirkvand...@ThankS-NO-SperM.hotmail.com> wrote
in message news:MNzag.430217$lz1.11...@phobos.telenet-ops.be...
[snip]
Mission accomplished.
Androfizzx
Spoonfed
No--it was just that I realized Tim and I had gone completely off the
topic of the star question. From what I can tell, Dirk and I
understand Special Relativity in just about the same way. And he's
willing to put the effort into explaining it as long as he thinks
you'll listen. But then if he does put in the effort, it's his pet
peeve if you just leave the thread dangling, and go posting the same or
similar question in another thread (which you did), he gets kind of
cranky. He takes his revenge by finding the stupidest thing you said
and immortalizing it on his website. He's fairly honest about it
though--he generally doesn't quote out of context.
There are several people that post on the newsgroup that really seem
bright enough to understand the theory but choose not to; repeating the
same thing time after time, and completely oblivious to the mistake
they are making even though it has been pointed out to them over and
over again, usually refusing to acknowledge even that anyone pointed
out the mistake. For me, Dirk performs a useful role by cataloguing
these posts so that I can distinguish between people who are completely
immune to logic, and people who might just be off the beaten path.
But on this specific topic of the current age of a distant object, Dirk
and I agreed to disagree some time ago.
http://groups.google.com/group/sci.physics.relativity/msg/cfc2edf4396876f5?&hl=en
And in so doing, I believe I also agreed to disagree with Gould, Baez,
Penrose, Hawking, John Nash, and most other famous physicists.
However, I was gratified to find Mike Fontenot in my court, who wrote a
paper on the Current Age of Distant Objects (CADO).
http://groups.google.com/group/sci.physics.relativity/msg/fb97d9745316a097?hl=en&
The problem is, there's not much of an argument I can make against a
flat denial of the reality of the current age of a distant object. I
can point at a tree all day and say it's there, but a determined cynic
could develop all manner of argument to say that it's not. I have to
figure out why someone would think it was meaningless.
I believe it has to do with Alexander Friedmann's 1922 paper "Uber die
Krummung des Raumes" (Zeitscrhift fur Physik 10:377-86) which I
wouldn't be able to read even if I saw it. Somewhere in this paper,
Friedmann does a boundary value problem, solving the Einstein Field
Equations, and comes up with three classes of solutions for the
geometry of the universe. The classes are closed, flat, and open
models.
In the flat and closed models the scale factor can be a complex or
imaginary number. In such a model the current age of distant objects
would have no physical meaning so Dirk and all the others would be
right. But in the open model the scale factor is always a real
positive number and the current age of distant objects is a valid
quantity; then Mike Fontenot and I would be right. I am quite
convinced that the open model is the correct one but I'm not a
brilliant physicist. I'm just a guy with a fondness for real valued
distances.
Dirk Van de moortel
"Spoonfed" <good4...@yahoo.com> wrote in message news:1147880095.1...@j33g2000cwa.googlegroups.com...
But all this has nothing to do with open, flat or closed models, or
boundary value problems, or with field equations, or with scale
factors or complex or imaginary numbers. It also has also nothing
to do with you or Mike or me or the 'others' being right or wrong.
It was about the *uselessness* of the concept of "*now* in a
remote location as judged from an *accelerating* frame".
So you don't "have to figure out why someone would think it was
meaningless". We already agreed that it is not meaningless and
that it is "perfectly valid, as long as things are made explicit".
The concept is just useless because I can fiddle with the *now*
of the remote location by simply adjusting my thrusters.
Riedt pretended that he wanted to know about the simple case
of the age of a remote inertial object as seen from inertial frame.
He claims he knows better, but he can't even explain what it is
'that he knows better than'.
The pig doesn't want to learn to sing. Pigs just snort and fart.
Dirk Vdm
Spoonfed
YOU CAN FIDDLE WITH THE NOW OF THE REMOTE LOCATION BY ADJUSTING YOUR
THRUSTERS. What about this do you think is useless or meaningless?
>
> Riedt pretended that he wanted to know about the simple case
> of the age of a remote inertial object as seen from inertial frame.
By skill or by luck, it's still a good question. Some people need to
see several examples before they can understand things clearly.
> He claims he knows better, but he can't even explain what it is
> 'that he knows better than'.
Riedt, what do you know better than?
Dirk Van de moortel
"Spoonfed" <good4...@yahoo.com> wrote in message news:1147921757.7...@u72g2000cwu.googlegroups.com...
I just said it's not meaningless.
Refresher:
http://groups.google.com/group/sci.physics.relativity/msg/cfc2edf4396876f5
>
> >
> > Riedt pretended that he wanted to know about the simple case
> > of the age of a remote inertial object as seen from inertial frame.
>
> By skill or by luck, it's still a good question. Some people need to
> see several examples before they can understand things clearly.
Riedt does not belong to that kind of people.
Refresher:
http://users.telenet.be/vdmoortel/dirk/Physics/Fumbles/RiedtTroll.html
>
> > He claims he knows better, but he can't even explain what it is
> > 'that he knows better than'.
>
> Riedt, what do you know better than?
Refresher:
http://users.telenet.be/vdmoortel/dirk/Physics/Fumbles/RiedtTroll.html
Dirk Vdm
Hexenmeister
[snip]
Mission accomplished.
Androcles
Hexenmeister
[snip]
Mission accomplished.
Androcles
Tim
Spoonfed wrote:
> I looked at it a little more... You say "the special properties of + do
> not extend beyond the two-signed domain." However, then you mention
> that (*1)(anything) = anything. In this case, at least in
> multiplcation, *1 takes the role of +1.
> Also, in the three-signed case, you can use complex numbers to mimic
> the properties.
> "-1" = e^(i 2 Pi /3) = cos(120)+i sin(120)
> "+1" = e^ (i 4 Pi / 3) = cos (240) + i sin (240)
> "*1" = e^(i 6 Pi / 3) = 1 = cos (360) + i sin (360)
>
> You probably already know this, but e^(i x), sin (x) and cos(x) as well
> as less familiar functions... bessel functions, legendre polynomials
> can be found with Taylor Expansion series.
>
> e^x = x^0/0! + x^1/1! + x^2/2! + x^3/3! +...
It would be really neat if e^x meant something for P4 and up!
> Plug an imaginary number in for x and you'll see a real and imaginary
> part. The real part is the definition of cosine and the imaginary part
> is the definition of sine.
>
> Anyway, I just wondered if you were aware that there is something
> similar in traditional mathematics to your multi-signed numbers in
> traditional mathematics. Four signed numbers would have angles 90,
> 180, 270, and 360. Five signed would have angles 72, 144, 216, 288,
> 360. It came as some surprise to me when I realized that any root of
> -1 could be described as a complex number.
You are imposing higher signs on a 2D space here. The polysigned
construction requires that four-signed numbers are 3D and five-signed
(P5) are 4D. Their angles come from the geometry of the simplex.
Projected down to 2D they could take on non harmonic values.
These systems do naturally generate unity roots and rotation.
> But this isn't similar to yours for four-signed numbers, because in
> your example you show four-signed numbers in three dimensions in a
> tetrahedral pattern.
Right
> Now it occurs to me that your three signed numbers and four-signed
> numbers are linearly dependent. That is, one of your unit vectors can
> be described as a linear combination of the other two. I don't know
> whether that's important to you or not.
I don't see it this way. I really appreciate you spending energy on
this construction. I see P3 and P4 as two individual number systems
that are part of the same family. They are adjacent in the natural
progression that is the family of polysigned numbers. This is where the
product behavior goes haywire so it is of interest. The notion of a
linear relation between P3 and P4 is doubtful. I have spent some time
on this and P4 does look a lot like
P2 X P3 ( or R X C in traditional terms)
but I have yet to define a product on P3 X P2 that transforms exactly.
Still, If i do a 3D Mandelbrot plot of P4 it looks like a thick
Mandelbrot set as if it were true.
P2 comes out as the identity axis of P4 with P3 perpendicular to that.
But there is still some error. You can see this a bit in
http://bandtechnology.com/PolySigned/Deformation/DeformationUnitSphereP4.html
Sorry for the complicated and unfinished answer. How do you see linear
dependence in P3 and P4?
> > The discrepancies with the accepted construction are suggestive.
> > Traditionally a zero-dimensional quantity is considered to be a point.
> That could use some clarification. Quantities have units, and that
> entails dimension. A location might be zero-dimensional. Also, a
> quantity might have zero uncertainty in its value, so in that way,
> maybe could be considered zero dimensional.
> > That is still what one-signed numbers render out to, but there is a
> > little bit more there.
> Can you add these one-signed numbers together? Or is the only
> one-signed number zero?
> > Just enough to get time with it's arrow and without any graphical
> > measure.
> *poof* what? how'd time get there? Because it is observed?
OK. This is an important one. I cover it pretty thoroughly at
http://bandtechnology.com/PolySigned/OneSigned.html
The polysigned numbers are intimately tied to dimensionality.
I take the one-signed numbers as the definition of zero dimensional.
We can do things like:
- 1.1 ( - 2.3 - 0.5 ) = - 3.08
in one-signed numbers.
Thinking in terms of superposition they can only get larger and larger.
So for example an integral of them should generate a nondecreasing
function.
Yet all the while when one graphs them they yield zero.
That is a funny thing about the cancellation law. You only actually
need it when you graph these things. All math can be done in any sign
level without ever invoking the cancellation.
In effect the components will keep building larger and larger values.
So long as they remain balanced you get a local answer.
The one-signed time correspondence is perfect because of this behavior.
It allows time to be part of spacetime without any graphical measure.
Do one-signed numbers have a magnitude? Yes:
| - 3.08 | = 3.08 .
But rendering that magnitude always yields zero.
To buy it you have to adopt the polysigned numbers and take them
literally.
One-signed numbers are time much much more convincingly than the real
numbers.
> > Also multidimensional spaces are developed without a Cartesian product.
> > These concepts are fundamental mathematics.
> > I am starting to believe that we exist not in a Cartesian product space
> > but in a particle product space.
> Is it possible there might be more than one valid representation of the
> same thing? This vaguely reminds me of the Lagrangian Method, which
> greatly simplifies a lot of mechanics problems by replacing the
> cartesian coordinate system with a system of generalized coordinates
> based on the interaction of objects.
The polysigned numbers are nonorthogonal so there are inherently many
representations for the same geometric position. I'd like to coin a
phrase like 'minimally nonorthogonal' for they barely waste information
and it can even be argued that they reduce information over their
Cartesian counterpart. A 2D cartesian representation takes 2.2 chunks
of information whereas the polysigned equivalent can do it in 2.15
chunks where 1 chunk is a magnitude and 0.1 chunks is a bit that
becomes sign information. Sorry if that is cryptic. I can expound if
you wish.
>
> > This would allow for a polysigned
> > substrate whose product yields spacetime. This view may rely solely on
> > the superposition principle i.e. we need only look at particles two at
> > a time and allow that more are satisfied combinatorically.
> > In this context a sole distance exists between two particles, not four
> > independent distances.
>
> Now you've lost me. Where do you get the idea anybody thinks there are
> four independent distances between particles?
This is almost a rhetorical question. When we instantiate particle
positions in traditional spacetime representations we are forced to
assign four values. All of them are distances and they are independent
of each other hence the 4D spacetime model.
The traditional model is:
1D + 1D + 1D + 1D.
The polysigned model is:
0D + 1D + 2D.
To instantiate a particle position the tatrix requires six magnitudes.
Three of these can be disappeared by cancellation leaving three
magnitudes.
But there is another way and it gets around the nonisotropic quality of
the topology.
It uses relativity. Consider a universe with just two generic point
particles a and b. One solitary distance. This implies that:
| a1 | = | a2 | = | a3 | = | b1 | = | b2 | = | b3 |
where these are the 0D, 1D, and 2D components.
Each particle presents to the other particle in a relative fashion.
Claiming invariant distance on the topology leaves a few degrees of
freedom since the original format was nonorthogonal.
In 2D we see that there is a sign choice.
Let's say the distance between a and b is 1.234.
This means that in 2D the distance is 1.234 and in 3D it is 1.234.
a2 could be -1.234 or +1.234 or it could even be - 1.1 + 2.334, etc.
b2 is valued likewise. But ultimately this freedom is merely a choice
of two signs. It is a binary quality. In 3D the freedom is one of an
angle. Each particle can freely present these qualities to the other
particle.
This same procedure in the 1D + 1D + 1D + 1D would yield four binary
choices per particle. Not very sensible.
The polysigned approach generates 3.2 chunks of information and the
traditional approach generates 1.4 chunks. The distance is included as
one chunk. Very different.
Thinking this way can open up questions about d(a,b) versus d(b,a) but
this analysis assumes that they are equal.
>
> > Nicely this perspective generates binary charge
> > with axis out of generic point particles when taken relatively through
> > T3(P1 + P2 + P3). I can even argue spin 1/2 and spin 1 behavior is
> > exhibited in P4 when this invariance is invoked, though my claim is
> > shaky because I don't have enough understanding of spin.
> >
> > To your original point...
> > If this product space is accurate it needs to be implemented in
> > arithmetical terms. Classical force equations look promising since they
> > provide a product relationship. However, the /r/r term is in conflict.
>
> In conflict with what? The surface area of a sphere is proportional to
> r^2. The part you are exposed to is 1 part of the surface area. As
> you back off, the part you are exposed to decreases by 1/r^2
The polysigned numbers generate spacetime as a product relationship.
It is the product behavior that allows the claim.
So this transformation is a step in that direction.
>
> Did you know that the force for an infinite cylinder is proportional to
> 1/r, and for an infinite plane is constant?
No I didn't know that. I don't follow the model. Are you instantiating
point charges on these structures? I don't see how that could yield a
constant force at any distance. What about a line? Does it blow up?
>
> > Consider the following transformation:
> > Y = 1 / ( 1 + X )
> > where X is traditional Cartesian distance and Y is now a measure
> > ranging from unity at what was the origin to zero at far away places.
>
> Kind of like taking a train track that extends infinitely to the
> horizon, and marking off the positions where you see it through a glass
> window?
No. Oddly the tape measure looks almost the same. It's just that the
units read:
1/1, 1/2, 1/3, 1/4, ...
instead of:
0, 1, 2, 3, ...
Cut off the first inch and flip the numbers.
>
> > In effect we take a tape meaure, chop off the first inch, and flip it
> > upside down. It's the same basic tape measure slightly modified so that
> > now the force equation becomes:
> > F = q1 q2.
>
> I'd have to ask what you started with. If F=(q1 q2)/x^2, this yields
> (q1 q2 Y^2)/(1-Y)^2
Let's let r be in X so it is the usual value.
The classical charge force equation reads:
F = ( q1 / r ) ( q2 / r )
where q1 and q2 are charges that have a magnitude and a sign and r is a
distance.
We believe that charge is actually quantized so we could forgo some
more constants and treat q1 and q2 as just signs. i.e. each of them is
+ 1 or - 1. Let's do that.
But the polysigned numbers want a geometrical product relationship.
We can make this product a geometrical product by transforming distance
by
1 / X
but even neater is to go to
1 / ( X + 1 )
So in Y we get:
F = (q1( r + 1 )) (q2( r + 1 )).
For large r (which still has the sense of X) the 1 will become
irrelevant.
Since the distance is unsigned we can combine it into the q which is
currently just a sign to get
F = q1 q2
where q's magnitude is its distance in Y and q's sign is its charge.
We now have a geometric product that is equivalent to the classical
equation for large r.
If we are discussing point particles there should not be any conflict.
Infinity is gone at near distances and at far distances.
I have no idea what this would say about strong and weak forces.
I'm just trying to get polysigned physics working.
I also like the transform because it has an intuitive sense about it.
Unity is local and zero is meaningless.
-Tim
Spoonfed
Dirk Van de moortel wrote:
[snipped for brevity]
> > >
> > > But all this has nothing to do with open, flat or closed models, or
> > > boundary value problems, or with field equations, or with scale
> > > factors or complex or imaginary numbers. It also has also nothing
> > > to do with you or Mike or me or the 'others' being right or wrong.
> > > It was about the *uselessness* of the concept of "*now* in a
> > > remote location as judged from an *accelerating* frame".
> > > So you don't "have to figure out why someone would think it was
> > > meaningless". We already agreed that it is not meaningless and
> > > that it is "perfectly valid, as long as things are made explicit".
> > > The concept is just useless because I can fiddle with the *now*
> > > of the remote location by simply adjusting my thrusters.
> > >
> >
> > YOU CAN FIDDLE WITH THE NOW OF THE REMOTE LOCATION BY ADJUSTING YOUR
> > THRUSTERS. What about this do you think is useless or meaningless?
>
> I just said it's not meaningless.
Oh, yes, so you did, sorry. My eyes skipped past the word "not" for
some reason when I read this earlier. Then I spent an hour yesterday
evening, wondering how it was that you thought I had agreed to such a
thing, LOL!
So that makes it a much simpler question, without capital letters. You
say, (and I agree) you can fiddle with the *now* of the remote location
by adjusting your thrusters. What do you mean by useless?
Are you saying that there is no possible way that anyone could ever
possibly find a way to exploit this capability? Or do you think it is
so unimportant that it should not be mentioned in the teaching of SR?
Because whether or not we can exploit the capability of changing
distant *now* to do anything useful, it certainly must affect our
observations of the cosmos.
Randy Poe
Tim wrote:
> This thread's got drift alright.
> > 1/r, and for an infinite plane is constant?
> No I didn't know that. I don't follow the model. Are you instantiating
> point charges on these structures?
The above statement is based on an assumption of uniform
charge density.
> I don't see how that could yield a
> constant force at any distance.
The quickest derivation of both results is with Gauss' Law.
http://hyperphysics.phy-astr.gsu.edu/Hbase/electric/gaulaw.html
Scroll down to "Applications of Gauss' Law" and click on
the sheet of charge and the conducting cylinder.
> What about a line? Does it blow up?
No, 1/r, just like the cylinder, by essentially the same
argument.
- Randy
Spoonfed
> > see several examples before they can understand things clearly.
>
> Riedt does not belong to that kind of people.
> Refresher:
> http://users.telenet.be/vdmoortel/dirk/Physics/Fumbles/RiedtTroll.html
>
> >
> > > He claims he knows better, but he can't even explain what it is
> > > 'that he knows better than'.
> >
> > Riedt, what do you know better than?
>
> Refresher:
> http://users.telenet.be/vdmoortel/dirk/Physics/Fumbles/RiedtTroll.html
>
> Dirk Vdm
Incidentally, Riedt, your question really doensn't have enough info in
it. But then again, look what you have to work with:
http://news.bbc.co.uk/1/hi/sci/tech/2381935.stm (Here is an awful
example. Old meaning young, meaning just coalesced out of primeval
material.)
http://www.newton.dep.anl.gov/askasci/ast99/ast99323.htm (Here's a
really bad answer to the question, given by a professional astronomer.)
Repeated verbatim elsewhere
(http://www.nameastargift.com/astronomydictionary/oldeststar.html)
http://www.abc.net.au/news/newsitems/200504/s1344811.htm (Without even
enough information to begin to guess what they are saying.)
Except for rare exceptions, they probably won't give enough information
to even ask the question, let alone answer it.
Dirk Van de moortel
"Spoonfed" <good4...@yahoo.com> wrote in message news:1147992477....@i39g2000cwa.googlegroups.com...
>
> Dirk Van de moortel wrote:
>
> [snipped for brevity]
>
> > > >
> > > > But all this has nothing to do with open, flat or closed models, or
> > > > boundary value problems, or with field equations, or with scale
> > > > factors or complex or imaginary numbers. It also has also nothing
> > > > to do with you or Mike or me or the 'others' being right or wrong.
> > > > It was about the *uselessness* of the concept of "*now* in a
> > > > remote location as judged from an *accelerating* frame".
> > > > So you don't "have to figure out why someone would think it was
> > > > meaningless". We already agreed that it is not meaningless and
> > > > that it is "perfectly valid, as long as things are made explicit".
> > > > The concept is just useless because I can fiddle with the *now*
> > > > of the remote location by simply adjusting my thrusters.
> > > >
> > >
> > > YOU CAN FIDDLE WITH THE NOW OF THE REMOTE LOCATION BY ADJUSTING YOUR
> > > THRUSTERS. What about this do you think is useless or meaningless?
> >
> > I just said it's not meaningless.
>
>
> Oh, yes, so you did, sorry. My eyes skipped past the word "not" for
> some reason when I read this earlier. Then I spent an hour yesterday
> evening, wondering how it was that you thought I had agreed to such a
> thing, LOL!
>
> So that makes it a much simpler question, without capital letters. You
> say, (and I agree) you can fiddle with the *now* of the remote location
> by adjusting your thrusters. What do you mean by useless?
My sister is 30 years old now. Let's take off.
4 years later...
How old is my sister now? Let's calculate.... ha, she must be 31.0 now.
3 years later...
How old is my sister now? Let's calculate.... ha, she must be 44.2 now.
3 years later...
How old is my sister now? Let's calculate.... ha, she must be 31.7 now.
3 years later...
How old is my sister now? Let's calculate.... ha, she must be 57.7 now.
...
That's what I call useless numbers.
>
> Are you saying that there is no possible way that anyone could ever
> possibly find a way to exploit this capability?
No, but I strongly doubt it.
You probably aren't saying that someone *will* find a way to exploit it
either.
> Or do you think it is
> so unimportant that it should not be mentioned in the teaching of SR?
What can (and apparently should) be mentioned and showed, is
that it's pretty useless ;-)
Dirk Vdm
Spoonfed
( am guessing R and C stand for Real and Complex.) So I gather you're
saying that P2, your 2-signed numbers is equivalent to the real
numbers, (since it goes + and - and covers an entire dimension) while
your P3 is equivalent to the complex numbers, since it covers two
dimensions.
I do see sort of a pattern from P1, P2, P3, to P4 in the sense that you
are taking vectors and pointing them as far away from each other
they'll go in as many dimensions as necessary to do so.
I now see that P2 represents a full number line extending linearly from
-infinity to +infinity, whereas P3 and P4 do not have any "straight"
number lines.
I was having some difficulty coming up with a picture of multiplication
in P4, myself. I see you mention that the "product behavior goes
haywire"
> but I have yet to define a product on P3 X P2 that transforms exactly.
> Still, If i do a 3D Mandelbrot plot of P4 it looks like a thick
> Mandelbrot set as if it were true.
> P2 comes out as the identity axis of P4 with P3 perpendicular to that.
> But there is still some error. You can see this a bit in
>
> http://bandtechnology.com/PolySigned/Deformation/DeformationUnitSphereP4.html
> Sorry for the complicated and unfinished answer. How do you see linear
> dependence in P3 and P4?
>
In Cartesian Math, of course, negative numbers are positive numbers
times negative one. This is also true of your P2, but not of P3, or
P4. You just don't have a *minus* in your arithmetic.
In which case, since you can't subtract, every coordinate in P2 might
be described uniquely by a(-1) +b(+1)+c(*1). where a, b, and c are...
hmmm. I want to say positive real numbers, but they don't exist, so
they would have to be * numbers. If a, b, and c could be positive OR
negative, then *1 would equal [negative (+1) + (-1)]. In which case
you would have linear dependence.
> > > The discrepancies with the accepted construction are suggestive.
> > > Traditionally a zero-dimensional quantity is considered to be a point.
> > That could use some clarification. Quantities have units, and that
> > entails dimension. A location might be zero-dimensional. Also, a
> > quantity might have zero uncertainty in its value, so in that way,
> > maybe could be considered zero dimensional.
> > > That is still what one-signed numbers render out to, but there is a
> > > little bit more there.
> > Can you add these one-signed numbers together? Or is the only
> > one-signed number zero?
> > > Just enough to get time with it's arrow and without any graphical
> > > measure.
> > *poof* what? how'd time get there? Because it is observed?
>
> OK. This is an important one. I cover it pretty thoroughly at
> http://bandtechnology.com/PolySigned/OneSigned.html
I'm sure I don't understand "Applying the additive identity to
one-signed numbers yields:
- x = 0." Then you say all one signed number are self cancelling,
which would seem to indicate -x -x = -0
I think I've heard there is a fairly elegant proof that the existence
of one and zero and addition are enough to prove the existance of
subtraction, multiplication, division, exponents, etc... If you could
locate such a proof and find a flaw or unjustified assumption, it might
help you to distinguish exactly what you mean, so as to communicate
your idea.
> The polysigned numbers are intimately tied to dimensionality.
> I take the one-signed numbers as the definition of zero dimensional.
> We can do things like:
> - 1.1 ( - 2.3 - 0.5 ) = - 3.08
> in one-signed numbers.
> Thinking in terms of superposition they can only get larger and larger.
> So for example an integral of them should generate a nondecreasing
> function.
> Yet all the while when one graphs them they yield zero.
> That is a funny thing about the cancellation law. You only actually
> need it when you graph these things. All math can be done in any sign
> level without ever invoking the cancellation.
> In effect the components will keep building larger and larger values.
> So long as they remain balanced you get a local answer.
> The one-signed time correspondence is perfect because of this behavior.
> It allows time to be part of spacetime without any graphical measure.
> Do one-signed numbers have a magnitude? Yes:
> | - 3.08 | = 3.08 .
> But rendering that magnitude always yields zero.
> To buy it you have to adopt the polysigned numbers and take them
> literally.
Or perhaps write the numbers on a page, and then view the page from its
edge. Then the numbers would be there, but. . . Okay, just kidding. I
really don't understand this at all.
> One-signed numbers are time much much more convincingly than the real
> numbers.
> > > Also multidimensional spaces are developed without a Cartesian product.
> > > These concepts are fundamental mathematics.
> > > I am starting to believe that we exist not in a Cartesian product space
> > > but in a particle product space.
> > Is it possible there might be more than one valid representation of the
> > same thing? This vaguely reminds me of the Lagrangian Method, which
> > greatly simplifies a lot of mechanics problems by replacing the
> > cartesian coordinate system with a system of generalized coordinates
> > based on the interaction of objects.
> The polysigned numbers are nonorthogonal so there are inherently many
> representations for the same geometric position. I'd like to coin a
> phrase like 'minimally nonorthogonal' for they barely waste information
> and it can even be argued that they reduce information over their
> Cartesian counterpart. A 2D cartesian representation takes 2.2 chunks
> of information whereas the polysigned equivalent can do it in 2.15
> chunks where 1 chunk is a magnitude and 0.1 chunks is a bit that
> becomes sign information. Sorry if that is cryptic. I can expound if
> you wish.
> >
I think you mean 2.2 is two dimensions with two signs positive or
negative, while 2.15 is . . .
> > > This would allow for a polysigned
> > > substrate whose product yields spacetime. This view may rely solely on
> > > the superposition principle i.e. we need only look at particles two at
> > > a time and allow that more are satisfied combinatorically.
> > > In this context a sole distance exists between two particles, not four
> > > independent distances.
> >
> > Now you've lost me. Where do you get the idea anybody thinks there are
> > four independent distances between particles?
>
> This is almost a rhetorical question. When we instantiate particle
> positions in traditional spacetime representations we are forced to
> assign four values. All of them are distances and they are independent
> of each other hence the 4D spacetime model.
No, actually, we aren't forced to assign four values. We can make the
choice to designate the axes of our coordinate system in any direction.
It is true that if we need four or more particles not in the same
plane then we need to have three spatial coordinates, and if they are
moving, we need to have time.
Your example uses a universe of two particles. I'm not quite
convinced that it wouldn't become more complicated than the Cartesian
system if you have more than two. Also, I'm still not quite clear
about what 3.2 chunks and 1.4 chunks means.
Okay, I'd better post this now, it's past my bed time.
Tim
Spoonfed wrote:
> I'm sure I don't understand "Applying the additive identity to
> one-signed numbers yields:
> - x = 0." Then you say all one signed number are self cancelling,
> which would seem to indicate -x -x = -0
something you wanted to keep.
The additive identity is a good thing to be musing.
While it alone allows for the graphical representation of polysigned
numbers it need only be invoked to perform that graphical
representation.
For instance in the reals ( P2 ) without this rule we could perform an
operation:
( - 2 + 3 ) ( - 3 + 4 ) - 5
= + 6 - 8 - 9 + 12 - 5
= - 22 + 18 .
We never used the law
- x + x = 0
and the answer is correct.
When we wish to give this answer graphical meaning as a singular entity
we invoke the additive identity and wind up on the real line in space.
This is the type of argument that allows the one-signed numbers to be
zero dimensional yet still carry meaning. They lose meaning when
rendered into a graphical representation. This behavior exactly matches
our notion of time. In effect the rendering process loses 1 chunk of
information, where a chunk is a magnitude.
> > The polysigned numbers are nonorthogonal so there are inherently many
> > representations for the same geometric position. I'd like to coin a
> > phrase like 'minimally nonorthogonal' for they barely waste information
> > and it can even be argued that they reduce information over their
> > Cartesian counterpart. A 2D cartesian representation takes 2.2 chunks
> > of information whereas the polysigned equivalent can do it in 2.15
> > chunks where 1 chunk is a magnitude and 0.1 chunks is a bit that
> > becomes sign information. Sorry if that is cryptic. I can expound if
> > you wish.
> > >
>
> I think you mean 2.2 is two dimensions with two signs positive or
> negative, while 2.15 is . . .
I mean 2 magnitudes and 1.5 bits so maybe I should have used the
representation 2.1.5.
If we are going to compare the reals and Cartesian product spaces to
polysigned spaces we should be coding in magnitudes and signs. So
struct Cartesian2D { magnitude m1, sign s1, magnitude m2, sign s2 }
where magnitude is like a float but without the sign, and sign is one
bit for the - or + that the reals render out to. So it takes two
magnitudes and two signs to pass a 2D Cartesian value. Let's not forget
that the information above is ordered so that we have an inherent
interpretation of m1 being x let's say and m2 being y, and s2 going
with m2 of course.
I am claiming that I can pass a three-signed value with less
information.
Due to the nonorthogonal nature of the system at least one magnitude
can always be zeroed out. So for instance if I start with:
-1.2 + 2.3 * 3.4
I can reduce to:
+ 1.1 * 2.2
Let's assume that the ordering of my two magnitudes will be just like
this example: lowest sign first. For sign information all that I have
to pass is the sign that isn't present:
struct Polysigned2D { magnitude m1, magnitude m2, polysign3 s1 }
s1 is a one of three choice and can be encoded in 1.5 bits. A modern
computer still needs two full bits to do the job but one of the
possibilities is a spare.
I hope that is sufficient for you.
The main objection that I can see is that some will insist that the
real number is the fundamental chunk rather than magnitude. I beg to
differ with them. Few truely fundamental continuous properties contain
both positive and negative properties.
While this flies in the face of any signed approach it opens one up to
questioning our usage of sign. Accepting magnitude as fundamental makes
mathematical sense and physical sense.
>
> > This is almost a rhetorical question. When we instantiate particle
> > positions in traditional spacetime representations we are forced to
> > assign four values. All of them are distances and they are independent
> > of each other hence the 4D spacetime model.
>
> No, actually, we aren't forced to assign four values. We can make the
> choice to designate the axes of our coordinate system in any direction.
> It is true that if we need four or more particles not in the same
> plane then we need to have three spatial coordinates, and if they are
> moving, we need to have time.
No, but that is what we do traditionally. And if we do a two particle
problem and claim that it exists in one dimension only we will get a
false answer. That is because the fundamental particle contains more
information. Even the yet to be discovered graviton is claimed to have
spin character. The new approach begets that character with generic
point particles.
<big snip here>
> Your example uses a universe of two particles. I'm not quite
> convinced that it wouldn't become more complicated than the Cartesian
> system if you have more than two. Also, I'm still not quite clear
> about what 3.2 chunks and 1.4 chunks means.
This is definitely a valid concern. I do not yet have a formal
mathematical proof for this concept. The idea is that superposition and
relativity alone should be enough to grant the degrees of freedom of
spacetime. In this approach the Cartesian product will never come into
use. You can look at the substrate (the family of polysigned numbers)
as existing in parallel joined together only by the distance between
particles.
We can say that the distance between two particle in meters is the same
as their distance in time. That is distance invariance in a nutshell.
We can add on to this that their distance in the polar space P3 is the
same, and possibly on up through high dimension.
All of physics relies upon superposition for its resolution. When we
try to do a three body problem we do it as a sum of its parts. Field
theories work out on this principle as well. Each part of the problem
is a two-body problem. Without this approach there is little to go on.
It is a usage of symmetry.
3.2 chunks is 3 magnitudes and 2 bits.
1.4 chunks is 1 magnitude and 4 bits.
This informational measure is how I see this attempt panning out.
The new approach generates a 3D compatible information measure through
this method. This measure is based on particles taken two at a time. It
is fairly abstract. It makes one ponder what one particle alone even
means.
-Tim
Spoonfed
Dirk Van de moortel wrote:
> "Spoonfed" <good4...@yahoo.com> wrote in message news:1147992477....@i39g2000cwa.googlegroups.com...
> >
> > So that makes it a much simpler question, without capital letters. You
> > say, (and I agree) you can fiddle with the *now* of the remote location
> > by adjusting your thrusters. What do you mean by useless?
>
> My sister is 30 years old now. Let's take off.
> 4 years later...
> How old is my sister now? Let's calculate.... ha, she must be 31.0 now.
> 3 years later...
> How old is my sister now? Let's calculate.... ha, she must be 44.2 now.
> 3 years later...
> How old is my sister now? Let's calculate.... ha, she must be 31.7 now.
> 3 years later...
> How old is my sister now? Let's calculate.... ha, she must be 57.7 now.
> ...
> That's what I call useless numbers.
>
Those aren't useless numbers because I could *use* them to partially
determine when and where you accelerated between each calculation. For
instance, sometime around year 7 you spent a very short period of time
moving back toward your sister, to measure that she was 44.2 years old,
then you returned to your original approximate velocity away from earth
for another three years to see her at 31.7.
A full plot of "sister's age" vs. time would be enough to exactly
determine the relative velocity throught the trip assuming the trip was
along a straight line.
> >
> > Are you saying that there is no possible way that anyone could ever
> > possibly find a way to exploit this capability?
>
> No, but I strongly doubt it.
> You probably aren't saying that someone *will* find a way to exploit it
> either.
>
Not in the sense that we'll find a way to travel backward in time.
> > Or do you think it is
> > so unimportant that it should not be mentioned in the teaching of SR?
>
> What can (and apparently should) be mentioned and showed, is
> that it's pretty useless ;-)
>
> Dirk Vdm
>
I used it at least twice in the last couple months in explaining SR.
I used it here
http://groups.google.com/group/sci.physics.relativity/msg/14dfcc6ba12b8dd8?&hl=en
I used it here:
http://groups.google.com/group/sci.physics.relativity/msg/582e47f7ea8109c7
Where the current age of distant objects--particularly the two ends of
a tree and a stick, could be changed by accelerating the observer.
So, do you feel I was wrong in using the concept of the objects' ages
in answering these questions?
> >
> > Because whether or not we can exploit the capability of changing
> > distant *now* to do anything useful, it certainly must affect our
> > observations of the cosmos.
Also, I used it here,
http://groups.google.com/group/sci.physics/msg/731d1e1e2188e13e?&hl=en
Where I calculated that IF the big bang expanded from a singular event,
then to create the Dipole in the CBR would have required an
acceleration of .781c to affect the current age of the distant object
(being the plasma at the edge of the universe).
Spoonfed
Tim
> But there is another way and it gets around the nonisotropic quality of
> the topology.
> It uses relativity. Consider a universe with just two generic point
> particles a and b. One solitary distance. This implies that:
> | a1 | = | a2 | = | a3 | = | b1 | = | b2 | = | b3 |
> where these are the 0D, 1D, and 2D components.
> Each particle presents to the other particle in a relative fashion.
> Claiming invariant distance on the topology leaves a few degrees of
> freedom since the original format was nonorthogonal.
> In 2D we see that there is a sign choice.
Above '2D' should be a '1D'
> Let's say the distance between a and b is 1.234.
> This means that in 2D the distance is 1.234 and in 3D it is 1.234.
Again '2D' -> '1D' and '3D' -> '2D'
> a2 could be -1.234 or +1.234 or it could even be - 1.1 + 2.334, etc.
> b2 is valued likewise. But ultimately this freedom is merely a choice
> of two signs. It is a binary quality. In 3D the freedom is one of an
Again '3D' -> '2D'
Sorry. I must have been thinking P3 which is 2D instead of 3D.
I should just stick to P's.
> angle. Each particle can freely present these qualities to the other
> particle.
> This same procedure in the 1D + 1D + 1D + 1D would yield four binary
> choices per particle. Not very sensible.
> The polysigned approach generates 3.2 chunks of information and the
> traditional approach generates 1.4 chunks. The distance is included as
> one chunk. Very different.
> Thinking this way can open up questions about d(a,b) versus d(b,a) but
> this analysis assumes that they are equal.
-Tim
Peri of Pera
speculation of astronomers who may use relativistic or non-relativistic
methods. What I am concerned about are the inconsistencies in SR. I
have explained my arguments against time dilation and length
contraction in this and other threads and think repeating them may not
serve any good for now.
Peter Riedt
Spoonfed
properly.
Tim wrote:
> Spoonfed wrote:
> > I'm sure I don't understand "Applying the additive identity to
> > one-signed numbers yields:
> > - x = 0." Then you say all one signed number are self cancelling,
> > which would seem to indicate -x -x = -0
> I did a lot of deleting to make this small again. Sorry if I lost
> something you wanted to keep.
>
> The additive identity is a good thing to be musing.
> While it alone allows for the graphical representation of polysigned
> numbers it need only be invoked to perform that graphical
> representation.
> For instance in the reals ( P2 ) without this rule we could perform an
> operation:
> ( - 2 + 3 ) ( - 3 + 4 ) - 5
> = + 6 - 8 - 9 + 12 - 5
> = - 22 + 18 .
So you represent the single number -4 by -22 + 18 or -21 + 17 or -20
+16 etc. You have multiple representations of the same number.
> We never used the law
> - x + x = 0
> and the answer is correct.
> When we wish to give this answer graphical meaning as a singular entity
> we invoke the additive identity and wind up on the real line in space.
> This is the type of argument that allows the one-signed numbers to be
> zero dimensional yet still carry meaning. They lose meaning when
> rendered into a graphical representation. This behavior exactly matches
> our notion of time. In effect the rendering process loses 1 chunk of
> information, where a chunk is a magnitude.
>
I'm not sure I agree that time loses its meaning when rendered
graphically.
Or in polar coordinates,
1.2 (120 degrees) + 2.3 (240 degrees) +3.4 (0 degrees)
= 1.1 (120 degrees) + 2.2 (0 degrees)
I thought this was interesting. Your P2 with one of the magnitudes set
to zero can represent any point on half the number line. P3 with one
magnitude set to zero can represent one third of a plane. P4 with one
magnitude set to zero can locate any point in one quarter of space.
> Let's assume that the ordering of my two magnitudes will be just like
> this example: lowest sign first. For sign information all that I have
> to pass is the sign that isn't present:
> struct Polysigned2D { magnitude m1, magnitude m2, polysign3 s1 }
> s1 is a one of three choice and can be encoded in 1.5 bits. A modern
> computer still needs two full bits to do the job but one of the
> possibilities is a spare.
Okay, yeah, I get it. You use the magnitudes to tell (in order) the
magnitudes of the two vectors you need, and the third number to tell
which of the three is zero. It's unusual, but these three numbers
would represent, uniquely, a point in two dimensions.
> I hope that is sufficient for you.
>
The main objection I have is that the 2D Cartesian coordinate system
can be represented by polar coordinates with just a positive magnitude
and a positive angle between 0 and 2Pi, which gives the location of the
point with 2.0 chunks of information. Another objection I would have
is that one can show that the set of real numbers between -infinity to
+infinity is the same size as the set of real numbers between zero and
infinity.
You can represent coordinates in three dimensions using spherical
coordinate system, with numbers representing the magnitude (0 to
infinity), azimuthal angle (0 to Pi) and polar angle (0 to 2 Pi). Here
is a three dimensional system with 3.0 chunks.
However, with a computer representation, of course, you have an
artificial representation of infinity because you've only got a finite
set of bits, and I'm not sure that the minimal information is really a
necessary part of where you're trying to go with your argument.
> The main objection that I can see is that some will insist that the
> real number is the fundamental chunk rather than magnitude. I beg to
> differ with them. Few truely fundamental continuous properties contain
> both positive and negative properties.
Momentum, velocity, rapidity, location, force. However, there may be
the same fundamental property behind all of these. Even so, all it
takes is one.
> While this flies in the face of any signed approach it opens one up to
> questioning our usage of sign. Accepting magnitude as fundamental makes
> mathematical sense and physical sense.
>
> >
> > > This is almost a rhetorical question. When we instantiate particle
> > > positions in traditional spacetime representations we are forced to
> > > assign four values. All of them are distances and they are independent
> > > of each other hence the 4D spacetime model.
> >
> > No, actually, we aren't forced to assign four values. We can make the
> > choice to designate the axes of our coordinate system in any direction.
> > It is true that if we need four or more particles not in the same
> > plane then we need to have three spatial coordinates, and if they are
> > moving, we need to have time.
>
> No, but that is what we do traditionally. And if we do a two particle
> problem and claim that it exists in one dimension only we will get a
> false answer. That is because the fundamental particle contains more
> information. Even the yet to be discovered graviton is claimed to have
> spin character. The new approach begets that character with generic
> point particles.
>
If I am not mistaken, spin has something to do with antisymmetric wave
functions.
If (psi*)(psi) = (psi)(psi*), where psi* is the complex conjugate of
psi, you have something with integer spin, so two of them can occupy
the same state of motion. If (psi*)(psi) = (psi)(psi*) you have an
antisymmetric wave function, and suddenly for some reason you have spin
and particles that can't occupy the same motion state. Now, psi has
something to do with density, (in particular |(psi)(psi*)| is the
probability density, and integrating this over a volume tells you how
many particles are likely to be in a region. Why would there be
something so complicated to describe density? Well, have you ever
tried to take a bunch of point particles and then determine the density
*at* a point somewhere on the map. It's a rather tricky business. Not
sure I was going anywhere with this. Maybe just wanted to say I don't
really understand spin.
I'm still not seeing this link between P1 and time. Also, you are
trying to find a multiplication technique between vectors in P4. In R3
you can't directly multiply these numbers but they have dot products
and cross products. Perhaps replicating these will give you some
insight into how to describe P4.
Spoonfed
I'm not familiar with the inconsistencies in SR.
Eric Gisse
Might as well skip the step of pretending to give a shit what this
group thinks, too.
It is obvious that you aren't going to change your mind.
>
> Peter Riedt
dda1
All you have "explained" to all of us is your ignorance and persitency
in your imbecility.
Spoonfed
As for the question of inconsistencies in SR
There's no way SR is mathematically inconsistent. If you want to argue
that SR does not properly represent reality, you might get a hearing,
but the insistence that it is mathematically inconsistent basically
shows you haven't done your homework.
Any set of events and any object moving under the speed of light can be
represented in a spacetime diagram, and the space time diagram can be
represented at any velocity. There is no way to establish a set of
events which is possible in one reference frame and impossible in
another. It is also impossible to to get an object to travel at or
over the speed of light or to get light to travel at any other speed
than the speed of light within any of the spacetime diagrams.
That being said, you and I will never spend any more than an instant in
a given reference frame. The earth is rotating, and revolving around
the sun, the sun around the galaxy, the galaxy falling toward Virgo,
etc. We are constantly entering a new reference frame, and in each
successive frame, the appearance of the entire universe is re-mapped,
so it may appear that light travels slower or faster than it should.
As for the question of the star, which you want to leave the
speculation to the astronomers,
In some cases, where you're given insufficient information,
"A star is measured by astronomers to be 10 billion years old--Peter
Reidt," all you can do is speculate as to what the astronomers meant
by that.
It could mean the image the astronomers see through the telescope is of
a star that formed 10 billion years ago.
I've seen it used meaning the image is of a star that looks as our
system did 10 billion years ago, when it was just forming.
Also, if you assume that our galaxy is 13.7 billion years old, and that
star's current age is 10 billion years, that means the time dilation
factor is 13.7/10 = 1.37, and assuming that we have not accelerated
during the trip (which is a silly assumption, by the way) we can
calculate that v/c = sqrt(1-1/gamma^2) = sqrt(1-1/1.37^2) = .683c.
This is where you assume the astronomers are giving the CADO (current
age of distant object) of the star, which is probably not actually the
case.
Note that while the distant star's acceleration has very little effect
on the distant star's current or apparent age, OUR acceleration could
have a HUGE effect on it's current and apparent age. It also has a
HUGE effect on its current and apparent distance from us. There are
plenty of texts on SR that gloss over, or misrepresent the effect of
the acceleration of the observer.
ste...@nomail.com
But what is the use of that? The "sister's age" is calcuated
using the relative velocity throughout the trip. That is the
only way it can be determined. The "sister's age" cannot
be determined if the velocity is not known. A full plot of
"sister's age" vs. time can only be made if the velocity is already
known. Using that plot to then determine the already known
velocity does not seem particularly useful.
Stephen
Tim
Time never gets measured directly on a tape measure as the other
dimensions do. So its lack of direct graphical measure puts it in a
different category from the other three; hence the term spacetime. The
polysigned construction is yielding this character yet builds time and
space as parts of one structured progression based on simple rules.
Are you arguing that
- x = 0
is not valid for one-signed numbers? I used to think it might be OK to
consider them an exception but there is no need to. As long as this
feature is considered as an operator (i.e. the render operation) it
does not destroy the usage of one-signed numbers algebraically.
Yes. I agree with all of that. Vector addition works just fine for them
as well.
Though to move around in those sections in general you will still need
the third component, unless you are willing to only travel outward away
from the origin.
I concur. In terms of magnitude and sign informationally the angular
coordinate systems are more efficient.
Thanks for the education. I read a lot of this stuff but have yet to
understand it. I've spent some time going back to the experiments too.
Stern-Gerlach is one of the basics and it is not straightforward.
Shooting things through apertures is tricky business. They had to widen
it in one dimension to a rectangular opening to get results and the
pattern is an ellipse rather than the two discrete dots that some
diagrams show. If in the limit of a small aperture that ellipse still
exists then the continuous nature that they claim to not be present may
be a broken argument. They say that because the center of the ellipse
is hollow (no silver atoms landed there) that the atoms are exhibiting
discrete angular momentum character. But if you consider the two dots
that some idealized diagrams make and connect them via that same
ellipse then the space in between them is taken with silver atoms
exactly where they should not be. Furthermore the process they use to
get the image involves chemical processes. I'm not saying that the
experiment is a farce but that a responsible skeptic would investigate
these details. That theory hinges on the experiment makes this a
troubling process. There must be better repeats of the experiment and
if anyone could point to them that would be great.
The P4 product is well defined. So is P5, etc.
But it is a mystery what it means.
It's fairly coherent. It has continuous qualities:
http://bandtechnology.com/PolySigned/Deformation/DeformationUnitSphereP4.html
Rotation is a natural part of the product but it also deforms a shape
due to magnitude variance. As I try to understand the P4 product it
starts to look alot like P2 X P3. It's very close when the P2 X P3
product is simply their independent products but that doesn't match
perfectly(or there is a bug in my code). We have to define a P2 X P3
product to claim the equivalence. I haven't gotten there yet. It
probably needs some dimensional mixing. Anyhow the procedure might not
be general (should P5 have an equivalent in P3 X P3 or P2 X P2 X P3?)
so it is probably a sidetrack.
These high sign (P4+n) products may be relevant to a physics theory.
They are simple to compute but difficult to visualize. The unit shell
analysis forms a complete mapping of the product. Fortunately for P4 it
is 3D so can be visualized. The P5 equivalent would be very garbled in
a 2D projection.
I'd like to convince you of the time correspondence but you will have
to either point out the conflicts you see or present your own version
of time to be scrutinized. Time as a real number doesn't make as much
sense as space as a real number. Time does not present a full degree of
freedom and seems to be unidirectional. space provides freedoms that
correspond well to three real numbers.
-Tim
Spoonfed
> Hi. I appreciate your persistence.
> > > we invoke the additive identity and wind up on the real line in space.
> > > This is the type of argument that allows the one-signed numbers to be
> > > zero dimensional yet still carry meaning. They lose meaning when
> > > rendered into a graphical representation. This behavior exactly matches
> > > our notion of time. In effect the rendering process loses 1 chunk of
> > > information, where a chunk is a magnitude.
> > >
> >
> > I'm not sure I agree that time loses its meaning when rendered
> > graphically.
> Perhaps you could expound on your thinking. How about the 0D claim?
> Time never gets measured directly on a tape measure as the other
> dimensions do. So its lack of direct graphical measure puts it in a
> different category from the other three; hence the term spacetime. The
> polysigned construction is yielding this character yet builds time and
> space as parts of one structured progression based on simple rules.
To me, the meaning of a quantity on a line plot or a scatter plot or
what-have-you is whatever you happen to put on the label on the axis.
Therefore, time is still time, pressure would be pressure, volume would
be volume, etc. Nothing loses it's meaning when plotted on a graph
unless you forget to label it's axis, or forget to include the units.
The plot is merely a method of precisely communicating relationships
between two variables.
And my concept of spacetime is nothing more than a representation--a
method of precisely locating events in space and time. A map.
Now, as for P0, I'm sure I don't understand, because these, in my mind,
are the properties of P0:
a) P0 is mathematically equivalent to the set of vectors in one
direction from the origin.
b) Vectors in P0 can have any magnitude (so long as it's 0. see
property e)
c) The additive identity in P0 is 0
d) If plotted all magnitudes of P0 look like zero.
e) All values -x in P0 are 0.
f) you can add incrementally to -x the value 0 over and over again it
somehow emulates time?
> Are you arguing that
> - x = 0
> is not valid for one-signed numbers? I used to think it might be OK to
> consider them an exception but there is no need to. As long as this
> feature is considered as an operator (i.e. the render operation) it
> does not destroy the usage of one-signed numbers algebraically.
>
Consider WHO an exception? What is a render operation? What feature?
To move around... You'd have to subtract to go toward the origin, but
subtraction is not available. So the next best thing is to add the
other available vector. Okay... I'm still thinking of this "no
subtraction" thing as somehow artificial--just an arbitrary rule that
we're following because it's against the rules of your game. But I
think you are seeing it as a fundamental property of reality brought
about by first principles.
[Big Snip]
> > However, with a computer representation, of course, you have an
> > artificial representation of infinity because you've only got a finite
> > set of bits, and I'm not sure that the minimal information is really a
> > necessary part of where you're trying to go with your argument.
>
> I concur. In terms of magnitude and sign informationally the angular
> coordinate systems are more efficient.
But efficient does not mean fundamental. Mathematical shortcuts often
hide more than they reveal. In any case, I don't think it's crucial to
your argument.
[snip]
> > > While this flies in the face of any signed approach it opens one up to
> > > questioning our usage of sign. Accepting magnitude as fundamental makes
> > > mathematical sense and physical sense.
I think your battle may be against the presumptive use of co-linear
negative numbers, rather than the use of magnitude, which we all use.
You would do better checking into a library than from asking me. But
apparently Stern-Gerlach sent a beam of oxygen atoms out of an oven
through a single slit, to get the beam very narrow, then through a
magnetic field which divided the beam into five distinct bands, one for
each of the l=2 quantum states--m={-2,-1,0,1,2} I remember being
rather surprised that these were the only values--how did no atoms end
up between one and the other? I now think that the atom somehow aligns
itself with the magnetic field in one of the five ways when it first
enters. But honestly, I would have to study it more. Now, note the
quantum states m={-2,-1,0,1,2} are all determined by boundary
conditions of the wave equation around a point. The boundaries are
psi=0 at r=0 (no electron can be in the nucleus), psi=0 at r=infinity
(the electron is not an infinite distance away) psi at theta=0 = psi at
theta = 2 Pi. You can't have two different wave functions at the same
point in space. And something similar for the azimuthal angle. Also,
some relation is determined based on potential energy, momentum, and
kinetic energy--I forget now what it is exactly, but then through some
bewildering but surprisingly straightforward mathematics, all the
possible wave functions can be determined around a single nucleus. And
then, it turns out that this solution gives a surprisingly good
prediction of the character of the electrons in the outer shell of the
atom. Except that for some atoms it didn't quite. For some atoms
there was an extra splitting of atoms that the quantum states did not
account for. And there was one other odd thing--instead of there being
one electron per quantum possibility, there seemed to be two. Putting
two and two together, they determined that there were actually two
states the electrons could be in within each quantum possibility, and
these two states caused the extra splitting. Some models attribute the
extra quantum state to the spin of the electron. Much like the the
earth orbits around the sun and rotates on its axis, they consider the
electron to orbit the nucleus and spin on its axis. However, a more
accurate description of its motion could probably be obtained by
analysis of the boundary problem from the beginning.
I see that any observer will always experience a forward progression of
time at the same rate, though an acceleration will change the location
of your future, and likewise push your past away from your physical
position, and will make the moments of your past longer ago as they are
further away.
To confuse the issue further :-) , here's an animation:
http://en.wikipedia.org/wiki/Image:Animated_Lorentz_Transformation.gif
But I don't have an argument against or for the time correspondence
with P0. I am still not quite getting it. I also don't think I
understand P2 X P3 or exactly what is going on with the animations you
are showing me.
The red dot represents a unit vector and it is being multiplied by all
the points on the sphere that are coiled around? So are you performing
something similar to a dot-product of all the vectors represented in
the coil with the vector represented in the red dot?
The Sorcerer
"Tim" <tttp...@yahoo.com> wrote in message
news:1148244349.5...@i39g2000cwa.googlegroups.com...
Perhaps this will help:
x^2 + y^2 + z^2 = c^2 t^2 Einstein (who blames Lorentz)
xi^2 + eta^2 + zeta^2 = c^2 tau^2 Einstein (who blames Lorentz)
Ref: http://www.fourmilab.ch/etexts/einstein/specrel/www/ Section 3.
tau = (t-vx/c^2)/sqrt(1-v^2/c^2) Einstein (who blames Lorentz)
tau_y = (t-uy/c^2)/sqrt(1-u^2/c^2) Androcles (who blames Einstein)
tau_z = (t-wz/c^2)/sqrt(1-u^2/c^2) Androcles (who blames Einstein)
xi = (x-vt)/sqrt(1-v^2/c^2) Einstein (who blames Lorentz)
eta = (y-ut)/sqrt(1-u^2/c^2) Androcles (who blames Einstein)
zeta = (z-wt)/sqrt(1-w^2/c^2) Androcles (who blames Einstein)
http://www.androcles01.pwp.blueyonder.co.uk/Smart/how_to3.JPG
Carry three watches or do not use escalators.
Personally I prefer witches.
Double, double, toil and trouble,
Fire burn and Einstein's bubble.... Pop!
Spoofed (and he has been) is ineducable.
http://www.m-w.com/dictionary/spoofed
Don't let that 'n' fool you, it stands for nitwit.
Androcles.
Tim
If you try to state them in the topology:
0D + 1D + 2D
instead of in:
1D + 1D + 1D + 1D
Thus far I have no claims about relative velocities modifying
measurements of time, distance, etc. It may be that those can be built
on top of this topology. I think it would be prefereable to get
electromagnetic behavior first. Maybe something will fall out of the
construction inherently.
-Tim
The Sorcerer
Androcles Vdm.
The Sorcerer
O clairvoyant one?
Dirk Van de Androcles
Tim
Spoonfed wrote:
> Tim wrote:
> > Perhaps you could expound on your thinking. How about the 0D claim?
> > Time never gets measured directly on a tape measure as the other
> > dimensions do. So its lack of direct graphical measure puts it in a
> > different category from the other three; hence the term spacetime. The
> > polysigned construction is yielding this character yet builds time and
> > space as parts of one structured progression based on simple rules.
>
> To me, the meaning of a quantity on a line plot or a scatter plot or
> what-have-you is whatever you happen to put on the label on the axis.
> Therefore, time is still time, pressure would be pressure, volume would
> be volume, etc. Nothing loses it's meaning when plotted on a graph
> unless you forget to label it's axis, or forget to include the units.
> The plot is merely a method of precisely communicating relationships
> between two variables.
>
> And my concept of spacetime is nothing more than a representation--a
> method of precisely locating events in space and time. A map.
>
> Now, as for P0, I'm sure I don't understand, because these, in my mind,
> are the properties of P0:
>
> a) P0 is mathematically equivalent to the set of vectors in one
> direction from the origin.
Their definition is down at the level that real numbers are defined at.
But that is nitpicking. Loosely I accept a).
> b) Vectors in P0 can have any magnitude (so long as it's 0. see
> property e)
zero-dimensional and so when graphed always yield zero.
> c) The additive identity in P0 is 0
Sum( x, a ) = x
a is zero at any sign level. So c) looks OK.
> d) If plotted all magnitudes of P0 look like zero.
> e) All values -x in P0 are 0.
> f) you can add incrementally to -x the value 0 over and over again it
> somehow emulates time?
generalize sign so that we can have any natural number (n) of signs. It
turns out that n-signed numbers are n-1 dimensional structures. So
one-signed numbers are zero-dimensional. Two-signed numbers which are
the reals are one-dimensional. Product and sum are defined in general
for n-signed numbers. So they are defined for one-signed numbers.
One-signed numbers suffer from the paradoxical statement
- x = 0 .
I will not deny the paradoxical nature. The point is that the paradox
exactly matches the paradox of time. The one-signed numbers form a
definition of zero-dimensional. We are stuck with the real numbers
having been burnt into our heads and these concepts go below that
structure. They generate the structure of real numbers, complex
numbers, and even of time.
>
>
> > Are you arguing that
> > - x = 0
> > is not valid for one-signed numbers? I used to think it might be OK to
> > consider them an exception but there is no need to. As long as this
> > feature is considered as an operator (i.e. the render operation) it
> > does not destroy the usage of one-signed numbers algebraically.
> >
>
> Consider WHO an exception? What is a render operation? What feature?
answer is a no due to the question. The render operation is just the
act of graphing a value. It is this operation which invokes the
identity law.
> > Yes. I agree with all of that. Vector addition works just fine for them
> > as well.
> > Though to move around in those sections in general you will still need
> > the third component, unless you are willing to only travel outward away
> > from the origin.
> >
>
> To move around... You'd have to subtract to go toward the origin, but
> subtraction is not available. So the next best thing is to add the
> other available vector. Okay... I'm still thinking of this "no
> subtraction" thing as somehow artificial--just an arbitrary rule that
> we're following because it's against the rules of your game. But I
> think you are seeing it as a fundamental property of reality brought
> about by first principles.
Right. good point. Subtraction is merely an inverse relation to
summation.
Superposition is also a related concept. We can eliminate subtraction
as a fundamental concept because in the reals:
a - b = a + ( - 1)( b )
Really subtraction is just sign modification via multiplication.
Rather than propose that in three-signed numbers there are three
operations of addition, subtraction, and intrication (fake word), it is
much easier simply to define the product rules and the sum rules. That
is a sufficient construction.
This sense of need for subtraction may still be valid for a vector
concept. I would like to be able to speak of generic n-signed values
z1, z2, z3, znd z4 and do things with them. So long as they are all n
signed they are effectively vectors and could still use the usual
notation so that:
( z1 + z2 )( z3 - z1 z2 ) = z1 z3 + z2 z3 - 2 z1 z1 z2 .
This is a valid statement for any sign level. But the signs of this
equation should not be mistaken for the signs of the components of the
vectors. To actually generate -z is sort of a neat process. Let's say
we're in four-signed and we have the value
-2 # 3
The inverse is the sum of the inverses of each component:
+ 2 * 2 # 2 - 3 + 3 * 3
= - 3 + 5 * 5 # 2
= - 1 + 3 * 3 .
So now
- 2 # 3 - 1 + 3 * 3
= 0 .
This is true simply because
- 2 + 2 * 2 # 2 = 0
and
- 3 + 3 * 3 # 3 = 0
Therefor
-2+2*2#2-3+3*3#3 = 0
-2#3 +2*2#2-3+3*3 = 0
-2#3 -3+5*5#2 = 0
-2#3 -1+3*3 -2+2*2#2 = 0
-2#3 -1+3*3 = 0.
Sorry to bore you if that was too many steps.
> > I concur. In terms of magnitude and sign informationally the angular
> > coordinate systems are more efficient.
>
> But efficient does not mean fundamental. Mathematical shortcuts often
> hide more than they reveal. In any case, I don't think it's crucial to
> your argument.
Right. Also if we want to move around in those angular coordinate
systems we may still need some notion of a negative angle to do it
cleanly. This goes towards differentials and calculus type concerns; an
area where the polysigned numbers need a lot of work.
> > > > While this flies in the face of any signed approach it opens one up to
> > > > questioning our usage of sign. Accepting magnitude as fundamental makes
> > > > mathematical sense and physical sense.
>
> I think your battle may be against the presumptive use of co-linear
> negative numbers, rather than the use of magnitude, which we all use.
>
> > Thanks for the education. I read a lot of this stuff but have yet to
> > understand it. I've spent some time going back to the experiments too.
> > Stern-Gerlach is one of the basics and it is not straightforward.
> > Shooting things through apertures is tricky business. They had to widen
> > it in one dimension to a rectangular opening to get results and the
> > pattern is an ellipse rather than the two discrete dots that some
> > diagrams show. If in the limit of a small aperture that ellipse still
> > exists then the continuous nature that they claim to not be present may
> > be a broken argument. They say that because the center of the ellipse
> > is hollow (no silver atoms landed there) that the atoms are exhibiting
> > discrete angular momentum character. But if you consider the two dots
> > that some idealized diagrams make and connect them via that same
> > ellipse then the space in between them is taken with silver atoms
> > exactly where they should not be. Furthermore the process they use to
> > get the image involves chemical processes. I'm not saying that the
> > experiment is a farce but that a responsible skeptic would investigate
> > these details. That theory hinges on the experiment makes this a
> > troubling process. There must be better repeats of the experiment and
> > if anyone could point to them that would be great.
> You would do better checking into a library than from asking me. But
> apparently Stern-Gerlach sent a beam of oxygen atoms out of an oven
It was siver atoms actually. They have one outer electron which is
supposed to cause the effect.
Orbital plus angular momentum, each quantized. In effect two electrical
currents generating superposed B-fields in a classical sense. You've
probably already viewed:
http://math.ucr.edu/home/baez/spin/spin.html
which is pretty compact. I don't see the use of the psi conjugation or
any wave functions.
I also don't see the geometrical argument of having to flip something
over twice (4pi) to get back to where it started( up / down spin ). I
see some utility of four-signed numbers when spin is considered in that
context as if this geometrical nonsense were caused by the usage of
real numbers where four-signed powers of -1 would allow it to make
sense.
I wonder if the Lorentz transform could simply be treated as a
projection algorithm.
This then allows us to abstract the whole argument to an issue of
dimensionality.
A multidimensional basis may inherently require projective properties.
Almost like the render operation but that is a very abstract linkage.
Another linkage is interdimensional operators, like the vector cross
product.
>
> But I don't have an argument against or for the time correspondence
> with P0. I am still not quite getting it. I also don't think I
> understand P2 X P3 or exactly what is going on with the animations you
> are showing me.
Don't worry about P2 X P3. In all likelyhood it is a sidetrack.
Do worry about the animation. It has nothing to do with P2 X P3; it is
pure P4.
>
> The red dot represents a unit vector and it is being multiplied by all
> the points on the sphere that are coiled around? So are you performing
> something similar to a dot-product of all the vectors represented in
> the coil with the vector represented in the red dot?
The arithmetic product is all that is involved. In 3D ( P4 ) this
product exists.
Because of this we can do wierd things like look at the square of a
sphere which turns out to be a cone. Because the product is continuous
in nature the graphing works out fine.
That swirl is really a series of points with lineto operations. I
forget how many actual points are involved, but the continuity of the
results demonstrates that the product is well behaved. It just becomes
dynamic.
The red dot travels the surface of a sphere. Upon covering the whole
sphere the animation has surveyed the P4 product. You are literally
looking at a frame by frame sphere multiplied by a point.
Probably this product exists in somebodies notebooks from a long time
ago. Wouldn't you think that Hamilton would have come across it in
getting to the quaternion? But he wanted a product that conserves
magnitude. Also the nonorthogonality of the construction is a turn-off
until you consider generalized sign.
That a product exists in general dimension and that its rules are
simple is a new thing.
You seem to be in denial, but upon understanding the rules of the
product and seeing that it generates the real numbers as well as the
complex numbers and that it doesn't stop there perhaps you will see.
For example, using the tetrahedral coordinate system of four-signed
numbers we can take two 3D points like:
z1 = - 1 * 3 # 5
z2 = - 2 + 1
and multiply them together to get:
( - 1 * 3 # 5 )( - 2 + 1 )
= + 2 # 6 - 10 * 1 - 3 + 5
= - 13 + 7 * 1 # 6
= - 12 + 6 # 5
= z1 z2 .
This is the type of math that generates the animation we are talking
about. One of the values is the red dot and the other is a series of
points that form the spiral sphere. The resulting series is the image
to the left.
There are Cartesian and inverse Cartesian transforms as well as
Cartesian projection to get the results, but the math that generates
the graphical behavior is the four-signed product.
Here is the general n-signed product in C++ code:
for( i = 0; i < n; i++ )
{
for( j = 0; j < n; j++ )
{
k = (i+j)%n;
x[ k ] += s1.x[i] * s2.x[j];
}
}
This is the algorithm without error checking, initialization, etc.
x are just arrays of n doubles. This is a snippet of my actual code.
I do not have a sphere algorithm in polysigned so I use a Cartesian
algorithm, transform to polysigned, do the math, transform back to
Cartesian, take the projection down to 2D, then graph using libGD(a 2D
graphics library). I'm getting ready to go to openGL( a 3D graphics
library) so that occlusion works (there is no depth in the current
graphics) but have not breached the hump of porting yet. Sorry if you
didn't want to know all of that, but I do feel some need to help you
believe what you are seeing.
-Tim
Spoonfed
Androcles is fooling with you, Tim. I assure you he doesn't believe a
word he's telling you. The equations he gave you would produce
completely different and incompatible results for two observers
watching the same events facing different directions.
Eric Gisse
Androcles invests way too much time and effort into this for it to be a
joke. He is mentally ill.
Tim
I'm carrying on from:
http://groups.google.com/group/sci.physics.relativity/msg/833bddf35dc28ec4
-Tim
The Sorcerer
available. Use it.
Androcles.
Spoonfed
I'll try to verify what I think you're saying about P1 again.
a) P1 can be represented by the non-negative real numbers.
b) The additive identity in P1 is 0. That is for all x in P1, x - 0 =
x.
{You would term this as - x - 0 = -x which puts the sign value outside
the variable... Of course since the - is the one sign, it seems that -x
= x.}
c) I'm not sure what operators are allowed in P1, but subtraction is
definitely not.
d) P1 when rendered can only. . .
No, I give up. If a number has a nonzero magnitude then it should be
plotted at that magnitude. I have never once seen someone create a
zero dimensional timeline, nor try to plot any other quantity in this
way.
I don't see any paradox with time, because it is what it is. Yes, it
always moves forward and we can't go back in it, but that does not
prevent us from adding to calculate a future date, or subtracting to
calculate a duration, period, interval, etc. Historians don't struggle
with the zero-dimensionality of time when creating timelines. They
draw it as a one-dimensional line.
On the other hand, an event is a zero dimensional quantity. Perhaps if
you tried to render "now" it would continue to render as zero. You
could draw a dot on a page with a little arrow to it that labels it
"Now" And then you wouldn't be drawing a timeline--you'd just have a
time-point. For a more exact representation of what the dot
represents, you could label it "Here and Now"
> >
> >
> > > Are you arguing that
> > > - x = 0
> > > is not valid for one-signed numbers? I used to think it might be OK to
> > > consider them an exception but there is no need to. As long as this
> > > feature is considered as an operator (i.e. the render operation) it
> > > does not destroy the usage of one-signed numbers algebraically.
> > >
> >
> > Consider WHO an exception? What is a render operation? What feature?
> Consider the one-signed numbers an exception. I will assume that the
> answer is a no due to the question.
Right, now that I understand the question, "I didn't consider the
one-signed numbers an exception to additive identity." I never would
have thought of such a thing.
> The render operation is just the
> act of graphing a value. It is this operation which invokes the
> identity law.
But rendering and "invoking the identity law" as you put it, isn't just
a matter of representing a point by itself with no reference.
Rendering uses relative positions or relationships. I still don't
understand how numbers in P1 have any less capacity to be graphed than
P2. If they have non-zero values they can be graphed and plotted at
non-zero positions.
> > > Yes. I agree with all of that. Vector addition works just fine for them
> > > as well.
> > > Though to move around in those sections in general you will still need
> > > the third component, unless you are willing to only travel outward away
> > > from the origin.
> > >
> >
> > To move around... You'd have to subtract to go toward the origin, but
> > subtraction is not available. So the next best thing is to add the
> > other available vector. Okay... I'm still thinking of this "no
> > subtraction" thing as somehow artificial--just an arbitrary rule that
> > we're following because it's against the rules of your game. But I
> > think you are seeing it as a fundamental property of reality brought
> > about by first principles.
>
> Right. good point. Subtraction is merely an inverse relation to
> summation.
> Superposition is also a related concept. We can eliminate subtraction
> as a fundamental concept because in the reals:
> a - b = a + ( - 1)( b )
> Really subtraction is just sign modification via multiplication.
Subtraction is so simple in the Reals, but in P1, P3, P4, etc,
subtraction does not exist. Can you also eliminate subtraction as a
fundamental concept in P3 and P4? Probably so. But only because you
can subtract in another way. (And I see you point out it is pretty
straightforward below)
Can you eliminate subtraction as a fundamental concept in P1? You have
only convinced me it is not allowed. You've not quite convinced me
that the concept does not exist.
> Rather than propose that in three-signed numbers there are three
> operations of addition, subtraction, and intrication (fake word), it is
> much easier simply to define the product rules and the sum rules. That
> is a sufficient construction.
>
> This sense of need for subtraction may still be valid for a vector
> concept.
> I would like to be able to speak of generic n-signed values
> z1, z2, z3, znd z4 and do things with them. So long as they are all n
> signed they are effectively vectors and could still use the usual
> notation so that:
>
> ( z1 + z2 )( z3 - z1 z2 ) = z1 z3 + z2 z3 - 2 z1 z1 z2 .
>
> This is a valid statement for any sign level. But the signs of this
> equation should not be mistaken for the signs of the components of the
> vectors. To actually generate -z is sort of a neat process. Let's say
> we're in four-signed and we have the value
> -2 # 3
> The inverse is the sum of the inverses of each component:
> + 2 * 2 # 2 - 3 + 3 * 3
The inverse of -2 is +2*2#2
and inverse of #3 is -3+3*3
Interesting properties of tetrahedral unit vectors.
+3 +2 *3 *2 #2 -3
> = - 3 + 5 * 5 # 2
I can't quite see what you did this step.
> = - 1 + 3 * 3 .
> So now
> - 2 # 3 - 1 + 3 * 3
> = 0 .
> This is true simply because
> - 2 + 2 * 2 # 2 = 0
> and
> - 3 + 3 * 3 # 3 = 0
> Therefor
> -2+2*2#2-3+3*3#3 = 0
> -2#3 +2*2#2-3+3*3 = 0
> -2#3 -3+5*5#2 = 0
> -2#3 -1+3*3 -2+2*2#2 = 0
> -2#3 -1+3*3 = 0.
> Sorry to bore you if that was too many steps.
>
> > > I concur. In terms of magnitude and sign informationally the angular
> > > coordinate systems are more efficient.
> >
> > But efficient does not mean fundamental. Mathematical shortcuts often
> > hide more than they reveal. In any case, I don't think it's crucial to
> > your argument.
>
> Right. Also if we want to move around in those angular coordinate
> systems we may still need some notion of a negative angle to do it
> cleanly.
touche'. And some notion of negative radius to move toward the origin.
Right. In Oxygen the spin of the electron didn't create a noticeable
effect. I forgot to change elements before I got to that part.
Right. I was just saying that I think that might be an
oversimplifcation. For one, because even with a high estimate of the
electron's diameter, in order to produce the magnetic fields by
spinning, the outer edge would have to spin faster than the speed of
light.
Also, because the concept of a literally spinning electron doesn't
really seem to mesh very well with the quantum mechanics which defines
electron spin.
> In effect two electrical
> currents generating superposed B-fields in a classical sense. You've
> probably already viewed:
> http://math.ucr.edu/home/baez/spin/spin.html
> which is pretty compact. I don't see the use of the psi conjugation or
> any wave functions.
If I'm not mistaken, our current model for the atom comes from solving
the Laplace Equation in spherical coordinates. The shape of the
electron shells comes from the spherical harmonics, complex valued
functions of the polar and azimuthal angles, sometimes denoted Y_lm. I
don't know if you can find any good treatments of this online or not.
If you mean by that, describing coordinates in space and time so that
they can be projected onto a screen, YES.
> This then allows us to abstract the whole argument to an issue of
> dimensionality.
> A multidimensional basis may inherently require projective properties.
> Almost like the render operation but that is a very abstract linkage.
> Another linkage is interdimensional operators, like the vector cross
> product.
> >
> > But I don't have an argument against or for the time correspondence
> > with P0. I am still not quite getting it. I also don't think I
> > understand P2 X P3 or exactly what is going on with the animations you
> > are showing me.
> Don't worry about P2 X P3. In all likelyhood it is a sidetrack.
> Do worry about the animation. It has nothing to do with P2 X P3; it is
> pure P4.
> >
> > The red dot represents a unit vector and it is being multiplied by all
> > the points on the sphere that are coiled around? So are you performing
> > something similar to a dot-product of all the vectors represented in
> > the coil with the vector represented in the red dot?
> The arithmetic product is all that is involved. In 3D ( P4 ) this
> product exists.
Well, in any case, I've heard of dot products and cross products in 3D,
but I've never heard of an arithmetical product.
> Because of this we can do wierd things like look at the square of a
> sphere which turns out to be a cone. Because the product is continuous
> in nature the graphing works out fine.
> That swirl is really a series of points with lineto operations. I
> forget how many actual points are involved, but the continuity of the
> results demonstrates that the product is well behaved. It just becomes
> dynamic.
> The red dot travels the surface of a sphere. Upon covering the whole
> sphere the animation has surveyed the P4 product. You are literally
> looking at a frame by frame sphere multiplied by a point.
> Probably this product exists in somebodies notebooks from a long time
> ago. Wouldn't you think that Hamilton would have come across it in
> getting to the quaternion?
I haven't studied quaternions, so I have no idea. It looks to me like
he just happened to notice some pretty properties of matrices and
complex numbers. I probably miss the point. Anything that takes on
the form of a matrix has the form of a "mapping" of one set of
coordinates to another. Or what you might call a "projection
algorithm" or an "operator". Whereas vectors in P4 would be things one
might operate upon.
> But he wanted a product that conserves
> magnitude. Also the nonorthogonality of the construction is a turn-off
> until you consider generalized sign.
> That a product exists in general dimension and that its rules are
> simple is a new thing.
Quite.
Now we're talking.
In P2 multiplying (x0,x1) by (y0,y1)
you get (x0y0+x1y1,x0y1+x1y0)
The first part is equivalent to the positive value, while the second is
equivalent to the negative value. The number written in terms of reals
would be
(x0 - x1) * (y0 - y1) = (x0y0+x1y1) -(x0y1+x1y0)
In P3, multiplying (x0,x1,x2) by (y0,y1,y2) yields
(x0y0+x1y2+x2y1 + x0y1+x1y0+x2y2 + x0y2+x1y1+x2)
And we determined earlier that (1,0,0) (0,1,0) and (0,0,1) in these
coordinates were equivalent to 1, 1e^i(2Pi/3), and 1e^i(4Pi/3)
In P4, if you want to multiply (x0, x1, x2, x3) by (y0, y1, y2, y3)
you would get
(x0y0+x1y3+x2y2+x3y1 , x0y1+x1y0+x2y3+x3y2 , x0y2+x1y1+x2y0+x3y3,
x0y3+x1y2+x2y1+x3y0)
P2 and P3 have two fairly convincing analogies in what little I know of
traditional mathematics. If this P4 has the same mathematical
properties as the quaternions, perhaps that would lead to something.
As far as I know, though, you might be creating something entirely new
with the concept of simple (not dot or cross) multiplication of three
dimensional quantities. All in all, I like the idea of such a
paradigmatic shift. Unfortunately, I think you'll find that your P4
and all Pn are going to be equivalent to e^i(2Pi/j) (0<=j<n) and can be
represented fully in two dimensions. Your approach may lead to further
insights, though.
Tim
> Tim wrote:
<Snipping away old stuff.>
If you accept spacetime as natural then there is the additional weight
of putting these qualities of time along side three dimensional space.
The zero dimensional context is almost nonexistence. But there is
slightly more to it.
Is there anything other than now? Show it to me. We need a dynamic.
Things move. Sometimes things don't move. There is a little yellow LED
resting on a bent lead sitting on a breadboard next to me that hasn't
moved in a month. Somehow it remains stable there. I just moved it.
When I ask where is the LED is the true answer a history of its
positions? Does the system really take on a 4D topology? Where is this
fourth measure? I can't measure it with a tape measure like I can the
others. Is it invisible? Then the future...
>
> > The render operation is just the
> > act of graphing a value. It is this operation which invokes the
> > identity law.
>
> But rendering and "invoking the identity law" as you put it, isn't just
> a matter of representing a point by itself with no reference.
> Rendering uses relative positions or relationships. I still don't
> understand how numbers in P1 have any less capacity to be graphed than
> P2. If they have non-zero values they can be graphed and plotted at
> non-zero positions.
I see your intuition. What does that say about the mathematical
construction?
If you are correct then the one-signed numbers are an exception to the
identity law.
But as I have pointed out the laws hold in general and so I will argue
that your intuition is wrong. I've gone there too and it took me a long
time to turn around. The first time I wrote it with a question mark. My
website even used to go your way, with the bifurcation noted. It is
still noted but now the general path is taken. It's funny that the most
fundamental can be so puzzling. We are discussing such a minimal
construction.
Let's consider this under a larger umbrella.
We know that the integral plays a fundamental role in our understanding
of physics.
Does it have an inverse? If it does it is the derivative, a form of
differencing that is quite different than mere subtraction. Yet the
integral is merely summation. Superposition is another powerful concept
that lies in similar territory. We accept that these concepts imply
that we can split things apart, add them together and get the same
thing again. But none of them rely on a definition of subtraction. The
notion of subtraction is implied by them. Subtraction is not
fundamental and so it need not be part of the definition. With the
exception of P1, polysigned numbers are capable of subtraction, but the
symbolic use of '-' to do the operation is in conflict with the
polysigned notation. The usual (real number) sense of the signs has to
be forgone for a general sign approach. In P3 if I write:
- ( + 2 * 3 )
I get
* 2 - 3 .
The signs become discrete rotational operators. They flip the system
around.
If we consider subtraction as rotation in the real numbers there is a
middle ground. The general inverse is something else entirely for
polysigned numbers, whereas for real valued vectors (nD) the usual real
flip works out by applying it to each component.
Subtraction implies superposition and inversion together. When an
inverse exists and superposition exists as an operation then
subtraction can take place. For the one-signed numbers no inverse
exists so no subtraction.
> > Rather than propose that in three-signed numbers there are three
> > operations of addition, subtraction, and intrication (fake word), it is
> > much easier simply to define the product rules and the sum rules. That
> > is a sufficient construction.
> >
> > This sense of need for subtraction may still be valid for a vector
> > concept.
> > I would like to be able to speak of generic n-signed values
> > z1, z2, z3, znd z4 and do things with them. So long as they are all n
> > signed they are effectively vectors and could still use the usual
> > notation so that:
> >
> > ( z1 + z2 )( z3 - z1 z2 ) = z1 z3 + z2 z3 - 2 z1 z1 z2 .
> >
> > This is a valid statement for any sign level. But the signs of this
> > equation should not be mistaken for the signs of the components of the
> > vectors. To actually generate -z is sort of a neat process. Let's say
> > we're in four-signed and we have the value
> > -2 # 3
> > The inverse is the sum of the inverses of each component:
> > + 2 * 2 # 2 - 3 + 3 * 3
>
> The inverse of -2 is +2*2#2
> and inverse of #3 is -3+3*3
>
> Interesting properties of tetrahedral unit vectors.
>
> +3 +2 *3 *2 #2 -3
>
> > = - 3 + 5 * 5 # 2
>
> I can't quite see what you did this step.
Just combine the like signs and reorder so
+ 3 + 2 = + 5
* 3 * 2 = * 5
Then
+ 5 * 5 # 2 - 3
= - 3 + 5 * 5 # 2
>
Right.
> The first part is equivalent to the positive value, while the second is
> equivalent to the negative value. The number written in terms of reals
> would be
>
> (x0 - x1) * (y0 - y1) = (x0y0+x1y1) -(x0y1+x1y0)
Yes.
>
> In P3, multiplying (x0,x1,x2) by (y0,y1,y2) yields
> (x0y0+x1y2+x2y1 + x0y1+x1y0+x2y2 + x0y2+x1y1+x2)
Forgot a y0 on the end but yes.
>
> And we determined earlier that (1,0,0) (0,1,0) and (0,0,1) in these
> coordinates were equivalent to 1, 1e^i(2Pi/3), and 1e^i(4Pi/3)
>
> In P4, if you want to multiply (x0, x1, x2, x3) by (y0, y1, y2, y3)
> you would get
> (x0y0+x1y3+x2y2+x3y1 , x0y1+x1y0+x2y3+x3y2 , x0y2+x1y1+x2y0+x3y3,
> x0y3+x1y2+x2y1+x3y0)
Yes!
>
> P2 and P3 have two fairly convincing analogies in what little I know of
> traditional mathematics. If this P4 has the same mathematical
> properties as the quaternions, perhaps that would lead to something.
It doesn't though. They are two very different constructions with
disparate properties.
Associative, commutative, distributive properties all work with the
polysigned family.
Algebraically what you can do to the reals you can do to any of them
under product and sum operations. Quaternions don't even commute.
>
> As far as I know, though, you might be creating something entirely new
> with the concept of simple (not dot or cross) multiplication of three
> dimensional quantities. All in all, I like the idea of such a
> paradigmatic shift. Unfortunately, I think you'll find that your P4
> and all Pn are going to be equivalent to e^i(2Pi/j) (0<=j<n) and can be
> represented fully in two dimensions. Your approach may lead to further
> insights, though.
Someone else suggested this two-dimensional breakdown also. In fact
several people have. But I am sure that it is wrong. For example in
four-signed numbers you would wind up with:
- x * x = 0
and
+ y # y = 0
which yield
- x + x * x # x = 0
but do not provide the symmetry that the polysigned numbers define
since:
- 2 + 3 * 2 # 3 = 0.
The identity law implies the n-1 dimensional coordinate system. Any
n-signed value always resolves to at most n - 1 components due to
nonorthogonality. One component can always be zeroed out. So it is an
n-1 chunk information system. They have to map to n-1 dimensional
space. I am happy to discuss this further and will try to remain open
to being wrong. I believe the construction is tight and consistent.
I'll try to come up with a more convincing defeat of the 2D concept,
but maybe you can come up with a proof that these things are really
two-dimensional. I don't see how it can be done. It is bound to lose
algebraic properties but I'm not seeing a solid instance at the moment
other than what is described above.
-Tim
Spoonfed
I accept spacetime as something which can be represented quite easily.
I don't know if I accept it as "natural" in the way you mean it. It is
quite a helpful way of thinking about velocity transformations, and it
is natural in the same way that it is natural to make little sketches
on the corners of consequent pages of a book and make it appear to move
like a cartoon. It's natural enough that a child could understand it.
> The zero dimensional context is almost nonexistence. But there is
> slightly more to it.
> Is there anything other than now?
> Show it to me. We need a dynamic.
Well, there is the past, because I can remember it, and the future,
because I can predict it to a limited extent. A considerable amount of
the past is available on film. But except for complex organisms and
memory storage devices, particles in bulk tend to occupy their most
likely momentum state. In these states, where you have a lot of
indistinguishable particles randomly colliding, perhaps in some way, it
could be argued that entropy has erased history.
> Things move. Sometimes things don't move. There is a little yellow LED
> resting on a bent lead sitting on a breadboard next to me that hasn't
> moved in a month. Somehow it remains stable there. I just moved it.
> When I ask where is the LED is the true answer a history of its
> positions? Does the system really take on a 4D topology?
If you ask for a history it's not too hard to generate a history, and
it takes on a 4D topology. If you ask for it's position now, it's
history is not relevant to the question, and you need a 3D topology.
> Where is this fourth measure?
>I can't measure it with a tape measure like I can the
> others. Is it invisible? Then the future...
>
Are you now asking whether or not time is actually "observed?" We
remember, we predict, we experience, but is the very moment of
experience a quantity of measurable duration? I would give it high
likelyhood of being related to how long it takes for a regular chemical
reaction in some neuron in our brains to tick over to tell our
conscious mind that was then, this is now.
> >
> > > The render operation is just the
> > > act of graphing a value. It is this operation which invokes the
> > > identity law.
> >
> > But rendering and "invoking the identity law" as you put it, isn't just
> > a matter of representing a point by itself with no reference.
> > Rendering uses relative positions or relationships. I still don't
> > understand how numbers in P1 have any less capacity to be graphed than
> > P2. If they have non-zero values they can be graphed and plotted at
> > non-zero positions.
>
> I see your intuition. What does that say about the mathematical
> construction?
Does it say P1 is not a well defined concept?
> If you are correct then the one-signed numbers are an exception to the
> identity law.
You have said that P1 has a magnitude but is graphed as zero. It seems
to me that you are saying that it has an identity, but that identity is
meaningless. I feel that the two qualities are incompatible. Either
P1 is {0} or P1 is {0,infinity} and can be graphed as {0,infinity}
Where by - you mean a rotation in the complex plane by 120 degrees.
The usual (real number) sense of the signs is still there in the sense
that
+ 1 - 1 = * 1
* 1 - 1 = + 1
*1 + 1 = -1
> The signs become discrete rotational operators. They flip the system
> around.
> If we consider subtraction as rotation in the real numbers there is a
> middle ground.
Subtraction is more like reflection than rotation. Rotation has too
many options.
> The general inverse is something else entirely for
> polysigned numbers, whereas for real valued vectors (nD) the usual real
> flip works out by applying it to each component.
>
> Subtraction implies superposition and inversion together. When an
> inverse exists and superposition exists as an operation then
> subtraction can take place. For the one-signed numbers no inverse
> exists so no subtraction.
>
Or is subtraction simply the difference between the locations of two
points? Do P3 and P4 have a simple difference operation? If P1 has no
inverse, it can also have no difference, and thus no magnitude. i.e.
P1 = {0}, or do you see something different?
Yes, that would be the case if
-1= e^(i Pi/2) = (i in standard notation)
+1=e^(i 2Pi/2) = (-1 in standard notation)
*1=e^(i 3Pi/2) = (-i in standard notation)
#1=e^(i 4Pi/2) = (1 in standard notation)
Or are you saying that -2+3*2#3 is not equal to zero in P4? Actually,
have you got the angles from the origin for a tetrahedron? I should be
able to figure this one out myself. Probably better to start with this
end where we can crunch some numbers in P4 than to argue over
paradoxical and natural nature of P1 and time at the other end.
> The identity law implies the n-1 dimensional coordinate system. Any
> n-signed value always resolves to at most n - 1 components due to
> nonorthogonality. One component can always be zeroed out. So it is an
> n-1 chunk information system. They have to map to n-1 dimensional
> space.
Ahhh, _One component can always be zeroed out_ Somehow that doesn't
sound like quite such an arbitrary rule as "subtraction isn't allowed."
> I am happy to discuss this further and will try to remain open
> to being wrong. I believe the construction is tight and consistent.
> I'll try to come up with a more convincing defeat of the 2D concept,
> but maybe you can come up with a proof that these things are really
> two-dimensional. I don't see how it can be done. It is bound to lose
> algebraic properties but I'm not seeing a solid instance at the moment
> other than what is described above.
>
> -Tim
I think the direction to go with this is to further develop the
arithmetic of P4.
There are two ways of plotting it; as a tetrahedron,
We know that it is indeed true that -1+1*1#1 = 0 when drawn as
tetrahedron.
The difficulty is when picturing it as a tetrahedron, it is difficult
to imagine how that (-1)(-1) = (+1), (-1)(+1) = *1, etc. If there was
some kind of connection you could establish with how these signs rotate
one another. And it were associative, distributive, commutative, etc.
Tim
questioning its character. Some will claim there is no time, some will
claim the 4D reference standard, that temperature is time, that the
ideal clock does not exist, etc.
I got bitten by this bug years ago when I was reading about string
theories and realizing the fakeness of forcing three extended
dimensions. The theoretical validity of such constructions is on weak
ground. The strongest theoretical ground will be from a system that
generates spacetime. The polysigned numbers are only one such
construction. There may be others. I don't see how else one would get
this time correspondence, but then believing the time correspondence is
also tough as you are evidence of.
Typically physics does a dance between empirical and theoretical. But
the best theory is the one that predicts the empirical rather than
relying upon it. So far spacetime has been beyond that level of
theoretical construction. But now almost all of physics concerns itself
with the qualities of spaces and allows that those spaces generate the
behaviors. Particle physics and string theory both do this. To claim
that these are two individual spatial constructions (spacetime versus
the things in spacetime) will not be theoretically as clean as one
construction that gets the whole thing. So physics theory needs to
generate spacetime. That to me is where we are at. Hopefully in the
process the conflicts of existing theories will be resolved. Polysigned
numbers are fundamental yet their complexity beyond P3 is not
understood.
>
> > The zero dimensional context is almost nonexistence. But there is
> > slightly more to it.
> > Is there anything other than now?
> > Show it to me. We need a dynamic.
>
> Well, there is the past, because I can remember it, and the future,
> because I can predict it to a limited extent. A considerable amount of
> the past is available on film. But except for complex organisms and
> memory storage devices, particles in bulk tend to occupy their most
> likely momentum state. In these states, where you have a lot of
> indistinguishable particles randomly colliding, perhaps in some way, it
> could be argued that entropy has erased history.
>
> > Things move. Sometimes things don't move. There is a little yellow LED
> > resting on a bent lead sitting on a breadboard next to me that hasn't
> > moved in a month. Somehow it remains stable there. I just moved it.
> > When I ask where is the LED is the true answer a history of its
> > positions? Does the system really take on a 4D topology?
>
> If you ask for a history it's not too hard to generate a history, and
> it takes on a 4D topology. If you ask for it's position now, it's
> history is not relevant to the question, and you need a 3D topology.
It gets pretty complicated before the LED left the factory.
>
> > Where is this fourth measure?
> >I can't measure it with a tape measure like I can the
> > others. Is it invisible? Then the future...
> >
>
> Are you now asking whether or not time is actually "observed?" We
> remember, we predict, we experience, but is the very moment of
> experience a quantity of measurable duration? I would give it high
> likelyhood of being related to how long it takes for a regular chemical
> reaction in some neuron in our brains to tick over to tell our
> conscious mind that was then, this is now.
Yes. People who scrutinize time have some fairly interesting arguments.
One of these is that the perfect clock does not exist. Whether it is a
metal piece of mechanics or a laser with mirrors and counters the
construction can be construed as subject to the environment. All
empirical systems work this way. If you want to observe an electron you
will have to influence it.
The cleanest way out is pure theory without observers.
Does this get around the uncertainty principle?
>
> > >
> > > > The render operation is just the
> > > > act of graphing a value. It is this operation which invokes the
> > > > identity law.
> > >
> > > But rendering and "invoking the identity law" as you put it, isn't just
> > > a matter of representing a point by itself with no reference.
> > > Rendering uses relative positions or relationships. I still don't
> > > understand how numbers in P1 have any less capacity to be graphed than
> > > P2. If they have non-zero values they can be graphed and plotted at
> > > non-zero positions.
> >
> > I see your intuition. What does that say about the mathematical
> > construction?
>
> Does it say P1 is not a well defined concept?
I don't know that we can make further ground on this one. We are caught
in our cycles and until we try a new context this one probably isn't
going to budge for either of us. I think that my best explanation is
that we all have the real numbers imprinted in us from traditioinal
education. They are the base that much of modern physics works from.
This construction goes below them and recovers them as P2. It gets a
whole lot of other things too. One of them is P1. In this context It is
not a personal choice. I follow the polysigned numbers around, chasing
after them to get a good view. I found them and have seen them quite a
few times. You have also seen them enough to know them fairly well.
But it is very difficult to see P1, and P4 and up for that matter. They
are not well understood yet. Even upon getting them modeled in software
with no special exceptions P1 still scares me.
>
> > If you are correct then the one-signed numbers are an exception to the
> > identity law.
>
> You have said that P1 has a magnitude but is graphed as zero. It seems
> to me that you are saying that it has an identity, but that identity is
> meaningless. I feel that the two qualities are incompatible. Either
> P1 is {0} or P1 is {0,infinity} and can be graphed as {0,infinity}
Yes. I understand the conflict. My best resolution is to consider
arithmetic versus graphing. We can do arithmetic without invoking
cancellation. It just makes us forgo the shortcuts. In P2 this works
like:
- 3 ( - 1 + 1 ) + 2 ( - 2 + 1 )
= + 3 - 3 - 4 + 2
= - 7 + 5
= - 2 - 5 + 5
-----------------
= - 2 .
While a realistic person looks at this and zeros the first term
immediately that is the step that can be forgone. Until the break
(-----) no use of
- x + x = 0
has been taken.
In P1 there is only one sign but arithmetic can still be done:
- 3 ( - 1 - 1 ) - 2 ( - 2 - 1 )
= - 3 - 3 - 4 - 2
= - 12
-------------------
= 0 .
It is a distinction between dimension and distance.
In any whole dimension (now including zero) we can instantiate a
singular (one chunk) distance.
We can also call that distance a magnitude. The measure of that
distance (in its dimension) is something else. We should be careful
when transferring the notion of a singular distance to different
dimensions. Our intuitive sense that we can draw a line on a piece of
paper in two dimensions (the paper) and measure it with a
three-dimensional ruler makes it all seem OK to mix them. Some trixter
comes along and folds the piece of paper over in two so that the
endpoints match. Now their ruler distance is zero. Perhaps this is just
a model of a 1D space (the line) and not the real thing. After all
where would I even put the ruler in 1D? To throw away the observer and
deal with the construction on its own terms may be necessary.
Mathematicians do not need observers and are capable of the same level
of complexity as physicists.
You need an inverse function here.
So define Sum( Inv ( a ), a ) = 0.
Then
+ 1 - 1 = Inv( * 1 )
etc.
since
+ 1 - 1 * 1 = 0 .
I know you knew that and I got your context. Carrying on.
>
>
>
> > The signs become discrete rotational operators. They flip the system
> > around.
> > If we consider subtraction as rotation in the real numbers there is a
> > middle ground.
>
> Subtraction is more like reflection than rotation. Rotation has too
> many options.
Not in the reals. In the reals you can flip once or twice and that is
it.
So rotation and reflection are the same.
And it is because the polysign operators are rotating (many options)
that we do not need to define subtraction, intrication, etc. in higher
signs.
>
> > The general inverse is something else entirely for
> > polysigned numbers, whereas for real valued vectors (nD) the usual real
> > flip works out by applying it to each component.
> >
> > Subtraction implies superposition and inversion together. When an
> > inverse exists and superposition exists as an operation then
> > subtraction can take place. For the one-signed numbers no inverse
> > exists so no subtraction.
> >
>
> Or is subtraction simply the difference between the locations of two
> points? Do P3 and P4 have a simple difference operation? If P1 has no
> inverse, it can also have no difference, and thus no magnitude. i.e.
> P1 = {0}, or do you see something different?
I've addressed this above (dimension versus distance).
Right. I'm trying to disprove your 2D claim. Taking the identity law
seriously will conflict with imposing the polysign system on a 2D
space.
The general angle formula between unit vectors is:
a = pi - acos( 1 / ( n - 1 ))
where acos is inverse cosine and n is the number of signs.
For n = 4 this will yield 1.9106 radians or 109.47 deg.
>
> > The identity law implies the n-1 dimensional coordinate system. Any
> > n-signed value always resolves to at most n - 1 components due to
> > nonorthogonality. One component can always be zeroed out. So it is an
> > n-1 chunk information system. They have to map to n-1 dimensional
> > space.
>
> Ahhh, _One component can always be zeroed out_ Somehow that doesn't
> sound like quite such an arbitrary rule as "subtraction isn't allowed."
>
> > I am happy to discuss this further and will try to remain open
> > to being wrong. I believe the construction is tight and consistent.
> > I'll try to come up with a more convincing defeat of the 2D concept,
> > but maybe you can come up with a proof that these things are really
> > two-dimensional. I don't see how it can be done. It is bound to lose
> > algebraic properties but I'm not seeing a solid instance at the moment
> > other than what is described above.
> >
> > -Tim
>
> I think the direction to go with this is to further develop the
> arithmetic of P4.
>
> There are two ways of plotting it; as a tetrahedron,
>
> We know that it is indeed true that -1+1*1#1 = 0 when drawn as
> tetrahedron.
>
> The difficulty is when picturing it as a tetrahedron, it is difficult
> to imagine how that (-1)(-1) = (+1), (-1)(+1) = *1, etc. If there was
> some kind of connection you could establish with how these signs rotate
> one another. And it were associative, distributive, commutative, etc.
They are associative, distributive, and commutative.
So for z in P4
z1( z2 + z3)
= z1( z3 + z2 )
= z1 z2 + z1 z3
= z2 z1 + z1 z3 .
where plus here is being used as vector summation, not the four-signed
arithmetic plus sign. These expressions hold at any sign level. Proving
it in general is not so easy notationally but it is provable for any
given dimension and becomes apparent.
Just instantiate a few simple instances and you should be convinced.
For example use
z1 = - 1 * 2
z2 = - 2 * 3 # 4
z3 = -2 + 2
You will find all four expressions to have the same result.
I get
+ 2 * 2 # 3
for the first one.
modify the values slightly and it will still work.
The proof in general for four-signed involves generalizing the values
so that
z1 = - a + b * c # d
z2 = - e + f * g # h
z3 = - i + j * k # l
and gets large. The general four-signed product has sixteen terms.
If you don't believe you probably should try to disprove which will
take you to these exercises. You have the rules down pretty good and
it's really easy going math.
This type of math a child could learn.
Could you look at:
http://groups.google.com/group/sci.physics.relativity/msg/b775775a41f83810
I have no idea how these scale factors would come to be but if they did
exist then it poses the relative topology that we are getting to with a
bit more complexity.
Perhaps that would get up to Lorentz Xforms.
Also I have not studied right hand / left hand systems. That would add
another bit to the original topological product where this handedness
has been overlooked for 2D. As you look at the tetrahedral system I
think it is still there also as a binary.
-Tim
Spoonfed
> Spoonfed wrote:
[snipping more old stuff]
Is it possible that your analogy of P1 is more akin to the pointlike
nature of an event or the zero space-time interval represented by the
emission and absorption of a photon? I think you may have a powerful
mathematical concept in the polysigned numbers, but I can't see any way
P1 represents the whole of time.
> >
> > > The zero dimensional context is almost nonexistence. But there is
> > > slightly more to it.
> > > Is there anything other than now?
> > > Show it to me. We need a dynamic.
> >
> > Well, there is the past, because I can remember it, and the future,
> > because I can predict it to a limited extent. A considerable amount of
> > the past is available on film. But except for complex organisms and
> > memory storage devices, particles in bulk tend to occupy their most
> > likely momentum state. In these states, where you have a lot of
> > indistinguishable particles randomly colliding, perhaps in some way, it
> > could be argued that entropy has erased history.
> >
> > > Things move. Sometimes things don't move. There is a little yellow LED
> > > resting on a bent lead sitting on a breadboard next to me that hasn't
> > > moved in a month. Somehow it remains stable there. I just moved it.
> > > When I ask where is the LED is the true answer a history of its
> > > positions? Does the system really take on a 4D topology?
> >
> > If you ask for a history it's not too hard to generate a history, and
> > it takes on a 4D topology. If you ask for it's position now, it's
> > history is not relevant to the question, and you need a 3D topology.
>
> It gets pretty complicated before the LED left the factory.
Complicated, but that's not to say it doesn't exist.
>
> >
> > > Where is this fourth measure?
> > >I can't measure it with a tape measure like I can the
> > > others. Is it invisible? Then the future...
> > >
> >
> > Are you now asking whether or not time is actually "observed?" We
> > remember, we predict, we experience, but is the very moment of
> > experience a quantity of measurable duration? I would give it high
> > likelyhood of being related to how long it takes for a regular chemical
> > reaction in some neuron in our brains to tick over to tell our
> > conscious mind that was then, this is now.
>
> Yes. People who scrutinize time have some fairly interesting arguments.
> One of these is that the perfect clock does not exist. Whether it is a
> metal piece of mechanics or a laser with mirrors and counters the
> construction can be construed as subject to the environment. All
> empirical systems work this way. If you want to observe an electron you
> will have to influence it.
> The cleanest way out is pure theory without observers.
> Does this get around the uncertainty principle?
>
Don't get me started. But, no, there are no perfect clocks, no perfect
rulers. Do exact quantities of time and space exist? Delta(x)
Delta(p) > hbar or whatever. The difficulty is that the p stands for
momentum. Momentum implies a body. A body is an observer. i.e. any
object from which a coordinate system can be determined or from which a
distance can be measured. I know that the word "observer" means
"somebody who can make a measurement and know what it means" in common
dialogue, but in relativity, it just means any object or particle whose
reference frame can be determined. At least, that's what it means to
me. I really can't vouch for anybody else.
Now, I've heard people apply Delta(E) Delta(t) > hbar to a vacuum,
saying that particles can appear and disappear if they do it quickly
enough. This application of the equation seems a little flaky to me.
I prefer when it is applied to when a photon collides or reflects off a
target can only be determined with an accuracy proportional to
1/frequency of the light.
[Snipping our conversation about the mysterious paradoxical nature of
One-signed numbers.]
In summary, I think P1 is best described simply by {0}. However, we
may have an analogy in the space-time interval in a photon-transmission
event.
x=c*t so the interval = sqrt(c^2 t^2 - x^2) = 0
For now, let's focus on P4 and the exact habits of poly-signed numbers]
That is just way cool.
I don't know how to disprove, but I have an idea how to prove. I hope
you don't mind it taking traditional mathematics for granted and
piggy-backing.
I am trying to put together a method of translating your arithmetic for
P4 into R3. I've just got a start, and if you are familiar enough with
the ideas, you might be able to finish.
Since #1 is identity, I choose it as my simplest looking basis vector.
#1 = i = (1,0,0)
>From here I have many options, I could choose +1 to be any point (-1/3,
.9428 cos theta, .9428 sin theta)
I choose:
-1 = j = (-1/3, .9428, 0)
Now having gone to this point, I have only two remaining options, and I
choose arbitrarily between them.
+1 = k = (-1/3, -.4714 , .8165)
#1 = l = (-1/3, -.4714 , -.8165)
Now, I am thinking that multiplying by #i, -j, +k, or *l can be
considered operators, where i is the identity operator, and
j:j-->k
j:k-->l (read "j maps k to l"
j:l-->i
k:k-->j
l:l-->k
etc.
I'm not a super-educated-mathematician, but it looks like the
quantities can be represented pretty well as vectors in R3, while the
multiplication operators could be well represented by matrices in R3X3.
When finding such an matrix, it is easy to find by figuring what the
operator would do to (1,0,0), (0,1,0) and (0,0,1). Once you've figured
that out then the operator is fully determined.
So we need to know what these basis vectors are to determine what our
operators do to them. Everything below is done in R3, so the signs
have their traditional meaning, and i, j, k, and l are defined as
above.)
(1,0,0) = i
(0,1,0) = (3j+i)/(3*.9428)
(0,0,1) = (2k+j+i)/(2*.8165) = (2l+j+i)/(-2*.8165)
Applying the mapping j to each of these
j:i-->k
j:(3j+i)/(3*.9428)-->(3k+j)/3*.9428
j:(2k+j+i)/(2*.8165)-->(2l+k+j)/(2*.8165)
j:(2l+j+i)/(-2*.8165)-->(2i+k+j)/(-2*.8165)
Just to check for consistency, these last two should evaluate the same,
so I'll work those out
Reminder:
i = (1,0,0)
j = (-1/3, .9428, 0)
k = (-1/3, -.4714 , .8165)
l = (-1/3, -.4714 , -.8165)
2l+k+j =(-4/3, -.4714, -.8165)
-(2i+k+j)=-(4/3, .4714, .8165)
Okay, it seems consistent this far.
If you follow this process, you should be able to produce four
operators in R3X3, for each of the unit vectors in P4. Then if it
comes out alright, you should be able to use a lot of well-established
mathematical rules to make your points.
Most R3X3 operators are not commutative, and I can't recall offhand the
easy way to determine comutativity. I think it is when the transpose
is equal to the inverse, or something like that.
Tim
Spoonfed wrote:
> Tim wrote:
> > Spoonfed wrote:
> [snipping more old stuff]
>
> nature of an event or the zero space-time interval represented by the
> emission and absorption of a photon? I think you may have a powerful
> mathematical concept in the polysigned numbers, but I can't see any way
> P1 represents the whole of time.
The 'whole' of time may be more like a hole of time (nonexistent). An
event concept like choosing when to push a button is consistent with
the degree of freedom. Are you saying it could imply the discrete
nature of photon emission? I don't really understand the zero
space-time interval concept. Is that another way of saying relative
simultaneity?
I'm glad that you are getting into polysign.
> > Yes. People who scrutinize time have some fairly interesting arguments.
> > One of these is that the perfect clock does not exist. Whether it is a
> > metal piece of mechanics or a laser with mirrors and counters the
> > construction can be construed as subject to the environment. All
> > empirical systems work this way. If you want to observe an electron you
> > will have to influence it.
> > The cleanest way out is pure theory without observers.
> > Does this get around the uncertainty principle?
> >
>
> Don't get me started. But, no, there are no perfect clocks, no perfect
> rulers. Do exact quantities of time and space exist? Delta(x)
> Delta(p) > hbar or whatever. The difficulty is that the p stands for
> momentum. Momentum implies a body. A body is an observer. i.e. any
> object from which a coordinate system can be determined or from which a
> distance can be measured. I know that the word "observer" means
> "somebody who can make a measurement and know what it means" in common
> dialogue, but in relativity, it just means any object or particle whose
> reference frame can be determined. At least, that's what it means to
> me. I really can't vouch for anybody else.
>
> Now, I've heard people apply Delta(E) Delta(t) > hbar to a vacuum,
> saying that particles can appear and disappear if they do it quickly
> enough. This application of the equation seems a little flaky to me.
> I prefer when it is applied to when a photon collides or reflects off a
> target can only be determined with an accuracy proportional to
> 1/frequency of the light.
>
> In summary, I think P1 is best described simply by {0}. However, we
> may have an analogy in the space-time interval in a photon-transmission
> event.
>
> x=c*t so the interval = sqrt(c^2 t^2 - x^2) = 0
>
> For now, let's focus on P4 and the exact habits of poly-signed numbers]
> > The general angle formula between unit vectors is:
> > a = pi - acos( 1 / ( n - 1 ))
>
> That is just way cool.
>
>
> I don't know how to disprove, but I have an idea how to prove. I hope
> you don't mind it taking traditional mathematics for granted and
> piggy-backing.
>
> I am trying to put together a method of translating your arithmetic for
> P4 into R3. I've just got a start, and if you are familiar enough with
> the ideas, you might be able to finish.
>
> Since #1 is identity, I choose it as my simplest looking basis vector.
>
> #1 = i = (1,0,0)
>
> >From here I have many options, I could choose +1 to be any point (-1/3,
> .9428 cos theta, .9428 sin theta)
>
> I choose:
>
> -1 = j = (-1/3, .9428, 0)
>
> Now having gone to this point, I have only two remaining options, and I
> choose arbitrarily between them.
>
> +1 = k = (-1/3, -.4714 , .8165)
> #1 = l = (-1/3, -.4714 , -.8165)
>
> Now, I am thinking that multiplying by #i, -j, +k, or *l can be
> considered operators, where i is the identity operator, and
> j:j-->k
> j:k-->l (read "j maps k to l"
> j:l-->i
> k:k-->j
> l:l-->k
> etc.
Your notation #i, etc. is probably better stated as:
# 1 = i
- 1 = j
+ 1 = k
* 1 = l
So now
j j = k
etc. but
k k = i (not j )
Yes. This all looks good.
>
> I'm not a super-educated-mathematician, but it looks like the
> quantities can be represented pretty well as vectors in R3, while the
> multiplication operators could be well represented by matrices in R3X3.
>
>
> When finding such an matrix, it is easy to find by figuring what the
> operator would do to (1,0,0), (0,1,0) and (0,0,1). Once you've figured
> that out then the operator is fully determined.
>
> So we need to know what these basis vectors are to determine what our
> operators do to them. Everything below is done in R3, so the signs
> have their traditional meaning, and i, j, k, and l are defined as
> above.)
>
> (1,0,0) = i
>
> (0,1,0) = (3j+i)/(3*.9428)
>
> (0,0,1) = (2k+j+i)/(2*.8165) = (2l+j+i)/(-2*.8165)
>
> Applying the mapping j to each of these
>
> j:i-->k
> j:(3j+i)/(3*.9428)-->(3k+j)/3*.9428
> j:(2k+j+i)/(2*.8165)-->(2l+k+j)/(2*.8165)
> j:(2l+j+i)/(-2*.8165)-->(2i+k+j)/(-2*.8165)
Nice work.
I'm not following the last three lines here.
I see the multiply by j on the right.
Anyhow I guess you are checking the inverse transform?
I think my inverse values agree with yours.
i.e. in your notation I have:
( 0, 1, 0 ) = 0.354 i + 1.061 j + 0 k + 0 l
We've got the same factors after a little arithmetic.
And the last one matches in the first form.
When you get negatives in the polysigned slots you can just offset them
upwards since they are nonorthogonal and will take any offset, even
negative ones though it is an ambiguity of the transform.
>
> Just to check for consistency, these last two should evaluate the same,
> so I'll work those out
>
> Reminder:
>
> i = (1,0,0)
> j = (-1/3, .9428, 0)
> k = (-1/3, -.4714 , .8165)
> l = (-1/3, -.4714 , -.8165)
>
> 2l+k+j =(-4/3, -.4714, -.8165)
> -(2i+k+j)=-(4/3, .4714, .8165)
>
> Okay, it seems consistent this far.
>
> If you follow this process, you should be able to produce four
> operators in R3X3, for each of the unit vectors in P4. Then if it
> comes out alright, you should be able to use a lot of well-established
> mathematical rules to make your points.
>
Is that the catch? We'll have to speak in terms of four operators? They
are one operation in polysign. Yet we know that we can transform a P4
to a C3 (or R3) so we should be able to express the product for general
Cartesian values. We should wind up with three functions:
R1( c1, c2 )
R2( c1, c2 )
R3( c1, c2 )
where the result is the product of c1 and c2:
[C3 R1, R2, R3 ] . (where C3 indicates a 3D Cartesian vector )
These expressions could be messy.
> Most R3X3 operators are not commutative, and I can't recall offhand the
> easy way to determine comutativity. I think it is when the transpose
> is equal to the inverse, or something like that.
Are you saying that a table of Cartesian unit vector products would be
helpful?
I can spit those out easily on my computer.
So if A = [C3 1, 0, 0], B = [ C3 0, 1, 0 ], C = [C3 0, 0, 1]
that their products in polysigned complete the description?
So we'd have
A B , A C , B C
That's a 3 x 3.
Is it that easy?
Is that a tensor and is it a complete description of the product?
I'm not fully seeing how this will get general product for c1 and c2 in
R3.
Still I can easily generate AB, AC, and BC.
I'll let you try first unless you want me to do it.
I can verify your work.
I see commutativity rules for vectors but not for tensors in some docs
I have.
Your transform values are consistent with
http://bandtechnology.com/PolySigned/CartesianTransform.html
which gives:
[P4 1, 0, 0, 0 ] : [C3 1, 0, 0 ]
[P4 0, 1, 0, 0 ] : [C3 -0.333333, 0.942809, 0 ]
[P4 0, 0, 1, 0 ] : [C3 -0.333333, -0.471405, 0.816497 ]
[P4 0, 0, 0, 1 ] : [C3 -0.333333, -0.471405, -0.816497 ]
and the inverse:
[C3 1, 0, 0 ] : [P4 1, 0, 0, 0 ]
[C3 0, 1, 0 ] : [P4 0.353553, 1.06066, 0, 0 ]
[C3 0, 0, 1 ] : [P4 0.612372, 0.612372, 1.22474, 0 ]
The math is really easy to do straight in P4 for instances and even in
general compared to this approach. But the direction you are going
could expose a match to some existing work in the Cartesian domain.
That would be a nice discovery.
-Tim
Tim
I have forgotten that besides
AB, AC, BC
that there are also
AA, BB, CC
There is a 3x3x3 tensor that fits with this data that forms a vector
product called Levi-Civita. It has six places that are normally unity
type values similar to a Kronecker delta. The rest are zeros.
See
http://infohost.nmt.edu/~iavramid/notes/m332btfrm.pdf
page 2. There is a definition of Vector Product using the E(i,j,k)
Levi-Civita constant.
Perhaps that is the form that the general solution will take its best
shape in.
The form seems right. We need 18 slots for numbers. Those should relate
to the E() form but I don't know how yet.
-Tim
Sue...
It isn't clear to me why one would build a formalism based on
a paradox. You might find some resolution to a 'paradoxical'
concept of time in this discussion:
"The dual electromagnetic field tensor"
http://farside.ph.utexas.edu/teaching/jk1/lectures/node23.html
"Advanced potentials?"
http://farside.ph.utexas.edu/teaching/em/lectures/node51.html
Time becomes very real when you have to burn fuel to save it.
http://en.wikipedia.org/wiki/Noether's_theorem
Good luck with your endeavour. It would be nice if Fitpatrick
did not have to include a chapter called "Advanced Potential?"
Sue...
http://web.mit.edu/8.02t/www/802TEAL3D/teal_tour.htm
Tim
> It isn't clear to me why one would build a formalism based on
> a paradox. You might find some resolution to a 'paradoxical'
> concept of time in this discussion:
> "The dual electromagnetic field tensor"
> http://farside.ph.utexas.edu/teaching/jk1/lectures/node23.html
> "Advanced potentials?"
> http://farside.ph.utexas.edu/teaching/em/lectures/node51.html
Thanks for the links.
P1 is paradoxical in that it defines a zero-dimensional space with
qualities that match time.
P2 is the real numbers.
P3 is the complex numbers.
P4 and above are poorly understood but clearly defined.
To say that the system is built on a paradox is not accurate. The rules
are simple and general:
http://bandtechnology.com/PolySigned/PolySigned.html
In terms of a generational progression the zero-dimensional P1 may be
attractive since it is close to starting with nothing. If you can get
to P2 via an operator then to P3, etc. then the generator would produce
support for spacetime. This generator is probably something akin to the
vector cross product taken on a different topology than we normally
think of it working in. We normally think of it as taking two 3D inputs
and yielding a 3D output. This new approach requires that we look at
the source of the problem as starting in
0D + 1D + 2D ...
rather than
1D + 1D + 1D + 1D
I hope you see the emag possibility in the new topology.
What happens if you extend it?
-Tim
Sue...
Yes... I think I do.
> What happens if you extend it?
First, we should see it work, before we extend it.
A simple RLC circuit is a good illustration of
the imaginary operator. It would be helpful to
see solution to a real world problem in both
conventional notation:
http://hyperphysics.phy-astr.gsu.edu/hbase/electric/impcom.html
and your poly-signed notation.
Then as you extend the use out into Maxwell (?) near
and far fields, you have some basis in the lumped
constant models... particularly where we describe
reactive power in the near-field.
IOW... How about a tutorial for your notation, based
on the simplest application of imaginary operators for
those of us who have little confidence in the mathematics of
others and no confidence at all in our own. ;-)
Sue...
>
> -Tim
Spoonfed
I decided you could probably use a hand, so I put some more time into
the idea that I had the other day.
i={1,0,0}
j={-1/3, sqrt(8/9),0}
k={-1/3,sqrt(2/9),sqrt(2/3)}
l={-1/3,sqrt(2/9),-sqrt(2/3)}
i+j+k+l=0
Op0=Transpose[{i,(3j+i)/sqrt(8),(2k+j+i)/sqrt(8/3)}]
={{1,0,0},{0,1,0},{0,0,1}}
Op1=Transpose[{j,(3k+j)/sqrt(8),(2l+k+j)/sqrt(8/3)}]
={{-1/3,-sqrt(2/9),-sqrt(2/3)},{sqrt(8/9),-1/6,-sqrt(1/12)},{0,sqrt(3/4),-1/2}}
Op2=Transpose[{k,(3l+k)/sqrt(8),(2i+l+k)/sqrt(8/3)}]
{{-1/3,-sqrt(2/9),sqrt(2/3)},{-sqrt(2/9),-2/3,-sqrt(1/3)},{sqrt(2/3),-sqrt(1/3),0}}
Op3=Transpose[{l,(3i+l)/sqrt(8),(2j+i+l)/sqrt(8/3)}]
={{-1/3,sqrt(8/9),0},{-sqrt(4/9),-1/6,sqrt(3/4)},{-sqrt(2/3),-sqrt(1/12),-1/2}}
I have verified this in Mathematica, but I can't copy and paste into a
text editor, so errors may have cropped up in the re-typing.
Anyway, I verified with these operators that
Op0.i=i
Op0.j=j
Op0.k=k
Op0.l=l
Op1.i=j
Op1.j=k
Op1.k=l
Op1.l=i
Op2.i=k
Op2.j=l
Op2.k=i
Op2.l=j
Op3.i=l
Op3.j=i
Op3.k=j
Op3.l=k
Det[Op0]=Det[Op2]=1
Det[Op1]=Det[Op3]=-1
This suggests that while Op0 is the identity matrix, Op2 is a rotation,
while Op1 and Op3 are reflections.
I'm not sure if this is what you were hoping for, but you might see if
this is consistent with the demonstrations in P4 you've made so far.
Spoonfed
> <numerous snips of old lines>
> Spoonfed wrote:
> >
> > I am trying to put together a method of translating your arithmetic for
> > P4 into R3. I've just got a start, and if you are familiar enough with
> > the ideas, you might be able to finish.
> >
> > Since #1 is identity, I choose it as my simplest looking basis vector.
> >
> > #1 = i = (1,0,0)
> >
> > >From here I have many options, I could choose +1 to be any point (-1/3,
> > .9428 cos theta, .9428 sin theta)
> >
> > I choose:
> >
> > -1 = j = (-1/3, .9428, 0)
> >
> > Now having gone to this point, I have only two remaining options, and I
> > choose arbitrarily between them.
> >
> > +1 = k = (-1/3, -.4714 , .8165)
> > #1 = l = (-1/3, -.4714 , -.8165)
Oops. I meant *1 = l
I had already stated that #1 = i
Check the math. I have defined i, j, k, and l as
#1 = i = (1,0,0)
-1 = j = (-1/3, .9428, 0)
+1 = k = (-1/3, -.4714 , .8165)
*1 = l = (-1/3, -.4714 , -.8165)
The last three lines show how to get (1,0,0), (0,1,0), and (0,0,1) as
linear combinations of i, j, k, and l.
Now, the operators say
j maps i into k.
That is... j maps (1,0,0) to (-1/3, -.4714, .8165)
j maps (0,1,0) to what?
First I do it the way I believe you did it when coding. Translate
(0,1,0) to a combination of i, j, k, and l, then
the i terms go to j,
the j terms go to k,
the k terms go to l,
the l terms go to i.
So (0,1,0) = :(3j+i)/(3*.9428) is mapped to (3k + j)/(3*.9428). Then
convert this back to the R3 representation using the definitions of k
and j.
Yeah, there seem to be two meanings of operator, at least what I've
heard. One meaning is just the sign between numbers, so add, multiply,
subtract, divide are all what are typically called operators. These
are not actually complete operators, though. They require a number.
For instance If I just told you to perform the addition operation, you
could add whatever you wanted. If I told you to perform the add 1
operation, you'd have to add 1.
So the four separate operators here are multiplying by -1, +1, *1, and
#1.
> Yet we know that we can transform a P4
> to a C3 (or R3) so we should be able to express the product for general
> Cartesian values. We should wind up with three functions:
> R1( c1, c2 )
> R2( c1, c2 )
> R3( c1, c2 )
> where the result is the product of c1 and c2:
> [C3 R1, R2, R3 ] . (where C3 indicates a 3D Cartesian vector )
> These expressions could be messy.
>
I'm not sure I follow, but see if you can make sense of what I did in
my post from earlier today.
> > Most R3X3 operators are not commutative, and I can't recall offhand the
> > easy way to determine comutativity. I think it is when the transpose
> > is equal to the inverse, or something like that.
>
> Are you saying that a table of Cartesian unit vector products would be
> helpful?
> I can spit those out easily on my computer.
> So if A = [C3 1, 0, 0], B = [ C3 0, 1, 0 ], C = [C3 0, 0, 1]
> that their products in polysigned complete the description?
> So we'd have
> A B , A C , B C
> That's a 3 x 3.
> Is it that easy?
> Is that a tensor and is it a complete description of the product?
> I'm not fully seeing how this will get general product for c1 and c2 in
> R3.
> Still I can easily generate AB, AC, and BC.
> I'll let you try first unless you want me to do it.
> I can verify your work.
Okay, I tried first. :-)
> I see commutativity rules for vectors but not for tensors in some docs
> I have.
>
It's easy to see that some operators in R3X3 are not commutative. For
instance a normal 90 degree rotation around the x axis, followed by a
normal 90 degree rotation around the z axis leads to a different
configuration than doing the same operations in the opposite direction.
> Your transform values are consistent with
> http://bandtechnology.com/PolySigned/CartesianTransform.html
> which gives:
>
> [P4 1, 0, 0, 0 ] : [C3 1, 0, 0 ]
> [P4 0, 1, 0, 0 ] : [C3 -0.333333, 0.942809, 0 ]
> [P4 0, 0, 1, 0 ] : [C3 -0.333333, -0.471405, 0.816497 ]
> [P4 0, 0, 0, 1 ] : [C3 -0.333333, -0.471405, -0.816497 ]
>
> and the inverse:
>
> [C3 1, 0, 0 ] : [P4 1, 0, 0, 0 ]
> [C3 0, 1, 0 ] : [P4 0.353553, 1.06066, 0, 0 ]
> [C3 0, 0, 1 ] : [P4 0.612372, 0.612372, 1.22474, 0 ]
>
> The math is really easy to do straight in P4 for instances and even in
> general compared to this approach. But the direction you are going
> could expose a match to some existing work in the Cartesian domain.
> That would be a nice discovery.
>
> -Tim
Hope so!
Spoonfed
Tim wrote:
> <numerous snips of old lines>
> Spoonfed wrote:
> > Tim wrote:
> > > Spoonfed wrote:
> > [snipping more old stuff]
> >
> > Is it possible that your analogy of P1 is more akin to the pointlike
> > nature of an event or the zero space-time interval represented by the
> > emission and absorption of a photon? I think you may have a powerful
> > mathematical concept in the polysigned numbers, but I can't see any way
> > P1 represents the whole of time.
>
> The 'whole' of time may be more like a hole of time (nonexistent). An
> event concept like choosing when to push a button is consistent with
> the degree of freedom. Are you saying it could imply the discrete
> nature of photon emission? I don't really understand the zero
> space-time interval concept. Is that another way of saying relative
> simultaneity?
> I'm glad that you are getting into polysign.
>
The space-time interval, s, between two events, as I understand it, is
defined
s=sqrt(c^2 t^2 - x^2) , where c=speed of light, t=the time measured
between the two events in a given reference frame, and x is the
distance between the two events as measured in the same reference
frame.
Notice that the speed of light is described by x = c t, so plugging
this into the equation yields zero. When the quantity, s is real, it
means that a photon can travel from the earlier event to the location
of the later event with time to spare, so the interval is called
timelike, because one definitely happened after the other.
When s is imaginary, it means that a photon cannot go from one event to
the location of the other event before the second event happens. This
type of interval is called spacelike, because there is no way for the
two events to happen at the same place.
When s is zero, it means either the two events happened at exactly the
same place at the same time, or that it is possible for a photon to
have traveled between the two events. The emission and absorption
event have a zero spacetime interval between them, even though there
may be years of time and light years of distance between the two
events.
By changing the velocity of the observer, (meaning: looking at the
situation in different reference frames) The space and time distances
between these events can be set to any value you want as long as x=ct.
By viewing it in possible reference frames, (v<c) you could see these
events be arbitrarily close to zero.
Anyway, I thought it sounded enough like your concept of P1 that I
should mention it, even though I still cannot see any direct
connection.
Spoonfed
And this, after a couple of weeks ago, I was getting after people for
calling Lorentz Transformation a rotation, "just because it had a
determinant of positive 1." I should say that these transformations
all preserve volume, and the ones with negative determinant would
result in mirroring everything, but as far as what other distortions
might develop, I haven't attempted any analysis.
Tim
Sue... wrote:
Hi Sue.
I'm not excited about the amount of work it would be to repeat all of
complex mathematics in the polysigned domain P3. It may be true that
there would be some benefit and gain of an intuitive feel by doing a
few practical examples. So far I have merely defined the rules and
proven that product and sum are homomorphic to the complex numbers. I
am happy to help you understand them to this level. There may be some
interesting consequences such as looking at the complex exponential and
attempting to construct that from the polysigned domain primitively.
The simplest construction that I see is just looking at ( * 1 - d ) in
the limit as d goes to zero. Powers of this will generate the unit
circle, just as they do for the complex numbers for ( 1 + i d ).
My website is the tutorial for the polysigned numbers.
Anything that is too difficult to understand there should be remedied
and I would appreciate any feedback you might have. So for example if
something doesn't make sense to you there perhaps I should modify it.
In that process we could both benefit.
I assure you that the math is very simple. Also the construction is
trivial; meaning that if you were to generalize sign you would wind up
with the same construction.
Unfortunately by publishing the identity relation I have spoiled that
process. Ideally you would come to that realization on your own. The
product relations are much more straightforward. But these two concepts
( the product and the sum ) are all that the polysigned numbers are.
They are purely arithmetical though they are also multidimensional.
Putting this arithmetical behavior into standard vector notation is not
so straightforward as you see Spoonfed and I struggle with it. The sum
is still the standard vector sum. But the product is weird in the
context of standard matrix math. Yet arithmetically it is just as
simple as the real numbers.
I am happy to clear up any confusion you may have about P3. Upon
getting P3 you will probably be comfortable with any sign level. In the
process perhaps my website will be improved. I am open to trying a
practical RLC circuit but the exponential form will have to be
considered in pure math before doing sinusoidal analysis. The common
form mentioned above doesn't really fit the usual analysis method. So
perhaps we have a little project. Perhaps Laplace xform is lurking
there?
-Tim
Sue...
There are some other examples that might accomplish the same thing.
For example, parametric amplification is analogous to pumping a
childs swing by periodcally changing the resonant frequency.
If you are steeped in mechanics you might find that a simpler
example to illustrate your imaginary notation. Most physics
students are exposed to the AC circuit examples to illustrate
the imaginary operator.
In pure math, everything is ~imaginary~. But in physics, the square
root of -1 doesn't just describe the space between our ears. We
can put our hand on the cables, feeding an improperly matched
induction motor and know that sqrt(-1) is very real when the hot
cable scalds us.
If you want your notation to describe what is presently written
as Maxwell's time dependent equations, it is necessary to
show that displacement on the imaginary axis, represents
real and measurable effects. An exercise with a simple
RLC circuit would give you some feel for whether your notation
is simplifying the description or complicating it.
The lack of a clear representation of the near-field imaginary
components is one of the most difficult things about learning
Maxwell's equations and the confusion surrounding the issue
is the primary motive for special relativity.
I think it goes with out saying, marathon runners are also capable
of walking. I think you'd find it a helpful exercise to
study how your notation represents imaginary numbers that
have real and predictable consequences. If you know an
example simpler than the RLC circuit, of course, the simpler the
better. As you are posting to a physics and not a pure math group,
I assume you have some expectation that your application is
something more than magic squares or pi to a zillion places. ;-)
Sue...
>
> -Tim
Tim
> Spoonfed wrote:
> > Tim wrote:
> >
O.K. There are some results here. Nice work.
I see that determinants of +1 are considered 'proper' and -1 are
'improper' transforms.
Does this mean that the coordinate system switches handedness?
That would be new to me. As I try to look at this in the tetrahedral
system I see two unit vector systems ( a and b ) which are labeled with
the signs and as we translate and rotate them we can match up -a to -b,
+a to +b, and *a to *b. At this point #a and #b may be in the same
position (same handedness) or they may be 4 units apart (opposite
handedness) This is 3D and is demonstrable with physical models. So
even though the system is perfectly symmetrical there is handedness.
This behavior certainly is not general since in P3 the unit products
will not flip handedness. The pattern of the unit vector products is
rich as you go up in sign. Some of them are quite flat (the primes).
There are harmonic relations based on the radix math of the sign. But
we want to just focus on P4 for now.
I've created a new thread on sci.math and sci.physics:
http://groups.google.com/group/sci.math/msg/08eb7a3195c7c09d
I'm hoping we can continue discussing this problem there. I've
cross-referenced this thread and I hope you will not take offense. I am
happy to keep working here also, but the problem is no longer a
relativity problem and I hope we can get some math people to input on
it. Vector math leads to tensor math and it is by definition
rotationally invariant. The algorithm that we are dealing with is
rotationally variant in P4 and above. I believe that this poses a
schism. The form exist but probably is not defined.
-Tim
Tim
Certainly. Physics is the motivation for the construction.
I am unaware of the near field conflict that you mention. Can this be
construed as two charges coming very close and generating infinite
force? This then gets us over towards particle physics and what
qualities a point particle model presents. The polysigned topology
offers some nifty things in that arena that are covered in this thread.
But the topological concept is still being developed. There is enough
there to see some possibilities. If the attempt being made can stand
then geometry will be at the heart, with generic points yielding spin
axes and so forth.
-Tim
Sue...
I have never seen it expressed that way. A particle physicist might
be able to make that connection but I can't. It usually involves
magnetic components that may *appear* faster than light in computations
but it really isn't because reactive components don't represent
the transport of mass or energy over space. To paraphrase
Jackson? Fitzpatrick?
~it is an absurdity... but it is an absurdity that propagates at
the speed of light~
Two charges moving together together can exert ~twice~ the
force on a target charge and seem to violate light speed over
a short (1/r^3) distance. You may have heard the question
what is the speed of magnetism? It is not necessarily c
in the short range but neither does it violate the speed of light.
Some is addressed here:
http://farside.ph.utexas.edu/teaching/em/lectures/node51.html
and some here:
http://arxiv.org/abs/physics/0204034
> This then gets us over towards particle physics and what
> qualities a point particle model presents. The polysigned topology
> offers some nifty things in that arena that are covered in this thread.
> But the topological concept is still being developed. There is enough
> there to see some possibilities. If the attempt being made can stand
> then geometry will be at the heart, with generic points yielding spin
> axes and so forth.
I always imagine the point as zero volume and representative only
to describe a point of interaction. IOW... an electon IS its associated
field. Like the earth's barycentre, you can't go dig it up and put it
in
a bottle. That is not a helpful concept for QM however. >:-)
Sue...
>
> -Tim
Sue...
snip
> I am unaware of the near field conflict that you mention.
This may be a better example after you digest the
Fitzpatrick and Jackson's comments in the parallel
response.
If instead of representing a dipole coupling structure
with Maxwell's equations, we show a 90 degree
impedance inverting transmission line section
which translates the 377 ohm impedance of
free space or the incident wave impedance,
to the feed point impedance,
then all the imaginary (reactive) components are
correctly represented in the lumped constant equivalents,
without resort to all the gymnastics associated with
Green's functions and Retarded potentials.
The disadvantage is that all the spatial information
necessary to to derive magnetic fields is discarded.
I have an idea that Maxwell's notion of a
'displacement current" is still expressing itself
somewhere that it shouldn't. It may carved in the
stone of eps_0 and mu_0 forever more to abrade
our behinds when we sit on it.
So... there is definitely a market for a better way
to represent the imaginary components in EM field
equations. I am not familiar enough with your
formalism to see if it applies, however. I'll lurk
in on some of your development to see if I
can put a brick in place but it seems more
abstraction than I can master in the time
availible.
Sue...
sinp
> -Tim
Spoonfed
#1 (identity) and +1 proper
-1 and *1 improper.
Yes, improper transforms switch "handedness" or turn everything inside
out. That was also a surprise to me.
Remember I made several decisions when putting together the matrices.
The first was during choosing the vectors representing #1, -1, +1, and
*1. I chose #1 should be along the x axis. I think it might be
possible to choose an arbitrary axis, but the transformation that maps
#1 to #1 would still be the {{1,0,0},{0,1,0},{0,0,1}}
-1 which turned out to be an improper transform involved a choice, but
not a real choice. Regardless of which way I chose it, it is only
based on the perspective it's viewed from.
+1 which is the only proper transformation, also involved my only real
choice in constructing the vectors. After having gone from #1 to -1, I
had to choose between turning right or turning left. This suggests
that there may be two viable alternatives for the matrix representing
the X(+1) operator. There may be both a left-handed and right-handed
representation of the operators.
Finally *1 was the only choice left, and it was an improper
transformation.
I'll come back and look at this again when I get a-ways through the
class I'm taking this summer. I'll be brushing up on symmetries and
transformations.
> As I try to look at this in the tetrahedral
> system I see two unit vector systems ( a and b ) which are labeled with
> the signs and as we translate and rotate them we can match up -a to -b,
> +a to +b, and *a to *b. At this point #a and #b may be in the same
> position (same handedness) or they may be 4 units apart (opposite
> handedness) This is 3D and is demonstrable with physical models. So
> even though the system is perfectly symmetrical there is handedness.
> This behavior certainly is not general since in P3 the unit products
> will not flip handedness.
This is a good thing. "Handedness" is intrinsic in three dimensions.
If you have a representation of three dimensions that doesn't have
handedness fall out somehow, you haven't accomplished a representation
of 3D.
Incidentally, magnetic north and south are arbitrary constructions of
handedness, much like my choice of whether to go right or left on the
choice of how to represent +1, or like the choice of whether to use our
right or left hand when figuring the direction of a cross-product. And
if you don't already know this, magnetite, everybody's favorite
naturally occurring magnet, has a rhombic-dodecahedron crystal
structure that precisely resembles your P4-gon.
> The pattern of the unit vector products is
> rich as you go up in sign. Some of them are quite flat (the primes).
> There are harmonic relations based on the radix math of the sign. But
> we want to just focus on P4 for now.
>
I'm sure they are. And I may come to appreciate them in the future. I
have been reluctant to study them, because I have seen no reason to
tack extra dimensions on those we observe for no better reason that we
can. The only use I'd ever seen made of multi-dimensional constructs
was in Lagrangians and generalized coordinates. But I think that this
approach lends a certain inevitability and consistency of construction
to the higher dimensions.
> I've created a new thread on sci.math and sci.physics:
> http://groups.google.com/group/sci.math/msg/08eb7a3195c7c09d
> I'm hoping we can continue discussing this problem there. I've
> cross-referenced this thread and I hope you will not take offense. I am
> happy to keep working here also, but the problem is no longer a
> relativity problem and I hope we can get some math people to input on
> it. Vector math leads to tensor math and it is by definition
> rotationally invariant. The algorithm that we are dealing with is
> rotationally variant in P4 and above. I believe that this poses a
> schism. The form exist but probably is not defined.
>
> -Tim
If an analysis of P5 somehow were to give five operators: identity,
three simple rotations, and one procrustian stretch, it would again
become a relativity problem. You mentioned that the prime numbered
polysigns were "quite flat" I wasn't quite sure what this meant. If
changing the parameter related to that "flatness" could look anything
like the dots in this diagram.
http://en.wikipedia.org/wiki/Image:Animated_Lorentz_Transformation.gif
Do you have a general pattern for finding the unit vectors of P_n?
a = pi - acos( 1 / ( n - 1 )) =
a(2)=180
a(3)=120
a(4)=109.47
a(5)=104.48
i={1,0,0}
j={-1/3, sqrt(8/9),0}
= {cos(a4), sin(a4),0)}
k={-1/3,sqrt(2/9),sqrt(2/3)}
={cos(a4), sin(a4)cos(a3),sin(a4)sin(a3)}
l={-1/3,sqrt(2/9),-sqrt(2/3)}
={cos(a4),sin(a4)sin(a3),-sin(a4)cos(a3)}
I noticed you had been talking about some kind of recursive definition,
of 0D + 1D +2D, etc. I wonder whether the same kind of recursivity
could be applied to describing things in cartesian coordinates.
Because I think we've established a fairly general method of
translating the operators of P(n+1) into RnXn. All we would need is a
general method of establishing the n+1 unit vectors, then you'll have
an explicit mathematical description of polysigned numbers for all
dimensions.
Jon
Spoonfed
isn't so much a tutorial as a definition, in my own words. This is
just a summary of the most informative things I saw in the conversation
that may have been buried.
Tim wrote:
> Spoonfed wrote:
> > Tim wrote:
> <Snipping away old stuff.>
> > > Here is the general n-signed product in C++ code:
> > > for( i = 0; i < n; i++ )
> > > {
> > > for( j = 0; j < n; j++ )
> > > {
> > > k = (i+j)%n;
> > > x[ k ] += s1.x[i] * s2.x[j];
> > > }
> > > }
> >
> >
> >
Note, (i+j)%n = (i+j)modulo n
This particular piece of code makes the n-signed numbers
indistinguishable from particular sets of complex numbers. However,
the other piece to the definition
By definition, P4 represents four vectors lying at angles of
Pi-arccos(1/3) radians from each other. These four vectors cannot lie
in the same plane, thus are not described by any set of four complex
numbers.
In my understanding, this is generalizable, so in one dimension, P2,
you have two vectors lying 180 degrees from one-another,
In 2 dimensions, P3 is three vectors lying 120 degrees from
one-another.
In 3 dimensions, P4 is four vectors lying 109.47 degrees from each
other.
Now the final key was that Tim said that multiplication was possible in
any level of dimension. Not cross-product, not dot product, but simple
multiplication. I have never heard of such a thing in 3D, but he's
given a general, reasonable set of rules for simple multiplication in
his C++ code, which reproduces multiplication in 1D and 2D, and seems
to flow into a pattern generalizable to any dimension.
Sue...
Is the imaginary operator for Hilbert space the same as
the imaginary operator for Lorenz (Minkowski) space?
I can buy lumped equivalents for the
imaginaries in Lorenz space, at the radio parts store.
I wouldn't be so sure about finding parts to work in
Hilbert space.
Feynman may have addressed this somewhere deep in bowels
of QED but it also may be one of the "shut up and calculate"
issues.
Thanks for tutorial in following post (yes... the sentence
violates causality). I'll send some time with it. :o)
Sue...
Spoonfed
Sue... wrote:
> Spoonfed wrote:
> > The space-time interval, s, between two events, as I understand it, is
> > defined
> >
> > s=sqrt(c^2 t^2 - x^2) , where c=speed of light, t=the time measured
> > between the two events in a given reference frame, and x is the
> > distance between the two events as measured in the same reference
> > frame.
> >
> > Notice that the speed of light is described by x = c t, so plugging
> > this into the equation yields zero. When the quantity, s is real, it
> > means that a photon can travel from the earlier event to the location
> > of the later event with time to spare, so the interval is called
> > timelike, because one definitely happened after the other.
> >
> > When s is imaginary, it means that a photon cannot go from one event to
> > the location of the other event before the second event happens. This
> > type of interval is called spacelike, because there is no way for the
> > two events to happen at the same place.
>
>
> Is the imaginary operator for Hilbert space the same as
> the imaginary operator for Lorenz (Minkowski) space?
> I can buy lumped equivalents for the
> imaginaries in Lorenz space, at the radio parts store.
> I wouldn't be so sure about finding parts to work in
> Hilbert space.
>
Now, I'm trying to figure out what you might mean by this. I reviewed
what Hilbert Space was on Wikipedia, and managed to recall vaguely what
I learned of it in my Quantum Mechanics. The main difference, I think
is that Minkowski spacetime is four-dimensional, and Hilbert Space is
infinite dimensional.
For instance, the coefficients in a Taylor series expansion of a
function might go on forever Sum[A_n x^n, {n,0,infinity}], then your
coordinates are {A_0,A_1,A_2,A_3,...} for an infinite dimensional
vector.
The Polysigned numbers, as I understand them, can be of unlimited
dimension, but never infinite dimensional.
As for an imaginary operator in Hilbert Space, I have no idea what that
might be, except that Taylor series can be used to relate exp[x], exp[i
x], sin[x], cos[x], sinh[x], and cosh[x] and sometimes Lorentz
transformation is written as
/cosh[A] -sinh[A]\
\-sinh[A] cosh[A]/
where A=.5 ln[(c+v)/c-v)]
I might be wrong, but mathematically, I don't think you can render this
into a rotation
/cos[B] sin[B]\
\-sin[B] cos[B]/
where B is some imaginary or complex number. So I haven't see any
connection between Lorentz Transformation and imaginary numbers.
Except for the spacetime interval s=sqrt(c^2 t^2 - x^2) which is an
arbitrary definition, and could just as well be defined s=sqrt(x^2 -
c^2 t^2) thus making the spacelike intervals real and timelike
intervals imaginary. The meaning of both types of intervals is quite
physical and quite real, one being the time you have to wait between
the events happening at the same place, and the other being the
distance between simultaneous events, and both having the option to
tack on an arbitrarily large positive number. i.e.
For instance, if two events happen with a spacetime interval of 1
nanosecond they might be separated in any of the following ways.
time=1 ns, distance = 0 feet
time = 2ns, distance = sqrt(3)=1/73 feet
time = 3ns, distance = sqrt(8)=2.82 feet
time = 4ns, distance = sqrt(15)=3.87 feet
time=1 ms, distance= sqrt(999999)=999.9995 feet
time = 1 second, distance = sqrt(10^18-1) = 1 light second or too close
to measure the difference.
The vague similarity I thought I saw in the Polysigned numbers is the
fact that you can always zero either the time or the distance between
any two events. However, there is a hyperbolic relationship between
distance and time for events with nonzero interval. There is a linear
relationship between distance and time for events with zero interval.
If events have interval 0, you can choose to view it in any of these
frames...
time=0, distance = 0
time = 1, distance=1
time=2, distance=2
etc.
> Feynman may have addressed this somewhere deep in bowels
> of QED but it also may be one of the "shut up and calculate"
> issues.
>
Hmmmm, yes, I think I remember having the sense of Feynman's ghost
telling me that after I read an article of his about the Least Action
Principle three times without detecting his point.
> Thanks for tutorial in following post (yes... the sentence
> violates causality). I'll send some time with it. :o)
>
LOL
> Sue...
>
> >
> > When s is zero, it means either the two events happened at exactly the
> > same place at the same time, or that it is possible for a photon to
> > have traveled between the two events. The emission and absorption
> > event have a zero spacetime interval between them, even though there
> > may be years of time and light years of distance between the two
> > events.
> >
> > By changing the velocity of the observer, (meaning: looking at the
> > situation in different reference frames) The space and time distances
> > between these events can be set to any value you want as long as x=ct.
> >
> > By viewing it in possible reference frames, (v<c) you could see these
> > events be arbitrarily close to zero.
> >
> > Anyway, I thought it sounded enough like your concept of P1 that I
> > should mention it, even though I still cannot see any direct
> > connection.
-Jon
Sue...
In Hilbert space we plot probabilty. So it is tricky to get a
good electrical connection between an inductor and the dice.
Gauss Law and normal distribution gives us a reason to
try tho.
>
> For instance, the coefficients in a Taylor series expansion of a
> function might go on forever Sum[A_n x^n, {n,0,infinity}], then your
> coordinates are {A_0,A_1,A_2,A_3,...} for an infinite dimensional
> vector.
>
> The Polysigned numbers, as I understand them, can be of unlimited
> dimension, but never infinite dimensional.
>
> As for an imaginary operator in Hilbert Space, I have no idea what that
> might be, except that Taylor series can be used to relate exp[x], exp[i
> x], sin[x], cos[x], sinh[x], and cosh[x] and sometimes Lorentz
> transformation is written as
It is one of the problems of QM. Probability works... but we
can't offer a deterministic mechanism to explain why.
>
> /cosh[A] -sinh[A]\
> \-sinh[A] cosh[A]/
> where A=.5 ln[(c+v)/c-v)]
>
> I might be wrong, but mathematically, I don't think you can render this
> into a rotation
> /cos[B] sin[B]\
> \-sin[B] cos[B]/
> where B is some imaginary or complex number. So I haven't see any
> connection between Lorentz Transformation and imaginary numbers.
You are using an imaginary time scale because a moving charge
must accelerate a distant charge through a squishy coupling
we call the Coulomb force.
http://physics.nist.gov/cuu/Images/alphaeq.gif
http://physics.nist.gov/cuu/Constants/alpha.html
> Except for the spacetime interval s=sqrt(c^2 t^2 - x^2) which is an
> arbitrary definition, and could just as well be defined s=sqrt(x^2 -
> c^2 t^2) thus making the spacelike intervals real and timelike
> intervals imaginary.
No it is not arbitrary at all. Coulomb coupled paths are
isolated to the degree their volumes are separated spatially
OR temporally. That is why we need to sometimes equate
spatial and temporal displacements. Ahh... for QM it is
is probably considered arbitrary.
> The meaning of both types of intervals is quite
> physical and quite real, one being the time you have to wait between
> the events happening at the same place, and the other being the
> distance between simultaneous events, and both having the option to
> tack on an arbitrarily large positive number. i.e.
That is the QM paradigm. An electron moves, or it does not.
There is no accounting of the many forces that might cause
it to move. Dice! LOL
>
> For instance, if two events happen with a spacetime interval of 1
> nanosecond they might be separated in any of the following ways.
> time=1 ns, distance = 0 feet
> time = 2ns, distance = sqrt(3)=1/73 feet
> time = 3ns, distance = sqrt(8)=2.82 feet
> time = 4ns, distance = sqrt(15)=3.87 feet
> time=1 ms, distance= sqrt(999999)=999.9995 feet
> time = 1 second, distance = sqrt(10^18-1) = 1 light second or too close
> to measure the difference.
>From a formalism based on Minkowski's light cone. I'll take your
word all that doesn't violate the speed of light. :o)
>
> The vague similarity I thought I saw in the Polysigned numbers is the
> fact that you can always zero either the time or the distance between
> any two events. However, there is a hyperbolic relationship between
> distance and time for events with nonzero interval. There is a linear
> relationship between distance and time for events with zero interval.
Lacking your insight, I can't confim your acessment. But I will note
that you are not using polysign as a description of space-time but
rather as an index to Minkowski's space-time. So you we are stuck
with Coulomb gauge 3D + 1T paradigm and statistics where field
equations would be more descriptive. Basically QM.
>
> If events have interval 0, you can choose to view it in any of these
> frames...
> time=0, distance = 0
> time = 1, distance=1
> time=2, distance=2
> etc.
>
> > Feynman may have addressed this somewhere deep in bowels
> > of QED but it also may be one of the "shut up and calculate"
> > issues.
> >
>
> Hmmmm, yes, I think I remember having the sense of Feynman's ghost
> telling me that after I read an article of his about the Least Action
> Principle three times without detecting his point.
>
> > Thanks for tutorial in following post (yes... the sentence
> > violates causality). I'll send some time with it. :o)
> >
>
> LOL
I don't think I am up for doing something 'Maxwellian' with
a number system I barely understand. I'll have to keep it
under my cap for a while because there IS something
attractive about it that smells like natural logarithms
that ~might~ smooth some of the rough edges off of Maxwells
or Weber's equations in the near field. When that's done
Fitzpatrick won't need a chapter titled "Advanced Potentials?".
http://farside.ph.utexas.edu/teaching/em/lectures/node51.html
Sue...
Tim
> Tim wrote:
> > Spoonfed wrote:
So you still want 4D spacetime. It may be worth a try but I wouldn't
reccomend it.
Don't forget that these systems suffer from rotational variance up
there above P3.
>
> Do you have a general pattern for finding the unit vectors of P_n?
Yes. I have it all coded forward and reverse with optimization.
It is general and works for any sign level.
I guess you'll be wanting some code.
Can you do C++ ?
The library I have made is easy to use.
You can do things like:
/******* C++ code example *********/
nSigned s1(5); // a five-signed number
nSigned s2; // don't need to declare size for assigned values
Cartesian c;
s1[0] = 2.3;
s1[2] = 3.3;
s2 = s1 * s1 + s1; // multiply and add
c = s2; // the cartesian transform
cout << s1 << s2 << c << "\n"; // print values
s2 = s2 * s2; // this works too
c = s2 * s1; // this works too
s1 = c; // inverse transform
cout << s1 << "\n";
// etc.
/******** eo C++ code **********/
There are also projections that do not require manual entry of vectors.
It orthogonalizes random values so if you don't like the way something
looks you can just run it again and it will present another projection.
The graphing will require libGD and some other related libraries.
The code currently works on cygwin or linux but should port easily to
another platform.
There is no fancy installation system or configuration scheme. Just one
makefile that doesn't even have header dependencies (maintaining those
can be as troublesome as not having them). My code largely ignores the
C++ protection mechanism making lots of members that should be
protected public. But still it is no worse than C code for it.
> a = pi - acos( 1 / ( n - 1 )) =
> a(2)=180
> a(3)=120
> a(4)=109.47
> a(5)=104.48
>
> i={1,0,0}
>
> j={-1/3, sqrt(8/9),0}
> = {cos(a4), sin(a4),0)}
>
> k={-1/3,sqrt(2/9),sqrt(2/3)}
> ={cos(a4), sin(a4)cos(a3),sin(a4)sin(a3)}
>
> l={-1/3,sqrt(2/9),-sqrt(2/3)}
> ={cos(a4),sin(a4)sin(a3),-sin(a4)cos(a3)}
>
> I noticed you had been talking about some kind of recursive definition,
> of 0D + 1D +2D, etc. I wonder whether the same kind of recursivity
> could be applied to describing things in cartesian coordinates.
> Because I think we've established a fairly general method of
> translating the operators of P(n+1) into RnXn. All we would need is a
> general method of establishing the n+1 unit vectors, then you'll have
> an explicit mathematical description of polysigned numbers for all
> dimensions.
>
> Jon
I don't think your operators are truly generalized yet. Are you able to
express things like:
( 1.1, 2.2, 3.3 ) ( 1.2, 3.4, 5.6 ) = ( x, y, z )
where these vectors are Cartesian?
If you can get x, y, and z I'll check it.
I'm afraid you'll wind up having to go through the steps that I put on
sci.math for the result. Whether we call this a nonlinear versus linear
process may not be the right context. In the polysigned domain we've
discovered a linear system whose geometry is not linear above P3.
Perhaps we can tack the definition of linearity on to the list of terms
that polysigned math redefines (along with dimension).
The recursive definition (i.e. the family of polysgined numbers)
suggests a format that I call a tatrix, for triangular matrix:
a11
a21 a22
a31 a32 a33
...
These a(n,m) are just magnitudes since their sign is implied by the
ordering of the system. I have tried to find use of this form in
existing math and probed for it on sci.math once but haven't found
anything.
I am very happy that you like this math. You are the first person in a
while that has grasped it. It is instructive what you go through as you
learn how it works. That same 2D scenario has struck three people,
maybe the only three who have grasped the math. I wonder if something
is there. I once played with 2D folded systems that can pack all the
information you want by folding sectors (like Lorentz but no
negatives). It can lead to some number patterns since one ray is shared
by two sectors. So by the time you go to describe a high dimension
(without exclusion in each sector) you wind up with interesting
patterns. There is no direct relation to polysign that I know of, but
the conception of a mutidimensional system is common to both.
-Tim
Here are some of the transform results in case you want to try some. I
stopped at P7 but can go to any level you like.
Polysigned Unit Vectors -> Cartesian
[P1 1 ] :
[P2 1, 0 ] : [C1 1 ]
[P2 0, 1 ] : [C1 -1 ]
[P3 1, 0, 0 ] : [C2 1, 0 ]
[P3 0, 1, 0 ] : [C2 -0.5, 0.866025 ]
[P3 0, 0, 1 ] : [C2 -0.5, -0.866025 ]
[P4 1, 0, 0, 0 ] : [C3 1, 0, 0 ]
[P4 0, 1, 0, 0 ] : [C3 -0.333333, 0.942809, 0 ]
[P4 0, 0, 1, 0 ] : [C3 -0.333333, -0.471405, 0.816497 ]
[P4 0, 0, 0, 1 ] : [C3 -0.333333, -0.471405, -0.816497 ]
[P5 1, 0, 0, 0, 0 ] : [C4 1, 0, 0, 0 ]
[P5 0, 1, 0, 0, 0 ] : [C4 -0.25, 0.968246, 0, 0 ]
[P5 0, 0, 1, 0, 0 ] : [C4 -0.25, -0.322749, 0.912871, 0 ]
[P5 0, 0, 0, 1, 0 ] : [C4 -0.25, -0.322749, -0.456435, 0.790569 ]
[P5 0, 0, 0, 0, 1 ] : [C4 -0.25, -0.322749, -0.456435, -0.790569 ]
[P6 1, 0, 0, 0, 0, 0 ] : [C5 1, 0, 0, 0, 0 ]
[P6 0, 1, 0, 0, 0, 0 ] : [C5 -0.2, 0.979796, 0, 0, 0 ]
[P6 0, 0, 1, 0, 0, 0 ] : [C5 -0.2, -0.244949, 0.948683, 0, 0 ]
[P6 0, 0, 0, 1, 0, 0 ] : [C5 -0.2, -0.244949, -0.316228, 0.894427, 0 ]
[P6 0, 0, 0, 0, 1, 0 ] : [C5 -0.2, -0.244949, -0.316228, -0.447214,
0.774597 ]
[P6 0, 0, 0, 0, 0, 1 ] : [C5 -0.2, -0.244949, -0.316228, -0.447214,
-0.774597 ]
[P7 1, 0, 0, 0, 0, 0, 0 ] : [C6 1, 0, 0, 0, 0, 0 ]
[P7 0, 1, 0, 0, 0, 0, 0 ] : [C6 -0.166667, 0.986013, 0, 0, 0, 0 ]
[P7 0, 0, 1, 0, 0, 0, 0 ] : [C6 -0.166667, -0.197203, 0.966092, 0, 0, 0
]
[P7 0, 0, 0, 1, 0, 0, 0 ] : [C6 -0.166667, -0.197203, -0.241523,
0.935414, 0, 0 ]
[P7 0, 0, 0, 0, 1, 0, 0 ] : [C6 -0.166667, -0.197203, -0.241523,
-0.311805, 0.881917, 0 ]
[P7 0, 0, 0, 0, 0, 1, 0 ] : [C6 -0.166667, -0.197203, -0.241523,
-0.311805, -0.440959, 0.763763 ]
[P7 0, 0, 0, 0, 0, 0, 1 ] : [C6 -0.166667, -0.197203, -0.241523,
-0.311805, -0.440959, -0.763763 ]
Cartesian Unit Vectors -> Polysigned
[C1 1 ] : [P2 1, 0 ]
[C2 1, 0 ] : [P3 1, 0, 0 ]
[C2 0, 1 ] : [P3 0.57735, 1.1547, 0 ]
[C3 1, 0, 0 ] : [P4 1, 0, 0, 0 ]
[C3 0, 1, 0 ] : [P4 0.353553, 1.06066, 0, 0 ]
[C3 0, 0, 1 ] : [P4 0.612372, 0.612372, 1.22474, 0 ]
[C4 1, 0, 0, 0 ] : [P5 1, 0, 0, 0, 0 ]
[C4 0, 1, 0, 0 ] : [P5 0.258199, 1.0328, 0, 0, 0 ]
[C4 0, 0, 1, 0 ] : [P5 0.365148, 0.365148, 1.09545, 0, 0 ]
[C4 0, 0, 0, 1 ] : [P5 0.632456, 0.632456, 0.632456, 1.26491, 0 ]
[C5 1, 0, 0, 0, 0 ] : [P6 1, 0, 0, 0, 0, 0 ]
[C5 0, 1, 0, 0, 0 ] : [P6 0.204124, 1.02062, 0, 0, 0, 0 ]
[C5 0, 0, 1, 0, 0 ] : [P6 0.263523, 0.263523, 1.05409, 0, 0, 0 ]
[C5 0, 0, 0, 1, 0 ] : [P6 0.372678, 0.372678, 0.372678, 1.11803, 0, 0 ]
[C5 0, 0, 0, 0, 1 ] : [P6 0.645497, 0.645497, 0.645497, 0.645497,
1.29099, 0 ]
[C6 1, 0, 0, 0, 0, 0 ] : [P7 1, 0, 0, 0, 0, 0, 0 ]
[C6 0, 1, 0, 0, 0, 0 ] : [P7 0.169031, 1.01419, 0, 0, 0, 0, 0 ]
[C6 0, 0, 1, 0, 0, 0 ] : [P7 0.20702, 0.20702, 1.0351, 0, 0, 0, 0 ]
[C6 0, 0, 0, 1, 0, 0 ] : [P7 0.267261, 0.267261, 0.267261, 1.06904, 0,
0, 0 ]
[C6 0, 0, 0, 0, 1, 0 ] : [P7 0.377964, 0.377964, 0.377964, 0.377964,
1.13389, 0, 0 ]
[C6 0, 0, 0, 0, 0, 1 ] : [P7 0.654654, 0.654654, 0.654654, 0.654654,
0.654654, 1.30931, 0 ]
Spoonfed
Tim wrote:
> I don't think your operators are truly generalized yet. Are you able to
> express things like:
> ( 1.1, 2.2, 3.3 ) ( 1.2, 3.4, 5.6 ) = ( x, y, z )
> where these vectors are Cartesian?
> If you can get x, y, and z I'll check it.
> I'm afraid you'll wind up having to go through the steps that I put on
> sci.math for the result. Whether we call this a nonlinear versus linear
> process may not be the right context. In the polysigned domain we've
> discovered a linear system whose geometry is not linear above P3.
> Perhaps we can tack the definition of linearity on to the list of terms
> that polysigned math redefines (along with dimension).
If I got the same answer I probably went through the same or equivalent
steps. My answer is (-13.5662, -15.8395,18.4878)
But my process was not uncomplicated. Err, it was kind of complicated.
1.1i+2.2*(3j + i)/Sqrt[8]+3.3*(2k + j + i)/Sqrt[8/3]
Optional step: Replace i with (#1), j=(-1),k=(+1), l=(*1) but there
are no l's
1.1*(#1)+2.2*(3(-1) + (#1))/Sqrt[8]+3.3*(2(+1) + (-1) + (#1))/Sqrt[8/3]
Required step:Replace i with Op0, j with Op1, k with Op2, l with Op3.
Each of these are 3X3 matrices, so we get:
{{1.1, -3.95788, -0.255256}, {2.2, 0.478494, -3.59043}, {3.3, 1.43747,
1.72151}}
When we do the same for (1.2, 3.4, 5.6), we get
1.2i+3.4*(3Op1 + Op0)/Sqrt[8]+5.6*(2Op2 + Op1 + Op0)/Sqrt[8/3]
= {{1.2, -6.54974, -0.144486}, {3.4, 0.0863979, -5.99078}, {
5.6, 2.13315, 2.3136}}
Finally I multiply the two results together, and whether I multiply the
first by the second, or the second by the first, I get the same answer:
-13.5662 -8.09117 22.9613
-15.8395 -22.027 -11.4912
18.4878 -17.8177 -5.10547
Now, we can multiply by our identity, #1 = {1,0,0} to get our final
answer
(-13.5662, -15.8395,18.4878)
Is that what you got?
Spoonfed
> Spoonfed wrote:
> > Tim wrote:
I see a fairly general pattern emerging: let
th1=Pi-ArcCos(1/1)=180
th2=Pi-ArcCos(1/2)
th3=Pi-ArcCos(1/3)
th4=Pi-ArcCos(1/4)
>
> Polysigned Unit Vectors -> Cartesian
>
> [P1 1 ] :
>
Can't help you there.
>
> [P2 1, 0 ] : [C1 1 ]
> [P2 0, 1 ] : [C1 -1 ]
>
P2 is homomorphic to the set of real numbers, so -1 and +1 are
described by R1X1
+1 = {1}
-1 = {Cos(th1)}
>
> [P3 1, 0, 0 ] : [C2 1, 0 ]
> [P3 0, 1, 0 ] : [C2 -0.5, 0.866025 ]
> [P3 0, 0, 1 ] : [C2 -0.5, -0.866025 ]
>
*1 = {1,0}
-1 = {Cos[th2],Sin[th2]}
+1={Cos[th2],Sin[th2]Cos[th1]}
>
> [P4 1, 0, 0, 0 ] : [C3 1, 0, 0 ]
> [P4 0, 1, 0, 0 ] : [C3 -0.333333, 0.942809, 0 ]
> [P4 0, 0, 1, 0 ] : [C3 -0.333333, -0.471405, 0.816497 ]
> [P4 0, 0, 0, 1 ] : [C3 -0.333333, -0.471405, -0.816497 ]
>
i = {1, 0, 0}
j = {Cos[th3], Sin[th3], 0}
k = {Cos[th3], Sin[th3]*Cos[th2], Sin[th3]*Sin[th2]}
l = {Cos[th3], Sin[th3]*Cos[th2], Sin[th3]*Sin[th2]*Cos[th1]}
>
> [P5 1, 0, 0, 0, 0 ] : [C4 1, 0, 0, 0 ]
> [P5 0, 1, 0, 0, 0 ] : [C4 -0.25, 0.968246, 0, 0 ]
> [P5 0, 0, 1, 0, 0 ] : [C4 -0.25, -0.322749, 0.912871, 0 ]
> [P5 0, 0, 0, 1, 0 ] : [C4 -0.25, -0.322749, -0.456435, 0.790569 ]
> [P5 0, 0, 0, 0, 1 ] : [C4 -0.25, -0.322749, -0.456435, -0.790569 ]
>
i = {1, 0, 0, 0};
j = {Cos[th4], Sin[th4], 0, 0}
k = {Cos[th4],
Sin[th4]*Cos[th3],
Sin[th4]*Sin[th3],
0}
l = {Cos[th4],
Sin[th4]*Cos[th3],
Sin[th4]*Sin[th3]*Cos[th2],
Sin[th4]*Sin[th3]*Sin[th2]}
m = {Cos[th4],
Sin[th4]*Cos[th3],
Sin[th4]*Sin[th3]*Cos[th2],
Sin[th4]*Sin[th3]*Sin[th2]*Cos[th1]}
Pn+1 (Iterative definition--not recursive. )
(n-dimensions, or (n+1) polysigned numbers)
i = {1,0,0,...,0} = {i0,i1,i2,...,i_n}
j = {i0*Cos[th_n], i0*Sin[th_n], 0,...,0}
k={j0, j1*Cos[th_n-1], j1*Sin[th_n-1],0,....,0}
l={k0,k1,k2*Cos[th_n-2], k2*Sin[th_n-2],0,...,0}
m={l0,l1,l2,l3*Cos[th_n-3],l3*Sin[th_n-3],0,...,0}
.
.
.
z= {y0,y1,y2,y3,y4,y5,...,y_n-2, y_n-1, y_n-1*Cos[th1]}
Each vector has two modifications on the last.
Tim
[C3 -13.5662, -15.8395, 18.4878 ]
so congratulations. I didn't think you'd be able to get it that way.
I am studying your method more closely. Are you working in the
polysigned domain embedded in the Cartesian domain? For example
> Op1=Transpose[{j,(3k+j)/sqrt(8),(2l+k+j)/sqrt(8/3)}]
> ={{-1/3,-sqrt(2/9),-sqrt(2/3)},{sqrt(8/9),-1/6,-sqrt(1/12)},{0,sqrt(3/4),-1/2}}
>
the transform step was forgone in exchange for four operators so that a
product like:
(C3 0, 1, 0 )(C3 2, 3, 4)
could be gotten and that you would try a linear superposition of these.
And now I see you would have had only three operators if it were so.
You do convert back to the polysigned domain. But I misunderstood.
In that case the only rule system that is really needed is that
j j = k
j k = l
j l = i
j i = j
So now a general polysign product:
( ai + bj + ck + dl )( ei + fj + gk + hl)
= aei + afj + agk + ahl
+ bej + bfk + bgl +bhi
+ cek + cfl + cgi + chj
+ del + dfi + dgj + dhk
It's just the distribution and sum. This is identical to the mnemonic
product.
Ahh. You don't have to covert back to Cartesian because your OPn are
doing it.
Could your method be called a mixed mode?
I was hoping you'd stay in the Cartesian domain completely and hence
solve the problem that was posted on sci.math. Anyhow it's all good
exercise.
But can it be done purely in Cartesian? The transforms are there. The
math is clearly defined. But the matrix math is nonexistent.
Your math is consistent and makes sense and if it helps you understand
the system then it is certainly of value. It satiates a need to
translate to the matrix. It's a good step.
Whether we are discussing 3D space or four-signed numbers seems to be
the divide we are trying to bridge. I do not think that P4 should be
confused with the three dimensions of spacetime. Do you see spin 1/2
and spin 1 behavior in the sign behavior? This is the operator
treatment that you are developing as Opn() which I just see as sign.
Repeated use of a sign becomes exponentiation so for example (-1)^n
goes like:
- + * # - + * # ...
+1 goes like:
+ # + # + # ...
or
- * - * - * ...
*1 goes like
* + - # * + - # ...
#1 goes like
# # # ...
or
* * * * ...
or
+ + + ...
or
- - - ...
This type of patterning is simple but has quite some variations as one
goes up in sign.
Anyhow I've been looking at the - and * signs of P4 as spin 1/2 and the
+ as spin 1.
The # would be neutral. I don't mind if you don't think much of it. I
hardly understand spin so the congruence is questionable. P4 is the
first dimension above spacetime and if it exists behaves differently
from spacetime. All parts of this statement assume that a product
operation is in the base of physics. To allow these higher dimensions
in poses the problem of how things can be so simple. In some regards
these higher dimensions are well behaved. See for example:
http://bandtechnology.com/PolySigned/Mandelbrot/MandelbrotStudy.html
It's just slices of the higher dimensions but at face value it shows
that things get simpler in higher dimension.
-Tim
Tim
I see and agree with your analysis. I've dodged the use of trig in my
transform but the concept helps explain the behavior. It is a
dimensional reduction.
We know that the system has perfect symmetry with n rays in n-1
dimensions.
The sum of the unit vectors is zero so each contributes 1/(n-1) in any
other unit vector direction. We can extract out a dimension lying along
one of these unit vectors. What is left is n-1 rays is n-2 dimensions
scaled down by a factor. I'm being lazy and not getting the factor here
but it's on my website as u and v factors, and you are exposing it as a
trig relation. Nice work.
-Tim
Spoonfed
I'm glad you got the same answer. I was checking back all day
yesterday. Is there any particular reason our number should have
grown? If you multiply the magnitudes of your original numbers, you
get a smaller number than the magnitude of the product. This is
probably an error due to rounding, but I'll check into it.
> I am studying your method more closely. Are you working in the
> polysigned domain embedded in the Cartesian domain?
I think maybe that's what I did.
I can say again what I did. I expressed the original vectors as a
linear combination of the unit polysigned vectors. Then I replaced
each of the unit polysigned vectors with the associated operator in
R3X3. Then added each set of operators together, then multiplied the
two sums. There were a lot of steps involved and I couldn't have done
it without the aid of good software. To build the four operators I
figured out what each of your four unit polysigned vectors would do to
the three basis vectors in R3, because each of the three basis vectors
in R3 were linear combinations of your unit polysigned vectors, and you
had given a very good definition of what they did to each other. So, I
think all I have done is taken every step that you do in your computer
code and done it all at once. It was a pure brute-force approach
disguised as elegance.
> For example
> > Op1=Transpose[{j,(3k+j)/sqrt(8),(2l+k+j)/sqrt(8/3)}]
> > ={{-1/3,-sqrt(2/9),-sqrt(2/3)},{sqrt(8/9),-1/6,-sqrt(1/12)},{0,sqrt(3/4),-1/2}}
> >
> I was seeing your operators as working in the Cartesian domain so that
> the transform step was forgone in exchange for four operators so that a
> product like:
> (C3 0, 1, 0 )(C3 2, 3, 4)
> could be gotten and that you would try a linear superposition of these.
> And now I see you would have had only three operators if it were so.
> You do convert back to the polysigned domain. But I misunderstood.
> In that case the only rule system that is really needed is that
> j j = k
> j k = l
> j l = i
> j i = j
> So now a general polysign product:
> ( ai + bj + ck + dl )( ei + fj + gk + hl)
> = aei + afj + agk + ahl
> + bej + bfk + bgl +bhi
> + cek + cfl + cgi + chj
> + del + dfi + dgj + dhk
ae(ii)+af(ij)+ag(ik)+ah(il)+be(ji)+bf(jk)... okay, I see it.. That is
a much simpler way to get the answer, I guess, and it stays in the
polysigned system the whole time.
> It's just the distribution and sum. This is identical to the mnemonic
> product.
> Ahh. You don't have to covert back to Cartesian because your OPn are
> doing it.
> Could your method be called a mixed mode?
If you like. I will see whether or not I can come to a better
understanding of what exactly it is. All I really know for sure is
that I haven't seen it before.
I like the comparison to the Mandelbrot set. This idea seems to give
me the same sense of a simple construction that leads to an unlimited
level of complexity. Anyway, it seems important to me, for whatever
that is worth.
I was planning to try to give these a little more analysis today, but I
ran out of time. Where you have defined what your polysigned numbers
do to each other, you were able to predict what they would do to a
particular pattern on a spherical shell. But the choice of that
pattern, I'm guessing was kind of arbitrary, whereas you might find
some other patterns or shapes which would be more informative. While
I'm not sure what form these patterns would take, I think you would
find these vectors helpful.
Eigenvectors of the #1 Multiplication operator
{{0, 0, 1}, {0, 1, 0}, {1, 0, 0}},
Eigenvalues 1, 1, 1
Eigenvectors of the -1 Multiplication operator (1 real, two complex)
{{Sqrt[2/3], -(1/Sqrt[3]), 1},
{(-1 + I)*Sqrt[2/3], (1 + 2*I)/Sqrt[3], 1},
{(-1 - I)*Sqrt[2/3], (1 - 2*I)/Sqrt[3], 1}},
Eigenvalues -1, I, -I
Eigenvectors of the +1 Multiplication operator (3 real vectors)
{{-Sqrt[3/2], 0, 1}, {1/Sqrt[2], 1, 0}, {Sqrt[2/3], -(1/Sqrt[3]), 1}},
Eigenvalues: -1, -1, +1
Eigenvectors of the *1 Multiplication operator (1 real, 2 complex)
{{Sqrt[2/3], -(1/Sqrt[3]), 1}, {(-1 - I)*Sqrt[2/3], (1 - 2*I)/Sqrt[3],
1},
{(-1 + I)*Sqrt[2/3], (1 + 2*I)/Sqrt[3], 1}}
Eigenvalues -1, I, -I
So, if you can translate these Eigenvectors from Cartesian to
Polysigned coordinates, you'll find some polysigned numbers whose
direction is preserved when the Eigenvalue is 1 and is reflected when
the Eigenvalue is -1.
I'm thinking it might provide some kind of insight into P4. Then again
maybe it won't.
Spoonfed
Tim wrote:
> Spoonfed wrote:
>
> I see and agree with your analysis. I've dodged the use of trig in my
> transform but the concept helps explain the behavior. It is a
> dimensional reduction.
>
> We know that the system has perfect symmetry with n rays in n-1
> dimensions.
> The sum of the unit vectors is zero so each contributes 1/(n-1) in any
> other unit vector direction. We can extract out a dimension lying along
> one of these unit vectors. What is left is n-1 rays is n-2 dimensions
> scaled down by a factor. I'm being lazy and not getting the factor here
> but it's on my website as u and v factors, and you are exposing it as a
> trig relation. Nice work.
>
> -Tim
The biggest thing that told me I was on the right track was matching
your numbers. I can envision 1, 2, and 3 dimensions, and that's it. 4
dimensions with time, but not anything that you could ascribe angles
and structure to. When you said the angle is Pi-ArcCos(1/(n-1)), that
was, to me, one of the most mind-blowing concepts I ever heard. So of
course I had to use it as much as I possibly could.
Tim
| A B | = |A||B|
is broken. Yet that breakage is necessary for the commutative product.
P4 and above are warped. That is why the unit sphere study is
informative. The warping is strictly a function of orientation. So for
two P4 vectors z1 and z2 we can make unit vectors out of them and
multiply those by their proper scales so we get a1s1 and a2s2, where
a's are magnitudes and s's are unit magnitude P4 vectors. the product
result
z1 z2 = a1 s1 a2 s2 = a1 a2 s1 s2.
and
z1 z2 = a3 s3 .
So
a1 a2 = a3
and
s1 s2 = s3 .
I've never bothered to write this out. It is basic to the polysigned
math.
But what I'm trying to say is that the product warp is a matter of
orientation, not of the magnitude of the sources. So the unit shell
becomes the defining object to study the product over.
Well it's a good exercise. I think it will be worth your while to get
comfortable with the polysign domain on its own terms since the
prospect of defining them in the Cartesian domain does not look
promising. Just picture what a mess you'd have doing the test product
for (a,b,c)(d,e,f). Maybe you're up for it though. Anyhow the result
will be sensetive to the choice of coordinate transfrom and in that way
the result will always have the polysigned system embedded in it. In
other words the ususal tensor principles are broken above P3.
The shape is a puzzle and that annoying planar cut that cuts across
every other image.
Yet the final shape of the image approaches the cut. It also is
approaching a square angle. That's not surprising since that is the
limiting angle of high sign systems. But to see all of these things
wrapped into one is neat. There is a parity behavior that is yet to be
understood. It may reach all the way down to P1.
I've been operating under the assumption that the only products in P4
to preserve magnitude would be -1, +1, *1, and #1 and my next step will
be to verify that. If that is not the case then the shape of the result
could be instructive and thinking of these values as unique would have
to be reconsidered.
-Tim
Spoonfed
uncovered one.
You might be familiar with these
First, take your P4 polysigned number {P4 a,b,c,d} then rewrite just
the first three terms with the last component zeroed as {a-d,b-d,c-d}
in R3. Now, notice that multiplying any of the following 3X3 matrices
gives a familiar output.
i= {{1, 0, 0},
{0, 1, 0},
{0, 0, 1}}
i.{a-d,b-d,c-d}={a-d,b-d,c-d}-->{P4 a,b,c,d}
j= {{0, 0, -1},
{1, 0, -1},
{0, 1, -1}}
j.{a-d,b-d,c-d}={d-c,a-c,b-c}-->{P4 d,a,b,c}
k= {{0, -1, 1},
{0, -1, 0},
{1, -1, 0}}
k.{a-d,b-d,c-d}={c-b,d-b,a-b}-->{P4 c,d,a,b}
l ={{-1, 1, 0},
{-1, 0, 1},
{-1, 0, 0}}
l.{a-d,b-d,c-d}={b-a,c-a,d-a}-->{P4 b,c,d,a}
This just shows how to multiply by the unit vectors, but if we had two
vectors in P4, say {P4 d,e,f,g}*{P4 a,b,c,d} we can multiply as
follows:
[(d-g)i+(e-g)j+(f-g)k]*{a-d,b-d,c-d}
Where (d-g), (e-g), (f-g) (a-d), (b-d), and (c-d)are all scalars, and
i, j, and k are all 3X3 matrices.
This could also be written [{d-g,e-g,f-g}.{i,j,k}].{a-d,b-d,c-d}
where {i,j,k} is a 3X3X3 matrix. I remembered that you expected a
3X3X3 to show up somewhere in the solution.
The treatment so far has been three dimensional, but not Cartesian. I
also remember you mentioning a triangular matrix in translating between
the two coordinate systems.
How about this:
Let th_n=Pi-ArcCos(1/(n-1))
T={{1, Csc[th3]/3, (Csc[th2]*Csc[th3])/2},
{0, Csc[th3], (Csc[th2]*Csc[th3])/2},
{0, 0, Csc[th2]*Csc[th3]}}
This looks pretty scary, but it is actually the same thing I was doing
before.
For instance T.{1.1,2.2,3.3}=
1.1(i) +2.2(i+3j/sqrt(8))+3.3(i+j+2k)/sqrt(8/3)
So T converts from R3 to {P4 a-d,b-d,c-d,0}
and Inverse[T] converts the other direction.
===================
I also went through and found the Eigenvectors for P5. You were right,
I was a little disappointed. There wasn't a single real-valued
Eigenvector for any of the operators except identity. Do you expect
later dimensionalities to lead to more conceptualizable geometries? I
don't know much about string theory, but IIRC there is something about
11-dimensional or 21 dimensional space.
Let me know if you want anymore details. I'm buried in the
nitty-gritty of it and now I'm not sure now where I was headed.
Tim
> I remember you mentioning a 3X3X3 matrix, and I believe I may have
> uncovered one.
>
> You might be familiar with these
>
> First, take your P4 polysigned number {P4 a,b,c,d} then rewrite just
> the first three terms with the last component zeroed as {a-d,b-d,c-d}
> in R3. Now, notice that multiplying any of the following 3X3 matrices
> gives a familiar output.
This is a really good approach. You've slid into a nonorthogonal
Cartesian domain with hardly any math. I've not looked at the system
this way. It is suggestive. The product will still generate values in
the last slot but again they can be eliminated. So rather than
concerning oneself with reduction leaving a zero value in one of four
places you have forced the zero value to one specific place and allowed
the others to become real valued.
Don't forget that these angles are existent in the other dimensions
also. It may make some sense to apply this thinking in P3 (2D) and see
what that generates and extend upward.
I do not see that maintaining the difference will be helpful. So in
effect you are saying:
(SP4 a, b, c, 0 )( SP4 d, e, f, 0 )
where SP means special polysigned since the components are now real
valued.
The standard product solution can be reused. We should admit that there
is a slight ambiguity in this reuse with real valued components but as
usual that ambiguity should not harm the system. This may need proof.
It is alternatively:
( SC3 a, b, c )( SC3 d, e, f )
Where SC means special Cartesian and are nonorthogonal with the
specific angle.
The product in SP4 will be the usual result:
(SP4 ad + cf, ae + bd, af + be + cd, bf + ce)
= (SP4 ad + cf - bf - ce, ae + bd - bf - ce, af + be + cd - bf - ce, 0
)
= ( SC3 ad + cf - bf - ce, ae + bd - bf - ce, af + be + cd - bf - ce )
So there is a three-form product in 3D!
It's just on a slant.
Wow.
What if reality is a slanted place? Would that put us in a Lorentz
environment initially at zero velocity? Perhaps the road to c as in
E=mcc is here. You can't straighten out beyond orthogonal. It would be
sort of flipped around from the usual. You do not rely on the product
for this transformation so the deformation is not inherently part of
it. Yet now after the product is taken you can still express it in the
three-form.
I think it is a really nice move that you have made.
I'll try to dig into your math below a little more but I tend to see
what you are doing as what I wrote above.
Well not in the transform. That's more the spacetime analogy of the
family
P1, P2, P3, ...
being a triangular format. I see your triangle below. In the sense of
dimensional reduction it would be clean if when you go up or down a
dimension your T (below) maintained itself. You know about the scaling
factor to do with dimenisonal reduction so it makes sense that it
would.
>
> How about this:
>
> Let th_n=Pi-ArcCos(1/(n-1))
>
> T={{1, Csc[th3]/3, (Csc[th2]*Csc[th3])/2},
> {0, Csc[th3], (Csc[th2]*Csc[th3])/2},
> {0, 0, Csc[th2]*Csc[th3]}}
>
> This looks pretty scary, but it is actually the same thing I was doing
> before.
>
> For instance T.{1.1,2.2,3.3}=
>
> 1.1(i) +2.2(i+3j/sqrt(8))+3.3(i+j+2k)/sqrt(8/3)
>
> So T converts from R3 to {P4 a-d,b-d,c-d,0}
> and Inverse[T] converts the other direction.
>
> ===================
>
> I also went through and found the Eigenvectors for P5. You were right,
> I was a little disappointed. There wasn't a single real-valued
> Eigenvector for any of the operators except identity. Do you expect
> later dimensionalities to lead to more conceptualizable geometries? I
> don't know much about string theory, but IIRC there is something about
> 11-dimensional or 21 dimensional space.
Certainly. Look at P6. It has symmetrical opposites like P3 but also
has the handedness flip of P4. Sign 3 will invert. Sign two takes three
flips to go around and sign 4 three flips the other way. Signs 1 and 5
take single flips around (six times to go all the way). Depending on
how you want to break out the symmetry you can wind up with several
classes of behavior. Taking the signs as differential is probably most
meaningful. This is really the same as treating them as operators as
you have done. So rather than worrying about the actual position we
just study how the signs move things differently. In P5 they all look
the same except for the neutral sign 0. But rather than go to the
trouble of vectorizing these things it is much simpler to just treat
the signs as numbers and modulo math. You can build up tables that
expose all of this stuff. By virtue of the modulo math we're changing
radix as we raise the dimension. We're sort of doing radix geometry.
>
> Let me know if you want anymore details. I'm buried in the
> nitty-gritty of it and now I'm not sure now where I was headed.
The productiveness of your new context may not be so much in the vector
math as it is a way of rethinking Cartesian space. If one were to exist
in such a space you'd have to accept a natural angle as inherent yet
have no ability to see that angle. Because the Cartesian system has
another inherent angle which we call the right angle there could be
some consequences. The directness of your transform is great. You have
not relied on the product to get it so it does not suffer the magnitude
problem inherently. Only when the product is taken will the strangeness
show. I think that the true spacetime correspondence is below P4 so the
application of this thinking in P3 is more relevant in my opinion. But
the form is general so it will work for them all.
In the limit of large n the specific angle approaches the right angle
so the difference between the two systems naturally falls off. This may
be an important aspect.
I see a localizing effect but cannot formalize it yet. This is what we
would like for particle physics but it should arise on its own.
-Tim
Tim
Tim wrote:
> > I'm glad you got the same answer. I was checking back all day
> > yesterday. Is there any particular reason our number should have
> > grown? If you multiply the magnitudes of your original numbers, you
> > get a smaller number than the magnitude of the product. This is
> > probably an error due to rounding, but I'll check into it.
> We are in P4. The law
> | A B | = |A||B|
> is broken. Yet that breakage is necessary for the commutative product.
> P4 and above are warped. That is why the unit sphere study is
> informative. The warping is strictly a function of orientation. So for
> two P4 vectors z1 and z2 we can make unit vectors out of them and
> multiply those by their proper scales so we get a1s1 and a2s2, where
> a's are magnitudes and s's are unit magnitude P4 vectors. the product
> result
> z1 z2 = a1 s1 a2 s2 = a1 a2 s1 s2.
> and
> z1 z2 = a3 s3 .
> So
> a1 a2 = a3
> and
> s1 s2 = s3 .
What was written here is an invariant solution.
If we force s3 to be a unit vector then a3 is not necessarily equal to
a1 a2 :
a3 <=> a1a2 .
meaning that a3 could be less, equal, or greater.
The usual analysis uses unit vectors for the sources so that the
magnitude of the results are the scaling factor in that direction. Ah,
that is why we wanted those a's. What I should have said is that given
that the product of two unit vectors equal to a third non-unit vector
that scaling the sources will scale the results.
So that
s1 s2 = a3 s3 = z3
where s's are unit vectors and z are not necessarily unit vectors
then any scalar multiples (a's) of those unit vectors will scale the
result.
z1 = a1 s1
z2 = a2 s2
z1 z2 = a1 a2 z3
or
z1 z2 = a1 a2 a3 s3 .
I hope that makes more sense.
Surveying this behavior is best done on the unit shell.
-Tim