? understanding the world by math
Cheng Cosine
Math represents a set of powerful tools to help us approach
the true nature of this world. Though linear system theory provides
many very power tools for us to approach the nature, the real world
in many cases is not linear. Then, except conducting linearization
to under a small part of the nature within a small range, whatelse
can we use to understand this world?
galathaea
the finite combinatorics of discrete abstractions
is a much more foundational use of mathematics
which underlies most logical thinking about the world
-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-
galathaea: prankster, fablist, magician, liar
Mariano Suárez-Alvarez
If you realize how much we understand this world (as
evidence by the way we are able to manipulate it
and make it do what we want, from sending satellites
to outside of the solar system to developing materials
from which we build 45nm memory cells, to myriad other
rather remarkable feats), you'll see that you are
painting a much more bleak picture than what reality is.
-- m
zzbu...@netscape.net
Mostly thinking. Since the assumption that the world is mostly
linear is also based on the assumption that the world is mostly
electrons. Which is obviously what most science cranks don't
understand about RISC, Lasers, Cell Phones, Holograms, Fiber
Optics,
Parallel Processing, USB, XML, GPS, E-Publisding, On-Line
Publishing,
CD+rw, DVD+rw, HDTV, Mini Hard Disks, PV Cells,
Post GM Robotics, Autonomous Vehicles, and Drones.
Tim BandTech.com
The polysign numbers inherently contain spacetime correspondence due
to the behavior of the math beyond sign three. These higher sign
systems are somewhat nonlinear, though I hesitate to use that word
strictly since the higher sign systems do still obey the associative,
commutative, and distributive laws.
The most general answer to your question is to construct freely. Cover
new ground whatever way it can be found. Who'd have thunk that a
graphic so simple as
http://bandtechnology.com/PolySigned/MagnitudeSweep/P3.png
could be undiscovered? Here is another:
http://bandtechnology.com/PolySigned/MagnitudeSweep/P6HighDensity.png
Even once a grand new theory is found that covers your desires fully
don't you think that science will still remain open? Otherwise it will
become like religion. The book has been written... One must preserve
the book...
Try breaking some rules and see what happens.
- Tim
Don Stockbauer
Tim BandTech.com
> On Jan 30, 4:09 pm, Cheng Cosine <asec...@gmail.com> wrote:
>
> > Hi:
>
> > Math represents a set of powerful tools to help us approach
>
> > the true nature of this world. Though linear system theory provides
>
> > many very power tools for us to approach the nature, the real world
>
> > in many cases is not linear. Then, except conducting linearization
>
> > to under a small part of the nature within a small range, whatelse
>
> > can we use to understand this world?
>
> The polysign numbers inherently contain spacetime correspondence due
> to the behavior of the math beyond sign three. These higher sign
> systems are somewhat nonlinear, though I hesitate to use that word
> strictly since the higher sign systems do still obey the associative,
> commutative, and distributive laws.
>
> The most general answer to your question is to construct freely. Cover
> new ground whatever way it can be found. Who'd have thunk that a
> graphic so simple as
> http://bandtechnology.com/PolySigned/MagnitudeSweep/P3.png
> could be undiscovered? Here is another:
> http://bandtechnology.com/PolySigned/MagnitudeSweep/P6HighDensity.png
These graphics are an iterated function:
z = - z z @ m ;
or if you prefer the more formal
z[n] = - z[n-1] * z[n-1] @ m
where z[0] = 0 and m is a swept magnitude 0->inf and '@' means
superposition.
I point to your strong awareness of linearity issues. These are
behaviors of the space themselves, but yes, through a function. Surely
this must be so since I've just created them and this is the context
in which I was thinking when I constructed them. So I ask that you
grant me a new language and read my writing in my context, which
requires a fair amount of flexibility from a reader. This is because
the new language does carry slight inconsistencies with the old
language.
For instance unidirectional time is natural as P1 (the one-signed
numbers) while being zero dimensional and so inherently carrying the
meaning of 'now' within their very geometry.
I could just ramble on and on about this stuff. I need a reader who
talks back. The math can be taught to a capable high school student I
believe, though I suppose I should put this to the test somehow. But
Cheng, look at the structures in these graphics. There are no
conditionals within their construction. They've got cores, they've got
strings; nuclear structure. Structures rich with life from next to
nothing. I plead with you to consider seriously the polysign numbers.
The P6 graphic above is like an explosion, but it is stable. It is
five dimensional with so much character that thinning it can expose
more structure beneath the fireball surface:
http://bandtechnology.com/PolySigned/MagnitudeSweep
- Tim
Cheng Cosine
Well, I cannot agree with this interpretation. Think this example,
at this very moment there are many many people manipulating a TV
to fit their desires. However, among them, how many know what is
really
going on inside that little box?
But went overy my original text, I felt that I did not empasize
clear enough that I am more interested in "know how" direction.
Or simply put in this way: how to understand the nature.
To manipulate or not to manipulate, that is not the problem. :p
Mariano Suárez-Alvarez
That most people who use TVs do not understand how it works
is irrelevant. My point is, in order for a TV to work and for
other things to be done, *someone* has had to be able to analyze
nature to a degree of precision which is simple amazing.
Your observation that our methods (linear or not---the obsession
with non-linearity is so 90s!) only allow us to understand a
tiny part of nature are clearly underestimating what we are able to
do with those methods.
-- m
Mariano Suárez-Alvarez
> On Jan 30, 4:09 pm, Cheng Cosine <asec...@gmail.com> wrote:
>
> > Hi:
>
> > Math represents a set of powerful tools to help us approach
>
> > the true nature of this world. Though linear system theory provides
>
> > many very power tools for us to approach the nature, the real world
>
> > in many cases is not linear. Then, except conducting linearization
>
> > to under a small part of the nature within a small range, whatelse
>
> > can we use to understand this world?
>
> The polysign numbers inherently contain spacetime correspondence due
> to the behavior of the math beyond sign three. These higher sign
> systems are somewhat nonlinear, though I hesitate to use that word
> strictly since the higher sign systems do still obey the associative,
> commutative, and distributive laws.
What does this paragraph possibly mean?!
What does it mean that a sign system is nonlinear? What
difference is there from that to its being only "somewhat"
nonlinear? What on earth is a "spacetime correspondence"?
In what way can an algebraic system "contain" a "spacetime
correspondence"?
The difference (or one of the differences) between math and,
say, poetry is that unless you make explicit the meaning of
the terms you use, you are plainly and simply not saying
anything. Deep sounding mumbo-jumbo only impresses... hmmm...
the editors of Social Text.
-- m
Cheng Cosine
<mariano.suarezalva...@gmail.com> wrote:
> On Jan 31, 12:39 am, Cheng Cosine <asec...@gmail.com> wrote:
>
>
>
> - Show quoted text -
Someone? Okay, following the way you thought, then you can simply
narrow down my question as how does one think or understand the
nature like your "someone". In this case, writing that my thinking
is so 90's or even 1800's is irrelevent.
Tim BandTech.com
Mariano I am so sorry for you that you cannot understand such a simple
thing.
Well I suppose all math is simple isn't it? That's why it's math. If
it can't be nailed down firmly then it can't be simple enough. Yet who
claims to understand all of math? It it were so simple why then can't
we construct it from naught? Clearly humans are limited in their
abilities. I am and you are and so was Hamilton, Einstein, Dirac,
Feinman, etc.
Especially I am sorry that our communications have been so poor. For
someone who claims to understand the polysign system to refuse their
fundamental behaviors is sign of refusal to accept the polysign
numbers as valid.
I'll go over the spacetime correspondence here again for you or some
other reader who thinks I am lame. The family of polysign numbers is
large. The family is
P1, P2, P3, P4, P5, ...
Yet of all these systems only three preserve the following behavior:
| z1 z2 | = | z1 || z2 |
This is the usual familiar conservation of magnitude of the reals and
the complex numbers. While the higher sign systems are well behaved
arithmetically they break this rule. Distances are no longer conserved
in P4+.
The well behaved members of the family are
P1 P2 P3
which form a sufficient representation of spacetime including
unidirectional time.
The suggestive nature of this structured spacetime has consequences
which carry far and wide. To open to this possibility is beyond most.
Thanks Mario for trying to corrupt my in with Cheng. What a cheap
shot. You are a loser Mario.
- Tim
Tim BandTech.com
Sorry but atomic stability is not a linear process. Since we build
these devices out of atoms then they are nonlinear systems. As
galathaea put it, discrete processes are of interest. The behaviors
are repeatable and so the mastery of the material world can and does
go forth accidentally as much as it does at the hand of the linear
engineer and his paper. Persistence is a huge factor in the human
accomplishments of technology. Better just stay in your little math
world where reality doesn't matter Mario.
- Tim
Mariano Suárez-Alvarez
By now you could have at least picked up my correct name! ;)
-- m
Mariano Suárez-Alvarez
I honestly cannot tell if I understand or not: as I have stated,
you have, as far as I know, never explained what meaning
the terms I mentioned (and quite a few others I have seen you
use elsewhere) have, and I can confidently say that there is no
standard, well-known meaning attached to a claim such as "The
polysign
numbers inherently contain spacetime correspondence due to the
behavior of the math beyond sign three".
> [paragraph on how all humans are limited, and on my refusal
> to accept the polysign numbers snipped]
> I'll go over the spacetime correspondence here again for you or some
> other reader who thinks I am lame. The family of polysign numbers is
> large. The family is
> P1, P2, P3, P4, P5, ...
> Yet of all these systems only three preserve the following behavior:
> | z1 z2 | = | z1 || z2 |
I do not recall your listing *what* properties you expect
an absolute value function to have. If the only one you are
interested in is the multiplicative property
(*) | z1 z2 | = | z1 | | z2 |
then you should know that all the Pn *do* have an appropriate
'absolute value function' which is multiplicative.
This follows trivially from the fact that the Pn, when
m >= 2, are isomorphic as rings to direct products of
copies of R and C. The actual formulas, though, are rather
messy (but they can be obtained in principle with simple
linear algebra)
For example, if x = a0 e0 + a1 e1 + a2 e2 + a3 e3 is an
element of P4 (with the e0, ..., e3 the 'signs' and
the a0, ..., a3 the coefficients, which as usual are real
numbers), you can define
|x| = sqrt( ((a0 - a2)^2 + (a1 - a3)^2) (a0 - a1 + a2 - a3)^2 )
Then if y is another element of P4 a computation will show that
|x y| = |x| |y|
This can be done by hand.
Things get scarier as n grows. For example, when n = 5 and
x = a0 e0 + a1 e1 + ... + a4 e4, you obtain a multiplicative
function putting
|x| = Sqrt[
(((-4*a[0] + a[1] - Sqrt[5]*a[1] + a[2] + Sqrt[5]*a[2] + a
[3] +
Sqrt[5]*a[3] + a[4] - Sqrt[5]*a[4])^2 +
(10*(-((1 + Sqrt[5])*a[1]) - 2*a[2] + 2*a[3] + a[4] +
Sqrt[5]*a[4])^2)/(5 + Sqrt[5]))*
((Sqrt[5 + Sqrt[5]]*(-a[2] + a[3]) + Sqrt[5 - Sqrt[5]]*
(a[1] - a[4]))^2/8 +
(a[0] - ((3 + Sqrt[5])*a[1] - 2*(a[2] + a[3]) +
(3 + Sqrt[5])*a[4])/(2*(1 + Sqrt[5])))^2))/16
]
Cute, isn't it? The ugliness comes from the fact that
pentagons are complicated beasts! Mathematica did not manage
to check multiplicativity in the 10 minutes I gave it, but the
formula does work. If you are so inclined, you can check (*)
on randomly generated elements.
When n = 6, the corresponding formula is much nicer:
if x = a0 e0 + ... + a5 e5, setting
|x| = Sqrt[
((a[0] - a[1] + a[2] - a[3] + a[4] - a[5])^2*
(3*(a[1] + a[2] - a[4] - a[5])^2 +
(2*a[0] + a[1] - a[2] - 2*a[3] - a[4] + a[5])^2)*
(3*(a[1] - a[2] + a[4] - a[5])^2 +
(-2*a[0] + a[1] + a[2] - 2*a[3] + a[4] + a[5])^2))/16
]
does the trick---and Mathematica can check the multiplicative
property (*) in a minute or so.
&c.
> This is the usual familiar conservation of magnitude of the reals and
> the complex numbers. While the higher sign systems are well behaved
> arithmetically they break this rule. Distances are no longer conserved
> in P4+.
>
> The well behaved members of the family are
> P1 P2 P3
> which form a sufficient representation of spacetime including
> unidirectional time.
See, you are here trying to explain what you meant by "The polysign
numbers inherently contain spacetime correspondence due to the
behavior of the math beyond sign three", yet you have not given
any explanation whatsoever of what a "representation of spacetime"
is, what the difference between such a representation that includes
unidirectional time (whatever that may be) and one which does not
include it is, and what it means for something (the polysign numbers,
in this case) to be a "sufficient" representation.
A great professor I had used to refer to the Principle of
Preservation of the Difficulty: your "explanation" is a great
example of that principle in action.
> [Paragraph concluding that someone named Mario is a loser snipped]
-- m
Tim BandTech.com
I've got to check in right here because you seem to think that I am
defining a new absolute value function. I am not. This is the usual
absolute value. It is the same in P2 as it is on the reals. It is the
same in P3 as it is on the complex numbers. It is the same in P4 as it
is on P3; simply generalize in sign. It is simply the distance
function. The nuances of difference for me come at a different stage
of awareness. That some believe this absolute value to be a higher
form than the types it applies to is a deep mistake. Instead the
polysign construction exposes magnitude as fundamental, sign being a
discrete type whose marriage to this continuous unsigned magnitude
yields the real numbers, the complex numbers, and a myriad of higher
forms... also let's not forget that little rascal P1.
You Mariano insist on applying the old language to a new language and
will claim that any inconsistencies are a failing of the new language.
You see though the consequences of this new language range widely.
There are unmistakeable gains to be had. Upon turning the new language
onto the old one then the inconsistencies become shuffled another way
such that the old language becomes suspect. Because the new language
is a compact form of the old language with additional consequences
that were not present in the old language the mismatch can be
validated from the progressive side looking askance at what has been.
It is a function of human judgement as to which side one will take.
This is no different than the tear between say a string theory and a
classical particle theory. There is room for the pragmatist in the
middle to arbitrate but for me my side is clearly chosen. Your attack
on my statement of the distance function is illogical, for the
geometry is well exposed and the ordinary distance function suffices
directly on the polysign math as is stated at my website. You may
treat this refutation as an answer as well so we can drop the topic
and move on with your other attacks if you see and accept my argument
here. For now I will remain merely at this one point until we fully
address it into congruity. If you insist that I redefine the ordinary
math then so be it. I simply reuse the ordinary system of absolute
value as distance as in Euclidean geometry of the superpositional
space.
We'll eventually come to a dispute over just what is meant by
P1 P2 P3
as a symbolic construction. There is a healthy discussion, but one
that few will undertake because to forsake isotropic space for a
structured spacetime seems beyond hope to that human judgement system,
though Einstein did come part way in his convincing usage of the
Minkowski metric. Here I have answers but first you would properly
have to ask the questions since if I try to preanswer them it's as if
I'm shoving a bunch of information down your throat and it would then
simply become regurgitant. So it goes for the human race. Most simply
gag on my attempts here and so my stream of information simply flows
into databanks. They don't seem to mind holding onto it since it is
just trivia to them.
> If the only one you are
> interested in is the multiplicative property
>
> (*) | z1 z2 | = | z1 | | z2 |
>
> then you should know that all the Pn *do* have an appropriate
> 'absolute value function' which is multiplicative.
>
> This follows trivially from the fact that the Pn, when
> m >= 2, are isomorphic as rings to direct products of
> copies of R and C. The actual formulas, though, are rather
> messy (but they can be obtained in principle with simple
> linear algebra)
I am suspect that these can be clearly instantiated in your system. My
own analysis of P4 as an RxC space exposes that the error in the
product is symmetrical to the resultant, suggesting that an infinite
series will be necessary to cleanly compute the correct resultant.
While the analysis on this webpage is partly graphical the linear
compensations that might make your claim clean in two iterations have
been tried:
http://bandtechnology.com/PolySigned/Deformation/P4T3Comparison.html
So perhaps you'll be able to state the problem to some precision
level. Until you've done this I think maybe you better keep this
argument in check a bit. You say it is possible, but who has actually
done it? The polysign space is new. Could it be that it can challenge
the ways that you preach? I do think it is possible, especially given
the reaching attempts of Grassman and the bunk that has become
acceptable in the name of progress. People have been reaching for this
new ground for some time. I should be more respectful, but when the
reachers have become accepted and the reached unaccepted then the
shaky footing of the reaching tower leaves one wondering what they are
doing hanging around in its midst.
- Tim
Mariano Suárez-Alvarez
If by that you mean that you are defining, say for P3,
(*) | a0 e0 + a1 e1 + a2 e2 | = sqrt(a0^2 + a1^2 + a2^2)
then you should observe that this does not make sense,
because it is not well defined: for example, it is part
of your definition of P3 that
e0 + e1 + e2 = 2 e0 + 2 e1 + 2 e2
but using formula (*) you get
| e0 + e1 + e2 + e3 | = sqrt(3)
and
| 2 e0 + 2 e1 + 2 e2 + 2 e3 | = 2 sqrt(3).
> It is the same in P2 as it is on the reals. It is the
> same in P3 as it is on the complex numbers. It is the same in P4 as it
> is on P3; simply generalize in sign. It is simply the distance
> function. The nuances of difference for me come at a different stage
> of awareness. That some believe this absolute value to be a higher
> form than the types it applies to is a deep mistake. Instead the
> polysign construction exposes magnitude as fundamental, sign being a
> discrete type whose marriage to this continuous unsigned magnitude
> yields the real numbers, the complex numbers, and a myriad of higher
> forms... also let's not forget that little rascal P1.
> You Mariano insist on applying the old language to a new language and
> will claim that any inconsistencies are a failing of the new language.
I have never mentioned any kind of inconsistencies in the "new
language":
I've just repeatedly tried to make the point that it does not add
absolutely anything to what the "old language" provides. In fact, I
have
shown you how your Pn is simply an instance of a general construction,
which has been studied and *understood* for a century by now, and
how what you term "the old language" is extraordinarily useful in
obtaining information.
> You see though the consequences of this new language range widely.
Quite the contrary: I have yet to see anything useful coming out
of it.
> There are unmistakeable gains to be had.
While that may indeed be the case, I have not seen any gain yet,
only claims that there are.
> Upon turning the new language
> onto the old one then the inconsistencies become shuffled another way
> such that the old language becomes suspect.
What inconsistencies are you talking about?
< [long paragraph going from the "tear between say a string theory and
a
> classical particle theory" to "superpositional space", elided]
> We'll eventually come to a dispute over just what is meant by
> P1 P2 P3
> as a symbolic construction. There is a healthy discussion, but one
> that few will undertake because to forsake isotropic space for a
> structured spacetime seems beyond hope to that human judgement system,
> though Einstein did come part way in his convincing usage of the
> Minkowski metric. Here I have answers but first you would properly
> have to ask the questions since if I try to preanswer them it's as if
> I'm shoving a bunch of information down your throat and it would then
> simply become regurgitant. So it goes for the human race. Most simply
> gag on my attempts here and so my stream of information simply flows
> into databanks. They don't seem to mind holding onto it since it is
> just trivia to them.
I simply love the almost endearing way in which you put yourself
in a position pretty much beyond humanity, although of course
you do acknowledge that a few select, among whom Einstein, may have
partaken in the higher insights you are so generously sharing
with us---sharing hopelessly, in view of the limitations of
our poor human judgement system which so hinders our understanding,
but sharing nonetheless.
I'm torn between thankfulness and laughter ;-)
> > If the only one you are
> > interested in is the multiplicative property
>
> > (*) | z1 z2 | = | z1 | | z2 |
>
> > then you should know that all the Pn *do* have an appropriate
> > 'absolute value function' which is multiplicative.
>
> > This follows trivially from the fact that the Pn, when
> > m >= 2, are isomorphic as rings to direct products of
> > copies of R and C. The actual formulas, though, are rather
> > messy (but they can be obtained in principle with simple
> > linear algebra)
>
> I am suspect that these can be clearly instantiated in your system. My
> own analysis of P4 as an RxC space exposes that the error in the
> product is symmetrical to the resultant, suggesting that an infinite
> series will be necessary to cleanly compute the correct resultant.
You are suspect that *what* exactly can be instantiated in my system?
What error in what product? In what way is that error symmetrical?
What "resultant"? What are you possibly talking about in "an infinite
series will be necessary to cleanly compute the correct resultant"?
> While the analysis on this webpage is partly graphical the linear
> compensations that might make your claim clean in two iterations have
> been tried:
> http://bandtechnology.com/PolySigned/Deformation/P4T3Comparison.html
> So perhaps you'll be able to state the problem to some precision
> level. Until you've done this I think maybe you better keep this
> argument in check a bit. You say it is possible, but who has actually
> done it? The polysign space is new. Could it be that it can challenge
> the ways that you preach? I do think it is possible, especially given
> the reaching attempts of Grassman and the bunk that has become
> acceptable in the name of progress. People have been reaching for this
> new ground for some time. I should be more respectful, but when the
> reachers have become accepted and the reached unaccepted then the
> shaky footing of the reaching tower leaves one wondering what they are
> doing hanging around in its midst.
I honestly do not know what this paragraph means.
-- m
Tim BandTech.com
No I do not mean this. This is cartesian thinking on your part. This
is not the proper solution for P3, whose sign poles emanate from the
center of a 3-simplex to its verices. This geometry is already layed
out so I see that you are on a hoaxing mission here; that or you are a
blundering idiot, which is alot different from being labelled a loser.
I know that you are not a blundering idiot. So I must ask what
happened to my statement on distance above here? The miscommunication
is too simple to overlook. You delay the inevitable. I predict that
you will cease to produce reasonable content on this thread shortly.
Already this is some.
>
> then you should observe that this does not make sense,
> because it is not well defined: for example, it is part
> of your definition of P3 that
>
> e0 + e1 + e2 = 2 e0 + 2 e1 + 2 e2
Right, I've already said no to the first part so none of this follows.
>
> but using formula (*) you get
>
> | e0 + e1 + e2 + e3 | = sqrt(3)
which we don't get because this thread of reason is wrong.
>
> and
>
> | 2 e0 + 2 e1 + 2 e2 + 2 e3 | = 2 sqrt(3).
>
> > It is the same in P2 as it is on the reals. It is the
> > same in P3 as it is on the complex numbers. It is the same in P4 as it
> > is on P3; simply generalize in sign. It is simply the distance
> > function. The nuances of difference for me come at a different stage
> > of awareness. That some believe this absolute value to be a higher
> > form than the types it applies to is a deep mistake. Instead the
> > polysign construction exposes magnitude as fundamental, sign being a
> > discrete type whose marriage to this continuous unsigned magnitude
> > yields the real numbers, the complex numbers, and a myriad of higher
> > forms... also let's not forget that little rascal P1.
> > You Mariano insist on applying the old language to a new language and
> > will claim that any inconsistencies are a failing of the new language.
>
> I have never mentioned any kind of inconsistencies in the "new
> language":
> I've just repeatedly tried to make the point that it does not add
> absolutely anything to what the "old language" provides. In fact, I
> have
> shown you how your Pn is simply an instance of a general construction,
> which has been studied and *understood* for a century by now, and
> how what you term "the old language" is extraordinarily useful in
> obtaining information.
No, you have not instantiated the product.
Well, to extend even further than you have gone I do include myself in
the category of human, including its flaws. I also include you there
Mariano. You are in denial of what is right under your nose. Most
amazingly is that you care to deny the simple distance function. These
are vector spaces we are in, including the polysign. Yes, it is
slightly different, yet still a vector space. Your act of intentional
misunderstanding is a sorry thing to witness. Denial runs strong in
the human race. This is how such bizarre belief systems as religions
have formed. I am sorry that you can be so careless. Now, shall we
carry on with the proper distance function or are you still in denial
of something? You maybe should actually use my website though I
thought you were already up to speed on polysign:
http://bandtechnology.com/PolySigned/DistanceFunction.html
http://bandtechnology.com/PolySigned/CartesianTransform.html
- Tim
That's OK Mariano, it'll come up again sometime. Let's just focus on
the first paragraph since as I've stated we're far off from each other
on the simplest of things back there.
>
> -- m
Do you refuse to use my website? Are you able to read it?
There is a cartesian transform that has been there for some time and
thanks to the work of Jonathan Doolin a more native distance function
is presented on polysign without the need to transform to cartesian.
Mariano Suárez-Alvarez
You surely use the work 'usual' in a funny way, then...
> This
> is not the proper solution for P3, whose sign poles emanate from the
> center of a 3-simplex to its verices. This geometry is already layed
> out so I see that you are on a hoaxing mission here; that or you are a
> blundering idiot, which is alot different from being labelled a loser.
> I know that you are not a blundering idiot. So I must ask what
> happened to my statement on distance above here? The miscommunication
> is too simple to overlook. You delay the inevitable. I predict that
> you will cease to produce reasonable content on this thread shortly.
I know myself that I will eventually get bored, too.
> Already this is some.
The formulas for P2 and P3 are the ones you'd get from the
isomorphisms
P2 ~ R and P3 ~ C, of course.
What is the formula for P4? What's given in <http://bandtechnology.com/
PolySigned/DistanceFunction.html> is
| - a + b * c # d |
= sqrt( aa + bb + cc + dd - (2/3)(ab + bc + cd + de + ac + bd)),
but this does not make sense for there is an extraneous 'e'...
The same error appears in <http://groups.google.com/group/sci.math/msg/
5855b7e5ce2cf074>.
> > > It is the same in P2 as it is on the reals. It is the
> > > same in P3 as it is on the complex numbers. It is the same in P4 as it
> > > is on P3; simply generalize in sign. It is simply the distance
> > > function. The nuances of difference for me come at a different stage
> > > of awareness. That some believe this absolute value to be a higher
> > > form than the types it applies to is a deep mistake. Instead the
> > > polysign construction exposes magnitude as fundamental, sign being a
> > > discrete type whose marriage to this continuous unsigned magnitude
> > > yields the real numbers, the complex numbers, and a myriad of higher
> > > forms... also let's not forget that little rascal P1.
> > > You Mariano insist on applying the old language to a new language and
> > > will claim that any inconsistencies are a failing of the new language.
>
> > I have never mentioned any kind of inconsistencies in the "new
> > language":
> > I've just repeatedly tried to make the point that it does not add
> > absolutely anything to what the "old language" provides. In fact, I
> > have
> > shown you how your Pn is simply an instance of a general construction,
> > which has been studied and *understood* for a century by now, and
> > how what you term "the old language" is extraordinarily useful in
> > obtaining information.
>
> No, you have not instantiated the product.
What on earth does that mean? Since I have no idea
what "instantiating the product" means, I really do
not know if I have or have not done it...
> [psychological analyses elided]
You have already managed to bore me with your analyses of my
motivations, fears and what not. I will simply not
read them anymore: it takes way too much effort to go through
them a couple of times in search of meaning and mathematical
content.
-- m
Tim BandTech.com
Thank you for pointing out a typo. I've edited my next website edition
though it will be awhile before I load it up. Certainly just remove
this e term:
| - a + b * c # d |
= sqrt( aa + bb + cc + dd - (2/3)(ab + bc + cd + ac + bd + ad)),
Ooh, that 'de' term is supposed to be an 'ad'.
You see the symmetry right? This is simply the simplex geometry.
Anyway your choice to preferentially sort the information in such a
way that is defeatist is clear. My website is a full presentation and
lays out the geometry carefully. You call yourself a mathematician but
it should be painfully clear that we are caught as humans doing math.
So the ideal of the mathematician is somewhat farcical. Your own human
nature leaks out of your posts here as does mine. We are free to build
this into something here and I do attempt to make use of rhetoric to
help rope the likes of you into the fray. Most of all it is clear that
the human mind is challenged by the simplest of things. While we grant
ourselves intelligence in its practice we observe mimicry at its
core.
When you are staring progress in the face and swearing it is wrong
then what? You see Mariano we've only just almost gotten through the
distance function, but you are not all there are you? We have not even
made it through the distance function have we. The impedance is all
yours I'm certain.
As I recall back a ways you were challenging emergent spacetime, but
now here we are in a scuttle over the distance function. A true
mathematician would take the challenges a different way, and since you
do represent the human race fairly well I think then we can make such
a statement as:
Humans are poor mathematicians.
Hence as we all struggle to read and understand what has been written
in the books. We should keep this in mind, and keep it in mind of the
one who wrote the book as well, who was just another human. The mix of
digested material with raw source material is a curious thing. There
are some outstanding works. Kant's Critique of Pure Reason I do put on
that shelf. And there we see statements of a nonmathmatical composure;
simply using grammar strictly and unmistakably. I attempt to do the
same and do hope that an onlooker can detect the difference in the
struggling interaction that you and I have undergone.
It sounds as if you'd like to stop, but if you stop at this point you
will have completely wasted your and my time. Having gotten through
the distance function we should then return to the study of
| z1 z2 | = | z1 || z2 |
in order to consider that emergent spacetime is near at hand. You
would like to refute this gain, yet your means of doing so is weak.
That is because the argument which I make is well supported. So you
are off here trying to blunder your way out on a backlog of distance
which was never very controversial in the first place. I suppose I
could consider that you are cooking something up, but my guess is that
your stew is not tasting so good. You see these chides here in words
are made to get you going. I can make them as offensive as I wish to,
or as gently as they need to be to get your reactionary energy up. A
true mathematician would see their way beyond all of that and simply
filter it out rather than complain about it. This side energy is
forever going on in the human domain. I don't really hate you Mariano,
but I'm not going to get anywhere if I don't rub a taste of the truth
into you. If the truth is like salt to you then so be it. What does
that say of you?
What does that say of the ground you inhabit?
- Tim
Tim BandTech.com
A brief for the sake of concise progress:
You and I are agreed on the native polysign distance function.
Here I would anticipate a clear yes/no response from you.
Following your dismissal here I would then step back to the cartesian
transform
and claim a coherent and identical distance function there.
Then following that we take you as a grade school child back to the
spaces of 1D, 2D, and 3D and hopefully domonstrate to you that these
distance functions are the very same of the space you actually
inhabit. These are identical to the complex distance function and the
real distance function. Essentially distance is fundamental. Distance
does not inherently carry a sign. This is ve5ry primitive logic but so
to stick with this primitive thinking we then must introduce sign in
order to generate the real numbers, though this has not been taken up
in the past.
I can go on and on here since the flow takes me very far.
- Tim
Mariano Suárez-Alvarez
The formula is indeed symmetric. I do not know what the "simplex
geometry" is, though; by now I do not expect an explanation:
you have not bothered to explain any of the other terms I ask about.
In any case, you may be interested in knowing that the non-vanishing
of the formulas I gave earlier is precisely the condition on an
element
of Pn to be invertible.
-- m
> [long text I did not read, elided]
-- m
victor_me...@yahoo.co.uk
Surely it's time to put all this "polysign" nonsense to rest.
> I'll go over the spacetime correspondence here again for you or some
> other reader who thinks I am lame. The family of polysign numbers is
> large. The family is
> P1, P2, P3, P4, P5, ...
Not that large: the range of possible structures in these
systems is quite limited. For n >= 2, P_n is just R Z_n/(R u)
where R is the field of reals, Z_n is the cyclic group of
n elements, R Z_n is the group ring, u is the sum of all
elements of Z_n (considered as a group ring element) and
R u is the set of real multiples of u (an ideal in R Z_n)
and R Z_n/(R u) is the quotient ring.
The group ring R Z_n is a commutative finite-dimensional
semi-simple algebra over R, and so is isomorphic to
a product of copies of R and C. For n even, there will be two
copies of R and (n-2)/2 of C, while for n odd there will be
one copy of R and (n-1)/2 of C. Forming the quotient
R Z_n/(R u) eliminates one of the copies of R. We conclude
that for n even, P_n is the product of R and (n-2)/2 copies
of C, and for n odd, P_n is the product of (n-1)/2 copies of C.
To be more precise let z = exp(2 pi i/n) be the standard
primitive n-th root of unity. For 0 < r <= n/2 there is
homomorphism from P_n to C given by taking the j-th "polysign"
to z^{rj}. This is surjective save when r = n/2 (in the case
where n is even) in which case the image is R. These
homomorphisms fit together to form the stated isomorphism from
P_n to R x C^{(n-2)/2} or C^{(n-1)/2}.
> Yet of all these systems only three preserve the following behavior:
> | z1 z2 | = | z1 || z2 |
That is because when n <= 3, P_n embeds in C where this property
holds, and for n>= 4 it doesn't.
Mr Golden, you would be well-advised to find a textbook on algebra,
look up "group rings" and study the rudiments of group representation
theory.
Tim BandTech.com
> On 31 Jan, 18:57, "Tim BandTech.com" <tttppp...@yahoo.com> wrote:
>
> Surely it's time to put all this "polysign" nonsense to rest.
>
> > I'll go over the spacetime correspondence here again for you or some
> > other reader who thinks I am lame. The family of polysign numbers is
> > large. The family is
> > P1, P2, P3, P4, P5, ...
>
> Not that large: the range of possible structures in these
> systems is quite limited. For n >= 2, P_n is just R Z_n/(R u)
I'm sorry Victor but I've got to stop here and feel certain that you
are not understanding the generalization of sign, for each of the
members of the list is a vector space in and of itself, their
dimensions being
0D, 1D, 2D, 3D, 4D, ...
For instance in the 1D case we have two signs: -, and +. These two
signs go by the rule
- 1 + 1 = 0
as a sort of balancing act. By the time we move to three signs we
witness the dimension of the system has grown by one, and so it goes
on up the spectrum such that
D( Pn ) = n - 1 .
where D returns the dimension of the space Pn. For instance in the
three-signed numbers P3 we see that
- 1 + 1 * 1 = 0
and that in P3 the old
- 1 + 1 = 0
of P2 does not apply any longer. Hence the form directly above is not
reducible. This then is the most simplistic informational way of
getting at the 2D nature of P3. The same argument goes on for each
higher sign system into higher dimensions. Please note that there is
no usage of the cartesian product in these dimensional descriptions
and this is a clean break with tradition as far as I can tell.
Staying at P3 let's go quickly over another instance to demonstrate
the 2D nature. We pick out of a hat:
- 1.1 + 2.3 * 4.5
and we see that we can reduce this value by
- 1.1 + 1.1 * 1.1
thusly eliminating the smallest component and essentially zeroing out
to a reduced value:
+ 1.2 * 3.4
which has no further reduction. Taking the full freedom of the
composition of such numbers we witness a 2D space consistent with a 3-
simplex (equilateral triangle); unit vectors emanating from the center
of this shape to its vertices, these unit vectors being the signs
themselves, their units superposing back to the origin.
- Tim
I recommend that you review my website carefully
http://BandTechnology.com/PolySigned
but I am willing to carry on here ad nauseum if you wish
- Tim
Tim BandTech.com
So is this the fizzle point? No big bang to end with... what a shame.
Well I did predicted it would end this way, as do most.
You are par for the course M. Well done. A bit salty though.
May your wounds heal and grow stronger for the next time.
- Tim
victor_me...@yahoo.co.uk
> On Feb 2, 11:58 am, victor_meldrew_...@yahoo.co.uk wrote:
>
> > On 31 Jan, 18:57, "Tim BandTech.com" <tttppp...@yahoo.com> wrote:
>
> > Surely it's time to put all this "polysign" nonsense to rest.
>
> > > I'll go over the spacetime correspondence here again for you or some
> > > other reader who thinks I am lame. The family of polysign numbers is
> > > large. The family is
> > > P1, P2, P3, P4, P5, ...
>
> > Not that large: the range of possible structures in these
> > systems is quite limited. For n >= 2, P_n is just R Z_n/(R u)
>
> I'm sorry Victor but I've got to stop here and feel
I prefer to discuss mathematics, not your feelings.
> certain that you
> are not understanding the generalization of sign,
I am afraid that the rest of your message indicates
that you do not understand your own system as well as
I do.
> for each of the
> members of the list is a vector space in and of itself, their
> dimensions being
> 0D, 1D, 2D, 3D, 4D, ...
Well, P_1 isn't a vector space, but the rest are. Recall
that in my previous message I pointed out that for n >= 2,
P_n is an (n-1)-dimensional algebra over R.
> Please note that there is
> no usage of the cartesian product
You have essentially an n-tuple of nonnegative reals:
that's an element of a Cartesian product. You then
impose an equivalence relation on them: essentially
(a_1, ... , a_n) ~ (a_1 + b, ..., a_n + b).
So your "polysign numbers" are essentialy the quotient set
(R+ x ... x R+)/~ where R+ is the set of nonnegative reals.
Because of this, one gets an isomorphic structure by taking
(R x ... x R)/~ . Now the set R x ... x R with your
multplication of "signs" is the group ring R Z_n, and factoring
by the ~ equivalence relation is the same as factoring
out the ideal generated by the element I called u.
Mr Golden, this is straightforward algebra. I did advise
you to study the rudiments of algebra.
> Staying at P3 let's go quickly over another instance to demonstrate
> the 2D nature. We pick out of a hat:
> - 1.1 + 2.3 * 4.5
> and we see that we can reduce this value by
> - 1.1 + 1.1 * 1.1
> thusly eliminating the smallest component and essentially zeroing out
> to a reduced value:
> + 1.2 * 3.4
So (1.1, 2.3, 4.5) ~ (0, 1.2, 3.4) in the above notation
(And so what?) As I pointed out one gets an isomorphism of
P_2 to C by mapping (a, b, c) to a w + b w^2 + c where
w = exp(2 pi i/3).
> I recommend that you review my website carefully
> http://BandTechnology.com/PolySigned
I did. Did you?
> but I am willing to carry on here ad nauseum if you wish
I don't wish, but you have certainly demonstrated your
willingness to carry on ad nauseam already. :-(
Tim BandTech.com
<victor_meldrew_...@yahoo.co.uk> wrote:
> On 2 Feb, 19:03, "Tim BandTech.com" <tttppp...@yahoo.com> wrote:
>
> > On Feb 2, 11:58 am, victor_meldrew_...@yahoo.co.uk wrote:
>
> > > On 31 Jan, 18:57, "Tim BandTech.com" <tttppp...@yahoo.com> wrote:
>
> > > Surely it's time to put all this "polysign" nonsense to rest.
>
> > > > I'll go over the spacetime correspondence here again for you or some
> > > > other reader who thinks I am lame. The family of polysign numbers is
> > > > large. The family is
> > > > P1, P2, P3, P4, P5, ...
>
> > > Not that large: the range of possible structures in these
> > > systems is quite limited. For n >= 2, P_n is just R Z_n/(R u)
>
> > I'm sorry Victor but I've got to stop here and feel
>
> I prefer to discuss mathematics, not your feelings.
>
> > certain that you
> > are not understanding the generalization of sign,
>
> I am afraid that the rest of your message indicates
> that you do not understand your own system as well as
> I do.
We'll see. Thanks for taking up where Mario could not go.
Shall we discuss emergent spacetime? Obviously you've made no claim on
it and I have so I welcome your disproof. Since consequences of the
mathematical construction should weigh along with the simplicity of
construction then I have two tools which are superior to your own. We
may consider the two together as forming a chisel and mallet whereas
you are pulling out a swiss army knife...
Do you see that your usage of the cartesian product is not necessary
in the development of polysign? Do you see that P1 can still do a bit
of algebra?
Can you please state clearly the product which is so obvious to you
and your cyclic ring construction? Could you please do so in terms of
its components, so for instance the 3D (four component) product of P4
is:
( - a + b * c # d )( - e + f * g # h )
= + ae * af # ag - ah
* be # bf - bg + bh
# ce - cf + cg * ch
- de + df * dg # dh
Thus far it has not been stated clearly in R x C, for the discrepancy
is self-similar to the resultant. Your algebra on your product will
not be equivalent to the polysign, unless you've broken through this
hump. As lwal is fond of pointing out this is an isometric match atop
the isomorphic one. Given that the product exists this additional step
will be important to understand the full behavior of the polysign
system, not just half of it. As I understand it then you are proposing
that you have an equivalent space out of an awful lot of
constructional steps. Since the polysign construction is slim and
tight doesn't it become the superior construction? Look at the backlog
of definitions that you've created for yourself. This is the lens
through which you are viewing a very simple critter. After all you are
even relying upon R and C as two unique sets. Polysign has already
swiped them into one thing. Who is ridiculous here Meldrew? You would
like to uphold the old system. I would like to push for some progress
in a new direction which supports emergent spacetime from seemingly
the same old algebra as it ever was... But you see it is not quite the
same old algebra.
- Tim
Tim BandTech.com
So now we get to some good stuff... But then again I have to wonder if
since the product you never bothered to define in a simple form then
is your invertible Pn applicable on the product operator? This is
certainly of interest to me and it might be to you as well. You stay
there and I'll stay here, but the lame communication in between is
your bad, not mine.
The superposition inversion is terribly simple so maybe it signals the
native form of what you are speaking of. For instance in P4 a value ~z
does exist
- z + z * z
such that
@ z - z + z * z = 0 .
Thusly the first expression is the inverse of z, generalizable to the
following general invert
~ z = sum from s = 1 to n - 1 ( s z )
Is this invert anything like your inversion? This is merely the
superpositional form so I wait to understand whether you are proposing
that your own inversion exists on the product operator since this
superpositional form is pretty trivial.
- Tim
Mariano, there is this other tension of two parallel systems existing
together side by side. For every step that you make I may make one as
well. This is how I know that your product is likely expressed as a
long series in the native P4.
victor_me...@yahoo.co.uk
> construction then I have two tools which are superior to your own.
Your obsession with "superiority" is quite infantile.
> Do you see that your usage of the cartesian product is not necessary
> in the development of polysign?
My usage? It is you who started working with Cartesian products,
defining operations on n-tuples of reals. Alas you lack the honesty
to admit this.
> Can you please state clearly the product which is so obvious to you
> and your cyclic ring construction? Could you please do so in terms of
> its components, so for instance the 3D (four component) product of P4
> is:
>
> ( - a + b * c # d )( - e + f * g # h )
The "signs" correspond to elements in a multiplicative
cyclic group of order n. In this example
- a + b * c # d
corresponds to a g + b g^2 + cg^3 + d
where g satisfies g^4 = 1. You obtain P_4 by imposing the
relation 1 + g + g^2 + g^3 = 0.
> Your algebra on your product will
> not be equivalent to the polysign,
No, As I pointed out, P_4 is ismomorphic as a ring
to R x C. Indeed I outlined an isomorphism in
my original posting (for general n). Unfortunately
this statement of yours prove that you are either
(i) a liar, or
(ii) algebraically incompetent.
Of course, this "or" may not be exclusive. I did suggest
you apprise yourself of the topic of group rings. Have you?
> Since the polysign construction is slim and tight
weaselling adjectives
> doesn't it become the superior construction?
Again the infantile obsession with superiority.
> But you see it is not quite the same old algebra.
I see P_n as isomorphic to a product of copies of R and C,
but constructed in an awkward manner. Mr Golden, you really
should study some grou representation theory. You will see
some beautiful constructions way beyound your hamfisted
attempts. For instance every group algebra RG for a finite
group G is isomorphic to a product of matrix algebras over R,
C and H. Doesn't this put your polysigns in the shade? Learn
some humility and mathematics, Mr Golden, you'll find
it more rewarding than posting your bumptious meanderings.
Tim BandTech.com
<victor_meldrew_...@yahoo.co.uk> wrote:
> On 3 Feb, 13:09, "Tim BandTech.com" <tttppp...@yahoo.com> wrote:
>
> > construction then I have two tools which are superior to your own.
>
> Your obsession with "superiority" is quite infantile.
>
> > Do you see that your usage of the cartesian product is not necessary
> > in the development of polysign?
>
> My usage? It is you who started working with Cartesian products,
> defining operations on n-tuples of reals. Alas you lack the honesty
> to admit this.
Hah ! ! !
What a lie.
The reals are the two-signed numbers Meldrew.
As such when we progress to sign three we leave the real numbers
back at sign two and regenerate the system at stage three. Likewise in
the one form they were just one-signed numbers, all of these
constructed by a discrete marriage of sign with magnitude. The
fundamental form
s x
has been denied by Mariano and now by you too. Your very refusal to
address this form directly is evidence that you have not studied my
website. Your talkback here then I can only take to mean one thing.
You wish to stifle me.
>
> > Can you please state clearly the product which is so obvious to you
> > and your cyclic ring construction? Could you please do so in terms of
> > its components, so for instance the 3D (four component) product of P4
> > is:
>
> > ( - a + b * c # d )( - e + f * g # h )
>
> The "signs" correspond to elements in a multiplicative
> cyclic group of order n. In this example
> - a + b * c # d
> corresponds to a g + b g^2 + cg^3 + d
> where g satisfies g^4 = 1. You obtain P_4 by imposing the
> relation 1 + g + g^2 + g^3 = 0.
Now you are getting there Victor!
Next you need to define a product of two values g1 and g2, or however
you are going to notate. Whatever you choose these will be
instantiable in coordinated form:
(a,b,c,d),(d,e,f,g)
Since I have yet to see this form from you I await your clean notation
of this product which will be performable in simple arithmetic.
>
> > Your algebra on your product will
> > not be equivalent to the polysign,
>
> No, As I pointed out, P_4 is ismomorphic as a ring
> to R x C. Indeed I outlined an isomorphism in
> my original posting (for general n). Unfortunately
> this statement of yours prove that you are either
> (i) a liar, or
> (ii) algebraically incompetent.
> Of course, this "or" may not be exclusive. I did suggest
> you apprise yourself of the topic of group rings. Have you?
Hah !!!
He still hasn't instantiated his product and he is already defending
it!!!
Look at the diversion on group rings. Ridiculous.
>
> > Since the polysign construction is slim and tight
>
> weaselling adjectives
>
> > doesn't it become the superior construction?
>
> Again the infantile obsession with superiority.
Yes, let's discuss this a bit more.
As parallel construction which has greater merit?
Does one simplify the other?
Does the other form an overly complicated system?
Which of these systems overlooks emergent spacetime?
>
> > But you see it is not quite the same old algebra.
>
> I see P_n as isomorphic to a product of copies of R and C,
> but constructed in an awkward manner. Mr Golden, you really
> should study some grou representation theory. You will see
> some beautiful constructions way beyound your hamfisted
> attempts. For instance every group algebra RG for a finite
> group G is isomorphic to a product of matrix algebras over R,
> C and H. Doesn't this put your polysigns in the shade? Learn
> some humility and mathematics, Mr Golden, you'll find
> it more rewarding than posting your bumptious meanderings.
You still have not instantiated a product.
I like your term 'hamfisted'. It is so nice to see just a little bit
of color from you. Please, bring it on. Your group theory stinks.
Simple things are constructed simply. The overhead of the space that
you inhabit hampers your own abilities. Come down to the
s x
form and you'll see that much work has yet to be done. You have thus
far avoided my argument on emergent spacetime. I can see that you are
struggling to recover this simple form along with others that are
reading on here and not corresponding with me. I would warn you that
as they read on the lack of content on your part will become more and
more evident. I will win over perhaps just one or two. But they will
be the right one and two. They will see that the human race has been
off by one for a very long time.
- Tim
- Tim
victor_me...@yahoo.co.uk
> You wish to stifle me.
This is paranoid. I wish you would study a little mathematics,
so that you can put your work, and its significance, in context,
just as I have.
> > The "signs" correspond to elements in a multiplicative
> > cyclic group of order n. In this example
> > - a + b * c # d
> > corresponds to a g + b g^2 + cg^3 + d
> > where g satisfies g^4 = 1. You obtain P_4 by imposing the
> > relation 1 + g + g^2 + g^3 = 0.
>
> Now you are getting there Victor!
I got there before you.
> Next you need to define a product of two values g1 and g2, or however
> you are going to notate. Whatever you choose these will be
> instantiable in coordinated form:
> (a,b,c,d),(d,e,f,g)
> Since I have yet to see this form from you I await your clean notation
> of this product which will be performable in simple arithmetic.
Are you really not capable of multiplying two expressions
like ag + bg^2 + cg^3 + d together on your own, remembering
to simplify using the rule g^4 = 1. You must be less
intelligent than I thought :-( Now for some handholding:
your first exercise: expand out the product
(ag + bg^2 + cg^3 + d)(sg + tg^2 + ug^3 + v)
using the distributive law, and the commutative law
for multiplication.
> Look at the diversion on group rings. Ridiculous.
I still advise you to study group rings, Mr Golden,
you will learn some maths, but you may find that your
ideas are less original than you supposed.
> > Again the infantile obsession with superiority.
>
> Yes, let's discuss this a bit more.
> As parallel construction which has greater merit?
The simpler (and more general) construction is
to take Cartesian products of copies of R and C.
Does "merit" equate to complication and obfuscation?
> You still have not instantiated a product.
"instantiate"? Multiplication in general group rings
is easy to describe. When the group is cyclic, as
here, it is even easier, as the group ring is quotient
of a polynomial ring in one variable.
> I like your term 'hamfisted'. It is so nice to see just a little bit
> of color from you. Please, bring it on. Your group theory stinks.
Now we see your true colours, Mr Golden. A little robust critique
of your meagre pretensions, and you descend to crude abuse.
> I will win over perhaps just one or two.
I fear you may be over-optimistic here. :-(
> But they will
> be the right one and two. They will see that the human race has been
> off by one for a very long time.
Blimey, a saviour of the human race too :-(
Tim BandTech.com
No it does not. Since the real numbers have traditionally been used to
define the complex numbers then your statement is no better than
studying RxRxR.
Yet what I have domenstrated is that these construction R and C are
within Pn already. Furthermore there is the underling P1 which does
not appear in the usual construction since it was born of R. By
generalizing sign and taking that generalization literally we land in
a simplex geometry of symmetry. You still have not defined the product
in the terms of this (R,C) claim have you? You've merely mimiced the
polysign numbers in R rather that attempt a statement of product in
the space that you highlight as the essential space. So you are
working in the space RxRxRxR which is not the space that I work in.
You are using the D+1 simplex construction whose origin lays off of
the solution space. You are nearby to the barycentric system which is
dangerously close to the polysign system, but polysign does not rely
upon the real number. It constructs the real number from a more
fundamental form. This form you have not touched and refuse to make
contact with.
What say you of the symbolic instaniation
- 3.456
and its symbolic meaning? What is that strange symbol on the front?
why is it a discrete symbol of choice [-,+]?
What if there were a third choice? Does your math answer to this
question? Then we would admit a direct translation, but upon your
denial of the generalizability of the discrete sign then the topic
dead ends. You are already at a dead end. This is beyond my ability to
resolve, for the burden lays in your half of the court. Here we have a
fair playing field but one character essentially hoarding the ball;
refusing to serve it up. I am free to construct and you are free to
construct. Your avoidance of emergent spacetime is the weakest portion
of you lacking address. You admit that the polysign construction is
frustrating to work with. What else will you admit? Nothing right? If
you admit its integrity then what? All your crap of R and C has not
generated a shred of evidence within your product because you've
merely instantiated the polysign product. You are stuck in orthogonal
thoughts, attempting to preserve an orthogonal space that does not
deserve preservation. Perhaps I could call you Vector Mildew or some
such silly name. There, now I am an ass too. I hope you can at least
get a chuckle out of this low blow. Where is the consequence of your
claim of RXC and a product in that space? Does it exist? Is it
instantiable?
>
> > You still have not instantiated a product.
>
> "instantiate"? Multiplication in general group rings
> is easy to describe. When the group is cyclic, as
> here, it is even easier, as the group ring is quotient
> of a polynomial ring in one variable.
>
> > I like your term 'hamfisted'. It is so nice to see just a little bit
> > of color from you. Please, bring it on. Your group theory stinks.
>
> Now we see your true colours, Mr Golden. A little robust critique
> of your meagre pretensions, and you descend to crude abuse.
Yes, nice. Please note my low blow above too.
>
> > I will win over perhaps just one or two.
>
> I fear you may be over-optimistic here. :-(
>
> > But they will
> > be the right one and two. They will see that the human race has been
> > off by one for a very long time.
>
> Blimey, a saviour of the human race too :-(
Well, I thought you were taking that job... saving them from the
misrepresentation of true sign generalization and emergent spacetime.
What a horror story...
Where is your stake of difference on my own claims?
As far as I can tell now you've mimiced the polysign product and
completely ignored the nonorthogonal nature of the construction. This
prevents you from breaching the hurdle of the two-signed real number.
You are insulating yourself unnecessarily and especially by insulating
yourself there at that spot you've prevented any meaningful
communication. So you are left mimicing the polysign construction in
all of your double talk and endless terminology, while you will
happily teach a ten year old that sign comes in only two forms. If the
ten year old attempts a new form here what will you say back? No, that
is wrong... Sign comes in only two forms...
Or rather will you school them that it can be done, but all forms must
inherently contain the binary sign within their continuous basis?
Magnitude is fundamental Mildrew. Some day you'll see it this way.
The
s x
representation does generalize mathematics nicely. This form is called
polysign.
It's laws are compactly
Sum over s ( s x ) = 0 ,
s1 x1 @ s1 x2 = s1 ( x1 @ x2 ) ,
( s1 x1 )( s2 x2 ) = ( s1 @ s2 ) x1 x2 .
Upon eliminating the redundancy between these laws and the real
numbers we see that what is left upon dissecting sign from the real
construction is magnitude x.
This is the context of polysign math. It forms geometry in every
dimension without the need of any cartesian product.
Listening to the signals in this path of thought, and especially in
the realm of emergent spacetime is a very leading path. It goes far
and widely too.
- Tim
Mariano Suárez-Alvarez
I have absolutely no idea what your post means.
-- m
Tim BandTech.com
>
> > construction then I have two tools which are superior to your own.
>
> Your obsession with "superiority" is quite infantile.
And again your lack of address of emergent spacetime is a sad lacking
of your anticoncessionary movement.
Again also now from the philosophical stage could address what you
would tell a ten year old who asks of a third sign?
As I see it there may be many options and I leave you free to code
your own.
I merely present these as reasonable predictions of your own response:
No, Johnny, there are only two signs.
Well, Johnny, there can be three but first you'll have to get
through group theory.
Why, yes, Johnny.
So glad you brought it up.
Yes, you can have three signs and it is what advanced
mathematicians of old called the complex numbers.
These days we just call them polysign and teach you the two-signed
version until grade five, of old called the 'real' numbers.
They did not discover spacetime as emergent until someone like you
Johnny asked that very question.
If you'd like to skip to grade five then you may leave this math
class on occassion and listen in there.
These I see as valid options which you might consider. Certainly we
could call this an open book quiz, so if you want to check your notes
that is fine by me.
Next though from the philisophical point of view we may even
instantiate quite a number of these children in the past and into the
present who have asked this very question. So you see what may start
out as a prank can become quite a serious issue can't it? Have these
children been lied to by mathematicians of your ilk?
- Tim
victor_me...@yahoo.co.uk
> Have these
> children been lied to by mathematicians of your ilk?
Would you rather them being lied to by yourself?
victor_me...@yahoo.co.uk
> No it does not. Since the real numbers have traditionally been used to
> define the complex numbers then your statement is no better than
> studying RxRxR.
I don't recall mentioning R x R x R.
> You still have not defined the product
> in the terms of this (R,C) claim have you?
I'm not sure what you mean here. The Cartesian product
of rings is a product in the categorical sense, and
arithmetic operations are defined pointwise. This
is the case for any category of algebras
(construed in the sense of universal algebras). I hadn't
realized that I has to spell out quite so many minutiae for you.
> You've merely mimiced the
> polysign numbers in R rather that attempt a statement of product in
> the space that you highlight as the essential space.
I made no reference to an "essential space"
> So you are
> working in the space RxRxRxR which is not the space that I work in.
Really?
> You are using the D+1 simplex construction whose origin lays off of
> the solution space. You are nearby to the barycentric system which is
> dangerously close to the polysign system, but polysign does not rely
> upon the real number. It constructs the real number from a more
> fundamental form.
From a more fundamental form? Your system relies on the positive
reals anyway. One is virtually at the real number system there
already:
all one needs do is adjoin additive inverses.
>
> What say you of the symbolic instaniation
"instaniation" isn't in my dictionary
> half of the court.
> fair playing field
> hoarding the ball;
> refusing to serve it up.
Are we playing tennis or football here?
> Perhaps I could call you Vector Mildew or some
> such silly name. There, now I am an ass too.
You should lay off the insults; you aren't very good at them.
> Where is the consequence of your
> claim of RXC and a product in that space?
The product is defined pointwise.
> What a horror story...
Your horror stories won't compete with Stephen King's.
> As far as I can tell now you've mimiced the polysign product and
> completely ignored the nonorthogonal nature of the construction.
What I did was explicate your construction, and explain the
isomorphism
of P_n with a direct product of copies of R and C. Your response
is not fruitful: instead of complaining about my pointing it
out why don't you *use* this isomorphism to prove things about your
P_n?
That would be a constructive response.
> So you are left mimicing the polysign construction in
> all of your double talk and endless terminology, while you will
> happily teach a ten year old that sign comes in only two forms.
Who is this ten-year old whom you are polysigning at?
Do the authorities know?
> representation does generalize mathematics nicely. This form is called
> polysign.
As I keep telling you, you should study group rings. They can do cool
thing that you've never dreamt of.
> The
> s x
> representation does generalize mathematics nicely.
Blimey, not just *doing* mathematics, but *generalizing* it!
> It forms geometry in every
> dimension without the need of any cartesian product.
But you are using Cartesian products, as I have pointed out.
You are manipulating n-tuples of numbers -- elements of a Cartesian
product.
victor_me...@yahoo.co.uk
I think he is saying that his system has additive inverses
even though all the coefficients he uses are non-negative.
Of course he may just as well have started with all
reals, rather than just the non-negatives, since one gets
the same system of "polysign numbers" either way.
Mind you, I can't tell what he's on about most of the time
amy666
well , i do agree with mariano actually.
you are defining a NEW absolute value , in the sense that your absolute value is - i assume* - just the distance to the origin.
( * correct me if im wrong about that assumption )
its kind of weird that you consider an algebra where
a * b = b * a = ab but abs(a)*abs(b) =/= abs(ab).
your ' distance observation ' is correct , although i doubt that only occurs for polysigned as you claim ...
> This is the usual
> absolute value.
no.
its not even the usual numbers that you consider :p
and picking distance from origin as absolute value seems arbitrary , even unlogical as explained above.
even your zero-divisors do not neccessarily have absolute value of 0 ??
thats far from usual.
dont get me wrong , i have nothing against your ' distance ' or even ' magnitude ' but i wouldnt call it absolute values ( abs ).
this does not mean that i am not open to several distinct interpretations of absolute value though.
in fact , personally , for P4 i use 2 absolute values.
and , in fact , one of those matches mariano's given below.
> It is the same in P2 as it is on the
> reals. It is the
> same in P3 as it is on the complex numbers. It is the
> same in P4 as it
> is on P3; simply generalize in sign. It is simply the
> distance
> function. The nuances of difference for me come at a
> different stage
> of awareness. That some believe this absolute value
> to be a higher
> form than the types it applies to is a deep mistake.
> Instead the
> polysign construction exposes magnitude as
> fundamental, sign being a
> discrete type whose marriage to this continuous
> unsigned magnitude
> yields the real numbers, the complex numbers, and a
> myriad of higher
> forms... also let's not forget that little rascal P1.
>
> You Mariano insist on applying the old language to a
> new language and
> will claim that any inconsistencies are a failing of
> the new language.
in a way it is neccesary to at least try to apply the old language to a 'new language' , for e.g. explaining the concepts.
chill dude :)
>
> We'll eventually come to a dispute over just what is
> meant by
> P1 P2 P3
> as a symbolic construction.
thats not fair , in fact there is no such dispute , mariano understands P1 P2 P3 perfectly well.
( i wont mention higher to be neutral )
> There is a healthy
> discussion, but one
> that few will undertake because to forsake isotropic
> space for a
> structured spacetime seems beyond hope to that human
> judgement system,
> though Einstein did come part way in his convincing
> usage of the
> Minkowski metric. Here I have answers but first you
> would properly
> have to ask the questions since if I try to preanswer
> them it's as if
> I'm shoving a bunch of information down your throat
> and it would then
> simply become regurgitant. So it goes for the human
> race. Most simply
> gag on my attempts here and so my stream of
> information simply flows
> into databanks. They don't seem to mind holding onto
> it since it is
> just trivia to them.
pardon my french , but mentioning einstein doesnt make you smarter or prove you are correct.
mariano wrote :
>
> > If the only one you are
> > interested in is the multiplicative property
> >
> > (*) | z1 z2 | = | z1 | | z2 |
> >
> > then you should know that all the Pn *do* have an
> appropriate
> > 'absolute value function' which is multiplicative.
YES !
i had this dispute with timothy golden too.
you are completely correct mariano.
> >
> > This follows trivially from the fact that the Pn,
> when
> > m >= 2, are isomorphic as rings to direct products
> of
> > copies of R and C. The actual formulas, though, are
> rather
> > messy (but they can be obtained in principle with
> simple
> > linear algebra)
there are 2 types of 3D numbers.
do you know that mariano ?
if you consider them to be :
1 ) P4 or R x C
2 ) R x R x R
then 2) is Beresford.
if you consider all 3d numbers isomorphic to R x C
you are wrong.
i want this clarified , because we might or might not agree on this.
and it would be silly to have a dispute based upon bad notation and misunderstandings while actually agreeing.
i asked above about an important opinion of mariano that might resolve this dispute and/or confusion.
>
> >
> > For example, if x = a0 e0 + a1 e1 + a2 e2 + a3 e3
> is an
> > element of P4 (with the e0, ..., e3 the 'signs' and
> > the a0, ..., a3 the coefficients, which as usual
> are real
> > numbers), you can define
> >
> > |x| = sqrt( ((a0 - a2)^2 + (a1 - a3)^2) (a0 - a1
> + a2 - a3)^2 )
> >
> > Then if y is another element of P4 a computation
> will show that
> >
> > |x y| = |x| |y|
> >
> > This can be done by hand.
this is - as mentioned above - one of the absolute values i use myself for P4.
Mariano chooses to have the property
abs ( zero-div ) = 0.
which is reasonable.
and i used myself.
but there are others too that satisfy :
|x y| = |x| |y|
one could even generalize to complex-valued abs and alike.
> >
> > Things get scarier as n grows. For example, when n
> = 5 and
> > x = a0 e0 + a1 e1 + ... + a4 e4, you obtain a
> multiplicative
> > function putting
> >
> > |x| = Sqrt[
> > (((-4*a[0] + a[1] - Sqrt[5]*a[1] + a[2]
> + Sqrt[5]*a[2] + a
> > [3] +
> > Sqrt[5]*a[3] + a[4] -
> Sqrt[5]*a[4])^2 +
> > (10*(-((1 + Sqrt[5])*a[1]) - 2*a[2] +
> 2*a[3] + a[4] +
> > Sqrt[5]*a[4])^2)/(5 + Sqrt[5]))*
> > ((Sqrt[5 + Sqrt[5]]*(-a[2] + a[3]) +
> Sqrt[5 - Sqrt[5]]*
> > (a[1] - a[4]))^2/8 +
> > (a[0] - ((3 + Sqrt[5])*a[1] - 2*(a[2]
> + a[3]) +
> > (3 + Sqrt[5])*a[4])/(2*(1 +
> Sqrt[5])))^2))/16
> > ]
i used that abs too.
so same comment as above.
> >
> > Cute, isn't it? The ugliness comes from the fact
> that
> > pentagons are complicated beasts! Mathematica did
> not manage
> > to check multiplicativity in the 10 minutes I gave
> it, but the
> > formula does work. If you are so inclined, you can
> check (*)
> > on randomly generated elements.
why doesnt mathematica find it in 10 minutes ??
and how did you find your abs for P4 and P5 ??
by computer ?
simple algoritm ?
fast algoritm ?
> >
> > When n = 6, the corresponding formula is much
> nicer:
> > if x = a0 e0 + ... + a5 e5, setting
> >
> > |x| = Sqrt[
> > ((a[0] - a[1] + a[2] - a[3] + a[4] -
> a[5])^2*
> > (3*(a[1] + a[2] - a[4] - a[5])^2 +
> > (2*a[0] + a[1] - a[2] - 2*a[3] - a[4]
> + a[5])^2)*
> > (3*(a[1] - a[2] + a[4] - a[5])^2 +
> > (-2*a[0] + a[1] + a[2] - 2*a[3] + a[4]
> + a[5])^2))/16
> > ]
> >
> > does the trick---and Mathematica can check the
> multiplicative
> > property (*) in a minute or so.
i cant confirm this , i havent computed abs(P6) myself , though it looks good at first sight.
these absolute values are intresting.
there are still open questions not ?
like , whats the pattern for absolute values of P n with n as parameter ?
how about the topology of those hypersurfaces ?
let P(n) denote the abs for dimension n.
how do the hypersurfaces of e.g. P(n) = 1 look like ?
topology ?
super mario is a loser.
super sonic rules :)
> >
> > -- m
>
regards
tommy1729
amy666
( i love to quote his " i dont believe it " )
( snip entire post , im only commenting on one sentense without taking an opinion , plz dont feel insulted whoever you are )
>
> Blimey, a saviour of the human race too :-(
this reminds me of JSH.
sometimes he calls himself the saviour of the human race.
and other times he threatens us saying that he will destroy the world !!
he cant make up his mind , is he going to save us , or is he going to destroy us ?
its so funny :)
regards
tommy1729
" i dont believe it " victor meldrew
amy666
especially mariano.
for the sake of a constructive dialogue.
Tim BandTech.com
I can see that I've put you on the spot here.
Would you please answer Johnny's question.
victor_me...@yahoo.co.uk
> Would you please answer Johnny's question.
I am willing to discuss mathematics, but I am not willing
to join in your fantasies about offering to show a
10-year old boy your "third sign". :-(
Tim BandTech.com
> Hi:
>
> Math represents a set of powerful tools to help us approach
>
> the true nature of this world. Though linear system theory provides
>
> many very power tools for us to approach the nature, the real world
>
> in many cases is not linear. Then, except conducting linearization
>
> to under a small part of the nature within a small range, whatelse
>
> can we use to understand this world?
I apologize Cheng for the abuse of your thread by others here. And I
apologize for myself too. I do want to discuss you linear/nonlinear
awareness. This is an area that I do not feel so strongly about any
more. I understand the tenants of linearity as well behaved systems,
yet in the polysign domain we have results which are somewhat ill-
behaved, yet still quite very well behaved algebraically speaking.
Their geometries come predefined yet their structural qualities have
patterning, exclusion zones, and dimensional collapse within their
being. Above all is their simplest P1 which mimics time in its
unidirectional nature and its behavior as a zero dimensional ray
encapsulating the unescapable NOW.
Is a system which used to obey:
| z1 z2 | = | z1 || z2 |
but no longer does (beyond say a fixed threshold of dicrete value
three) a linear system? It's a strange puzzle that needs new answers.
There will be new language because the old language was built of
precepts such as the law above. I do not wish to coin some term
'semilinear' but would like to see somebody else take a stab at this
way of thinking. These guys that are mathematicians have goggles on
which takes them to their professional format but as I see it this new
form is so far below their radar in its simplicity that they cannot
get it. As much as I preach the freedom to construct I should take the
next stage of critiquing such construction. If it be that an ape can
arrange sticks in order of ascending length then should magnitude be
granted as fundamental? Surely this is a superior choice to granting
sticks of inverse length which the ape has no means to handle.
For the pro of modern day they like to give the ape the invisible
sticks and then take them away from the ape. Specifying all of this to
the ape is very important and has become an old habit, much as a roman
catholic might swing an old pot of incense for no apparent reason, or
feed people teensy little crackers, and just one per person.
To put such construction of uninstatiable inverts at the base of all
that is constructed (of continuum principles) is a farce. The
magnitude which mathemagicians seem so afraid of is a raw and
primitive being. They steer clear of it for its stark cold nature.
There can't be anything there... It's too generic... Gee, so we tack
on some discrete structure in front of these ghostly beings and lo and
behold out comes traditional mathematics with a twist. A discrete
twist at that, but forming pretty continuum twists as well without any
inherent reliance upon the binary twist which is merely the stage two
form. Thusly the preservationists will be pickled in their own jelly,
with a healthy dose of salt cooked into them so they may age the
better. In my dreams... really a truce could be called. The healthy
coexistence can be had though the progression must be of its inherent
chronological nature. These attempts to swoop polysign up into the
fold of group theory are a sad story that can be taken sadly several
ways. I am not the one who is responsible for the behavior of the
other side on such threads as this. They only aid the process of
breaking their own skins down to absorb some salt. Others will follow
with a more sober outlook. Still, to put tradition in its place these
early defeatists serve their purpose well. These are to date the best
arguments against polysign. It is my hope that this list of attempts
will grow longer. Yet here the lack of attention to detail and the
turning of a blind eye to the most simplistic concepts is evident in
Mariano's and Victor's approaches to the topic of polysign. In their
attachment to tradition they take the freddom of an n+1 dimensional
space in the production of an n-1 dimensional space. Simplex geometry
as fundamental answers this deficit cleanly and leaves us with exactly
n components composing a balanced n-1 dimensional space, where
dimension means exactly what it has traditionally meant, that being in
the sense of cartesian products of reals. I do not care to challenge
this definition of dimension though I can submit that signature
carries the same meaning, just off by one e.g. a P3 system is 2D.
victor_me...@yahoo.co.uk
> Is a system which used to obey:
> | z1 z2 | = | z1 || z2 |
> but no longer does
Your P_2 and P_3 satify this, because they are R and C. You P_n
for n >=4 don't, and can't possibly do, since they have zero-divisors.
You have your "magnitude analysis" of values of |z_1 z_2|/|z_1||z_2|
but this is entirely empirical and you don't bother to tell
your readers how you generate the random values of your variables
(i.e., what probability ditsribution you were sampling). I suspect
if you were to essay a theoretical analysis of the probability
distribution of this ratio, the isomorphism of your P_n with
R^a x C^b might be useful, but I suspect this might be too much
like actually doing maths for you :-(
> I do not wish to coin some term 'semilinear'
Already taken.
<rest of Golden's flatulent screed deleted>
Tim BandTech.com
What is new about this Tommy? This is the same old stuff, just on a
simplex representation. Distance, magnitude, absolute value: in terms
of geometry these three take the same meaning. This is the Euclidean
sense I believe, which I so call in that it is more primitive than the
Cartesian sense that we would have to accomodate Newton in somewhat
more, though he is Euclidean in many of his arguments. We must take
out our dividers and ask whether a full coordinate system has been
specified without the usage of perpendiculars. The answer is simple:
yes.
>
> ( * correct me if im wrong about that assumption )
>
> its kind of weird that you consider an algebra where
> a * b = b * a = ab but abs(a)*abs(b) =/= abs(ab).
>
> your ' distance observation ' is correct , although i doubt that only occurs for polysigned as you claim ...
>
> > This is the usual
> > absolute value.
>
> no.
>
> its not even the usual numbers that you consider :p
>
> and picking distance from origin as absolute value seems arbitrary , even unlogical as explained above.
>
> even your zero-divisors do not neccessarily have absolute value of 0 ??
>
> thats far from usual.
>
> dont get me wrong , i have nothing against your ' distance ' or even ' magnitude ' but i wouldnt call it absolute values ( abs ).
Well, the absolute value on the reals and complex numbers must match
and this is the way to do it on the simplex representations. It's
plain old geometry in n-1 dimensions.
>
> this does not mean that i am not open to several distinct interpretations of absolute value though.
>
> in fact , personally , for P4 i use 2 absolute values.
>
> and , in fact , one of those matches mariano's given below.
>
I'm happy to agree to disagree here since on P4 there is a simple
distance function that can be applied from a point position to the
origin, or if you wish from one point position to another as a
generalized vector format. This distance function fits the same
description as it did in P3 and P2, and continues to carry the same
meaning on up into Pn. These spaces are ordinary in terms of vector
superposition. The fascination comes in the study of the product. They
are only out of the ordinary in terms of product. Yes, this behavior
may be mimiced by the R ^ n spaces that Victor likes, but there is a
reason to use polysign: it is more primitive.
Emergent spacetime again is the topic everyone wishes to dodge and
that is fine. Until we get past this reliance upon R there is no point
in discussing P1 since the only result will be utter denial, which
takes the silent form that the discussions with M. and V. exemplify.
>
>
> > It is the same in P2 as it is on the
> > reals. It is the
> > same in P3 as it is on the complex numbers. It is the
> > same in P4 as it
> > is on P3; simply generalize in sign. It is simply the
> > distance
> > function. The nuances of difference for me come at a
> > different stage
> > of awareness. That some believe this absolute value
> > to be a higher
> > form than the types it applies to is a deep mistake.
> > Instead the
> > polysign construction exposes magnitude as
> > fundamental, sign being a
> > discrete type whose marriage to this continuous
> > unsigned magnitude
> > yields the real numbers, the complex numbers, and a
> > myriad of higher
> > forms... also let's not forget that little rascal P1.
>
> > You Mariano insist on applying the old language to a
> > new language and
> > will claim that any inconsistencies are a failing of
> > the new language.
>
> in a way it is neccesary to at least try to apply the old language to a 'new language' , for e.g. explaining the concepts.
Certainly I have already done this. But here Tommy we witness
attachment to the old language impeding the new language. This refusal
to budge is a dead end. I ask that a reader grant a bit of new room in
which to work. This is anyone's requirement to bring to life a new
construction. This freedom must be granted. Instead what we see here
are a couple of stodgy goats being nay-sayers. I am happy to put them
in their place and know that more serious consideration will follow. I
don't know how far into the future this will be. I honestly don't
understand how to communicate well to people of this closed form who
may be the people in power. It is like a communist attempting to
communicate with trust with a capitalist: it just doesn't work. The
capitalist will be the one you can't trust. Will a Catholic treat an
Atheist with respect? Certainly a true Catholic will treat the Atheist
as his lesser. This I offer as one explanation of the behaviors we
witness here. M. ranks very high on the biblical front. I think less
so with Victor... he may be of a more flexible branch. We'll see. So
far he refuses to leave his RxR world while the three-signed numbers
are an easily grasped concept without much overhead at all. These
types of people stifle the world when they are the ones in power.
Witness this effect far and wide and it becomes clear that I do not
wish to be one of these. If chaos ensues then surely there will be a
settling out period. I announce a rennaissance in mathematics just as
we are about to have with global economics. What we have here is a
very open court in which either side may misbehave. I do take this
misbehavior clause and apply it so that things might stay interesting.
Such silly misbehaviors as the apt naming of Vector Mildew I have to
say are quite good. I am fearless and open to making mistakes. Simply
point them out clearly, but do not refuse my freedom to construct sign
as a discrete entity of a more primitive form than the two-signed
number, from which that two-signed number then resolves as one
possibility. Clearly V and M have the cart before the horse, along
with rather a lot of others.
- Tim
Well, since of two of you have some deep agreement on an absolute
value function other than the usual distance function then why not
spec it here for P4? Is it so hard to instantiate it?
>
>
> > > This follows trivially from the fact that the Pn,
> > when
> > > m >= 2, are isomorphic as rings to direct products
> > of
> > > copies of R and C. The actual formulas, though, are
> > rather
> > > messy (but they can be obtained in principle with
> > simple
> > > linear algebra)
>
> there are 2 types of 3D numbers.
>
> do you know that mariano ?
>
> if you consider them to be :
>
> 1 ) P4 or R x C
>
> 2 ) R x R x R
>
> then 2) is Beresford.
>
> if you consider all 3d numbers isomorphic to R x C
>
> you are wrong.
>
> i want this clarified , because we might or might not agree on this.
>
> and it would be silly to have a dispute based upon bad notation and misunderstandings while actually agreeing.
We are in thirds here and on this one I am perking up a bit more than
on the other issues. Victor's recent digression is much as Tommy
points out here and as I have pointed out to Victor. He seems to think
that expressing things in the simplest terms will somehow flaw the
system, whereas to me to deal in instantiable goods is a strength of
any system. The RxC version of the product still has not been spec'd
by either M or V, so I have no idea why they push so hard on that
detail, then come back around to the RxRxR representation.
> ...
>
> read more »
Tim BandTech.com
> interested in is the multiplicative property
>
> (*) | z1 z2 | = | z1 | | z2 |
>
> then you should know that all the Pn *do* have an appropriate
> 'absolute value function' which is multiplicative.
>
> m >= 2, are isomorphic as rings to direct products of
> copies of R and C. The actual formulas, though, are rather
> messy (but they can be obtained in principle with simple
> linear algebra)
>
> element of P4 (with the e0, ..., e3 the 'signs' and
> the a0, ..., a3 the coefficients, which as usual are real
> numbers), you can define
>
> |x| = sqrt( ((a0 - a2)^2 + (a1 - a3)^2) (a0 - a1 + a2 - a3)^2 )
>
> Then if y is another element of P4 a computation will show that
>
> |x y| = |x| |y|
>
> This can be done by hand.
>
> x = a0 e0 + a1 e1 + ... + a4 e4, you obtain a multiplicative
> function putting
>
> |x| = Sqrt[
> (((-4*a[0] + a[1] - Sqrt[5]*a[1] + a[2] + Sqrt[5]*a[2] + a
> [3] +
> Sqrt[5]*a[3] + a[4] - Sqrt[5]*a[4])^2 +
> (10*(-((1 + Sqrt[5])*a[1]) - 2*a[2] + 2*a[3] + a[4] +
> Sqrt[5]*a[4])^2)/(5 + Sqrt[5]))*
> ((Sqrt[5 + Sqrt[5]]*(-a[2] + a[3]) + Sqrt[5 - Sqrt[5]]*
> (a[1] - a[4]))^2/8 +
> (a[0] - ((3 + Sqrt[5])*a[1] - 2*(a[2] + a[3]) +
> (3 + Sqrt[5])*a[4])/(2*(1 + Sqrt[5])))^2))/16
> ]
>
> pentagons are complicated beasts! Mathematica did not manage
> to check multiplicativity in the 10 minutes I gave it, but the
> formula does work. If you are so inclined, you can check (*)
> on randomly generated elements.
>
> if x = a0 e0 + ... + a5 e5, setting
>
> |x| = Sqrt[
> ((a[0] - a[1] + a[2] - a[3] + a[4] - a[5])^2*
> (3*(a[1] + a[2] - a[4] - a[5])^2 +
> (2*a[0] + a[1] - a[2] - 2*a[3] - a[4] + a[5])^2)*
> (3*(a[1] - a[2] + a[4] - a[5])^2 +
> (-2*a[0] + a[1] + a[2] - 2*a[3] + a[4] + a[5])^2))/16
> ]
>
> does the trick---and Mathematica can check the multiplicative
> property (*) in a minute or so.
>
This is interesting work you are doing Mariano. But why attempt to
preserve the behavior by defining a counterintuitive metric? It may be
that there is something to find there. Never the less, in terms of
ordinary geometrical distance the product does not conserve distance
by the behavior
| z1 z2 | = | z1 || z2 |
where these bars imply the usual magnitude, absolute value, or
distance. Because this usual form exists I think what you are doing
needs to have a bit more stress in this regard. Still, it may prove
useful. Nice work.
>
> > This is the usual familiar conservation of magnitude of the reals and
> > the complex numbers. While the higher sign systems are well behaved
> > arithmetically they break this rule. Distances are no longer conserved
> > in P4+.
>
> > The well behaved members of the family are
> > P1 P2 P3
> > which form a sufficient representation of spacetime including
> > unidirectional time.
>
> See, you are here trying to explain what you meant by "The polysign
> numbers inherently contain spacetime correspondence due to the
> behavior of the math beyond sign three", yet you have not given
> any explanation whatsoever of what a "representation of spacetime"
> is, what the difference between such a representation that includes
> unidirectional time (whatever that may be) and one which does not
> include it is, and what it means for something (the polysign numbers,
> in this case) to be a "sufficient" representation.
It's pretty simple Mariano. P1 is a unidirectional algebra that mimics
time. P2 is one dimensional and P3 is two dimensional. These are the
well behaved members of the family of polysign numbers under product.
Without the breakpoint such a claim cannot be made. This is somewhat
the problem with string theories that impose a higher dimensional
breakpoint all over again. If we ask the simple question:
Why spacetime?
and we get back an answer of three dimensional space then ask
Why 3D?
then get back an answer of a ten dimensional space the obvious next
question is
Why 10D?
The polysign breakpoint alleviates this philosophical burden. It does
still have a 10D stage up at
P1 P2 P3 P4 P5
but I'm not going to go there since I have a hard time seeing where
string theory begins.
Quantum gravity people seem to have taken up the puzzle of emergent
spacetime whereas the string theorists have dodged it. That's a pretty
broad brush stroke so I'm sure there is a list of exceptions. I'm just
relating my own research which is by no means exhaustive. The polysign
structured spacetime does seem sympathetic to brane type theories. As
you have so astutely stressed even the higher dimensions beyond P3 can
jump back down through the ill-named 'zero divisors'. So to leave them
completely out of a phyisical theory or to attempt to include them can
be left an open problem.
Importantly P1's algebra is demonstrable:
- 1 - 1 = - 2
( - 2 )( - 2 ) = - 4
while still rendering as zero dimensional by the rules of polysign.
This is coherent with physical observation of time, for we have no
freedom in traversing it as we do with the spatial dimensions.
The unidirectional nature of time is born out by P1; the one-signed
numbers.
Taking P1,P2,P3 as spacetime leads into thick physics and I don't
think this is the right time to enter a discussion on structured
spacetime physics. I would prefer to leave the argument just here in
terms of mathematics as a pure though arguably nonphysical support for
spacetime from simplistic arithmetic. This would be a math-centric
boundary point which provides a leading signal that I have followed
and am happy to discuss, but it's a long journey that is unfinished.
Mariano you must see that the three (P1, P2, P3) together form a
sufficient representation of spacetime.
Yet as you speak here you seem in denial, or rather blind to the
argument. Why not take off your blindfold and actually have a look at
the argument. Then we could perhaps get to some finer level of logic
rather than simply combatting that which cannot be constructed out of
ignorance of the construction. As the law pleads: ignorance is not a
valid excuse. Here as you have entered into the discussion please
attempt to avoid this pitfall which does haunt your posts.
- Tim
Tim BandTech.com
Hi Victor. I really appreciate you taking a look at those weird
probablility patterns.
They are from a standard rand() gnu function so nothing too fancy.
Just scaled random components in every component by successive calls
to rand() as in this hot copy of my C++ source code where x[] are the
components of a polysign value:
void nSigned::Random( double mag )
{ // a random value whose magnitude is random from zero to mag
Randomize();
if( mag > 0.0 )
{ // scale value by a random between 0.0 and mag
double scale = ( (double)rand() / ((double)(RAND_MAX)+(double)
(1)) );
mag = scale * mag;
*this = *this * mag;
}
}
void nSigned::Randomize()
{
int i;
int r;
for( i = 0; i < n; i++ )
{
r = rand();
x[i] = r;
}
Unitize();
return;
}
/* end of source code snippet */
I do see the work that you point out. Still Victor you insist upon an
RxC representation whilst you instantiate in RxRxRxR for P4. What
gives? When will you come down to dimension three and when will we see
a meaningful and direct presentation of computations in RxC that you
so fondly talk of?
I have considered and computed on the simplistic RxC product of two
values (r1,z1), (r2,z2) where of course the r is the real component
and the z is the complex component, their product being:
(r1,z1)(r1,z2) = ( r1 r2, z1 z2 )
however this product is not quite the dead ringer for P4 that it seems
on first investigation. It is fascinatingly close by, but upon
investigating the discrepancy of the two a self similar image is
found. Linear compensation of the error image does not lead to a
resolution of the two products. You can squeeze out one portion (eg
the real portion) but are still left with the other portion as I
recall. You see also that the product defined above is arbitrary so
far as I can tell. Within the polysign definition of product we see
dimensional mixing without any state if independence whereas in this
simple product that I have instantiated for you (that I've been asking
for from you for some time) does not intermix. Its components remain
independent of each other. So from this ground I can already argue
that the product you speak of is a false equivalent. Of course this
can be altered by you by presenting another definition of this
arithmetic product, but you've not gotten there yet have you? In the
end this definition is arbitrary and will be built arbitrarily to
match the polysign product, which is far from arbitrary. I suspect
that when this process is completed you'll have quite a scary looking
beast, and then your statements on RxC can be fully measured as to
their validation. I understand there is some proof that you all are
keen on, but it doesn't say just how clean such constructions are does
it?
The polysign are clean from the ground on which they were developed
and are a native form that does not require the complexity that your
form will. No advanced theorems on group theory are required. Rather
the series of spaces which are polysign
P1, P2, P3, P4, P5, ...
deserve investigation. Thus far two are standard and well known. These
are P2 which are the real numbers and P3 which are the complex
numbers, though the P3 format is quite different from C. All other of
this list are new constructions. I challenge you to show me
historically where their equivalents lay. I have been over much ground
and feel very clear that the statement I make here is correct. Yes, we
will have some neat surprises along the way and there are
interdisciplinary signals of significance. I do not ask that group
theory be demolished. I merely ask that polysign theory be granted the
room to grow. So long as you cannot stomach the generalization of sign
on its own terms then you have nipped a significant new construction
in the bud in favor of your old crop. This comes at the cost of a
yield of emergent spacetime with unidirectional time, something the
old ways should not have overlooked but somehow did.
- Tim
victor_me...@yahoo.co.uk
>
> Hi Victor. I really appreciate you taking a look at those weird
> probablility patterns.
> They are from a standard rand() gnu function so nothing too fancy.
Just relying on a C random generator. What distribution
is that meant to simulate?
>
> I do see the work that you point out. Still Victor you insist upon an
> RxC representation whilst you instantiate in RxRxRxR for P4.
I've never "instantiated in R x R x R x R" in my life :-(
>
> I have considered and computed on the simplistic RxC product of two
> values (r1,z1), (r2,z2) where of course the r is the real component
> and the z is the complex component, their product being:
> (r1,z1)(r1,z2) = ( r1 r2, z1 z2 )
Very simple, n'est-ce pas?
> however this product is not quite the dead ringer for P4 that it seems
> on first investigation.
Well there is an isomorphism, that's all.
> I do not ask that group theory be demolished.
That's awfully sweet of you.
Tim BandTech.com
Fine then. Let's make him a tenth grader. Is that better? Now it
sounds as if I have a small opening into your granting me the freedom
to build sign as fundamental. This is a new topic rather than an old
and stale thingy with mildew growing upon it. These vectors have
something just so slightly different in their unidirectional balanced
build that they get away with something and while it seems to be at
the cost of an additional component it quickly becomes clear that the
additional component can always be zeroed. Informationally then the
form is as compact as the old form. In fact I can argue that it is
more compact informationally than the cartesian representation since
the lack of sign in the components leaves the sign that was zeroed as
the only informational signage necessary. This code takes up less
space than does the binary n-tuple of sign needed in the cartesian
form, which will take n bits to code. Here we need merely to code 1 of
n in order to pass the same information. The savings in bits rises
into high dimension. For instance a 1024-signed space requires 1kbit
of signage in the cartesian representation whereas the polysign
representation needs merely- well let's simply be generous and see
that just 32 bits well outperforms the cartesian sign code. Of course
the magnitudes of these systems are matched in quantity and
informationally quite large. However for certain applications the high
dimensional polysign representation will outperform its cartesian
counterpart. Strangely too in high dimension the polysign vertices
approach perpendicular- the cartesian and the polysign coordinate
systems are similar in high dimension. I don't mean to make too much
of this similarity but these are some nuances that might get
overlooked at first glance.
The simplex as fundamental I'll assume you've accepted by now. This is
simply the balancing of the signs as in an n-signed space. These n
vectors cancel to yield zero just as do the rays from the center of an
n-simplex to its vertices, by way of their inherent symmetry. There is
no need of a direct inverse and so there is no need of the real
number. I am happy to admit that the real number does work
consistently, however it is against the spirit of the logic to grant a
two-signed number within the construction of sign. This would be a
paradoxical statement. The sort that someone would love for me to flub
the dub on.
Well guess who has been flubbing the dub? I bet even a ten year old
could figure this one out.
Perhaps you are sharp enough to see how far this could go. I've
already gone a ways.
There may be a new form of calculus awaiting. Perhaps it's been done
as you wish things were done within group theory or some such high
level language that most people will never arrive to understand.
The polysign is highly tangible once the construction is granted.
Again, I jusst drewel on and on here. Sorry. I used to be quite
compressed in my communications but that never really helped get
through very well.
- Tim
Mariano Suárez-Alvarez
It is not so much the expansiveness of your communications,
but the sheer amount of terms you use which mean pretty much nothing
to anyone but yourself.
-- m
Tim BandTech.com
<victor_meldrew_...@yahoo.co.uk> wrote:
> On 5 Feb, 18:31, "Tim BandTech.com" <tttppp...@yahoo.com> wrote:
>
>
>
> > Hi Victor. I really appreciate you taking a look at those weird
> > probablility patterns.
> > They are from a standard rand() gnu function so nothing too fancy.
>
> Just relying on a C random generator. What distribution
> is that meant to simulate?
It is intended to be a geometrical survey. By taking random values and
scaling them to unit magnitude an (n-1)-sphere in n-signed is covered
as a square or whatever the graph is spec'd as; the resulting graphs
are strictly in terms of magnitude. We could likewise study some
angular measures as raw data without getting too carried away. I guess
finally somebody sees that these probability graphs are well behaved
and have content. Somehow this is the product speaking. It does seem
to whimper out after awhile and become more redundant. Still these are
all well behaved systems algebraicially. There is nothing actually
broken about the higher sign numbers. They just don't do what alot of
people think they should in terms of distance behaviors. Whether they
play a part in physics or not the support for spacetime remains in the
systems whose distance behaviors under product are straight and
linear, or whatever terminology ought to go here.
>
>
>
> > I do see the work that you point out. Still Victor you insist upon an
> > RxC representation whilst you instantiate in RxRxRxR for P4.
>
> I've never "instantiated in R x R x R x R" in my life :-(
Yeah, whatever. You know what I mean.
>
>
>
> > I have considered and computed on the simplistic RxC product of two
> > values (r1,z1), (r2,z2) where of course the r is the real component
> > and the z is the complex component, their product being:
> > (r1,z1)(r1,z2) = ( r1 r2, z1 z2 )
>
> Very simple, n'est-ce pas?
>
> > however this product is not quite the dead ringer for P4 that it seems
> > on first investigation.
>
> Well there is an isomorphism, that's all.
Well, then, so what?
Why are people thinking this a gag?
When you want isometric correspondence then we just don't know do we?
Yet already the polysign built the complex numbers, the real
numbers, ...
Really, Mariano's defense has been poor, but good enough for you to
come in and save the day.
I suspect that without all of the crass crap, which I can argue is
well deserved, this conversation would not be going on.
- Tim
Mariano Suárez-Alvarez
Does he? I for one have *absolutely* no idea
what you mean by that!
> > > I have considered and computed on the simplistic RxC product of two
> > > values (r1,z1), (r2,z2) where of course the r is the real component
> > > and the z is the complex component, their product being:
> > > (r1,z1)(r1,z2) = ( r1 r2, z1 z2 )
> >
> > Very simple, n'est-ce pas?
> >
> > > however this product is not quite the dead ringer for P4 that it seems
> > > on first investigation.
> >
> > Well there is an isomorphism, that's all.
>
> Well, then, so what?
> Why are people thinking this a gag?
> When you want isometric correspondence then we just don't know do we?
> Yet already the polysign built the complex numbers, the real
> numbers, ...
> Really, Mariano's defense has been poor, but good enough for you to
> come in and save the day.
FWIW... I did not try to defend anything. I presented you some facts
(which are, in fact, essentially the very same Victor gave you,
apart from the fact that I made explicit the isomorphisms and
a few other details), and that was it. Whatever it was that
was poor, it was not a defense.
-- m
amy666
isomorphic to each other ?
is your opinion :
1) R x C = P4
and
2) R x R x R
and no other.
yes or no ?
you havent answered that.
regards
tommy1729
victor_me...@yahoo.co.uk
> On Feb 4, 9:41 am, "victor_meldrew_...@yahoo.co.uk"
>
> > to join in your fantasies about offering to show a
> > 10-year old boy your "third sign". :-(
>
> Fine then. Let's make him a tenth grader.
What on earth is a "tenth grader"? Is it like a flour grader?
> Sorry. I used to be quite
> compressed in my communications but that never really helped get
> through very well.
Your present logorrhoea doesn't aid your communication. If you
split up your diatribes into shortish paragraphs I might read some
of them. Oh yes, it would help if you also included some mathematical
content.
victor_me...@yahoo.co.uk
> On Feb 5, 3:08 pm, "victor_meldrew_...@yahoo.co.uk"
>
> > is that meant to simulate?
>
> It is intended to be a geometrical survey.
Do you mean it's simulating a geometric distribution?
> > > I do see the work that you point out. Still Victor you insist upon an
> > > RxC representation whilst you instantiate in RxRxRxR for P4.
>
> > I've never "instantiated in R x R x R x R" in my life :-(
>
> Yeah, whatever. You know what I mean.
Whatever what? I really don't know what you mean; you introduced
R x R x R x R into this thread, not me, but you never
explained its purpose.
As an aside, it really would help if you could post
in a much closer approximation to the English language.
> When you want isometric correspondence then we just don't know do we?
We do know; at least we who are competent to have worked it out
do. It's easy to write down the distance function on R x C
corresponding to yours on P_4, viz. the square of your
distance corresponds to the map (x,z) |--> (1/3)(x^2 + 2|z|^2).
It's easy to do this for all P_n.
Mariano Suárez-Alvarez
> mariano , how many numbers are there in 3d ?
>
> isomorphic to each other ?
>
> is your opinion :
>
> 1) R x C = P4
>
> and
>
> 2) R x R x R
>
> and no other.
>
> yes or no ?
Those are (up to isomorphism) the only two
reduced commutative 3-dimensional real algebras.
The complete list of 3-dimensional commutative
real algebras is:
A1) R x R x R
A2) R x C
A3) R x R[x] / (x^2)
A4) R[x, y] / (x^2, x y, y^2)
A5) R[x] / (x^3)
-- m
Tim BandTech.com
OK then. From scratch in advanced sx form:
s1 x1 @ s1 x2 = s1 ( x1 @ x2 ) ,
( s1 x1 )( s2 x2 ) = ( s1 @ s2 ) ( x1 x2 ) ,
sum over s ( s x ) = 0 .
These three may be called laws if you wish of the construction.
Certainly these three cannot be broken and so no exceptions should be
made to them in any way. They are slightly cryptic in that the '@'
symbol implies superposition. The motivation for this symbol is as the
zero sign which in hindsight is appropriate to sign superposition
since the identity sign shifts upward as one progresses into higher
sign otherwise. Thusly the zero sign is the same sign as the n sign.
This forms a modulo arithmetic such that the superposition
s1 @ s2
is a modulo n sum.
Beyond this slightly cryptic line these laws may be viewed from an
ordinary (ie real valued) thinking, except that every instance of sign
within these equations is contained in instances of s such as s1 and
s2 whose actual values may vary such as
s1 = 1, s2 = 2; s1 @ s2 = 3
where all such sign values are taken mod n where n is the signature of
the system Pn so for instance a common task space P4 has n=4.
There are two types within the construction. So far the first has been
addressed more carefully. The second whose representative in the
equations above is x does not contain a sign since that part has been
demoted as a prefix to this unsigned part. Since our ordinary sense of
numbers is that such a thing is a magnitude leads me to use this word
as my reference term for this x portion. Now composing such parts in
the general form
s x
requires still that we stipulate which n we are working in. So for
instance every usage of an equation such as
- 1 + 2 * ( - 3 )( - 4 @ 2 ) = - 1 + 2 @ 12 # 6
must be accompanied with a specification of what signature the math
was done in, though for this case we can reverse compute that it was
done in P5. Such a reverse computation requirement would be considered
obnoxious, as is this optional text you are reading right here within
this attemp at a complete description without running on at the mouth
in useless ways.
- Tim
victor_me...@yahoo.co.uk
> On Feb 6, 3:01 am, "victor_meldrew_...@yahoo.co.uk"
> > content.
>
> OK then.
<snip>
Logorrhoea indeed. Another descrpition of your P_n system
adding no more to what is on your website.
secondmouse
There's very little of merit (math content anyway) on your
website, e.g. what self-indulgent nonsense -
"I have created polysign numbers".
You don't seem to be taking on board the comments
made to date; i.e. consult some basic algebra textbooks
Tim BandTech.com
Yes, that's correct. To the best of my knowledge my website is a
complete rendition and all that I write here is diarrhea. It is within
this interactional diarrhea that I hope to eventually ferment and
culture a future crop. The seeds are here and the fertilizer is here
and perhaps you just take a leak on them and germinate them.
Supposedly there are quite some interesting elements in the huma
urine, at least some progressive people believe this to be so. My own
experiments do show that the grass in any greener there, but who is to
say over time what an accumulation of these nutrients could do?
My experiments in polysign are markedly different than the
interactions that I've had here on usenet. I've found the number
system to be leading and have gone where it has taken me. It has taken
down into areas that need more work. Such is the quality of the
construction that it will be some time before it is fully built out
and that it will take more than a handful to do it. Existing math
weaves a tapestry of cross tied terminologies and misuse of the
english words such that my own misuses are within the tolerance of the
system. I plead to you by refering you back to the greats who are
caught in the same difficulty. Particularly in defense of me I refer
you to the guy whose name starts with a P. and is fond of magnitude. I
myself named my little dog Magnitude for this term is so pretty and
primitive as my dog.
In further defense of magnitude I would present to you the gorilla
conjecture:
If a gorilla can sort sticks by length with out any mathematical
training at all then can we accept magnitude as fundamental? Must we
train the gorill to cut a stick to be in between the lengths of two
other sticks to procede or is this simplistic conjecture good enough?
I think the simplest form is good enough but if you wanted to garnish
it up a bit it would still pass just fine.
I'm letting go of Johnny's question for now since it is clear that you
are uncomfortable answering it. This is good. You have integrity and I
must apologize for any name calling that I performed earlier.
Please take your time answering that question. I cannot anticipate an
overnight modification. I'm in it for the long haul. Caught a ride on
the freight train... Wasn't going my way, but what the hey, I'm going
somewhat that way. Having gone too far I'll just hop off, trace the
tracks back to the last good crossing where a road sign reads the way
with a nice neat arrow pointing my way. I read it on the way but was a
bit on the edge as to whether I read it right. If I read it wrong then
the next train would be a real pain at 30 knots on the hour. Even if
you didn't get the units right the scale is sliding so providing he
unit sphere is a sensible study for the behaviors we seek to
understand. This gets over to another guy with an R. in his name, but
instead of just getting bent up there we'd be much better off
considering structure since this is a new paradigm with new language
and that new language is well understood. Let's not get so fraught on
such a word as instantiation. Let's just accept that your ilk are poor
at it. They're so busy getting abstracted that they haven't bothered
to look down at the foundation. That foundation was a lattice of poles
lashed together in perpendiculars on which they've built too much.
They forgot to triangulate that thing and so it is folding up in front
of your eyes.
I don't know where your jumping off point is. There are some creaking
groans going on I think inside of your head right now. What about
Johnny? Will he ever get a clear answer from you? If you were just a
street bum it wouldn't matter much. But since I see that you hesitate
your sorry speech so far has a glimmering of hope. That mildew odor on
your clothes must be explicable. There must be quite a story there.
Nice to meet you on the street Victor.
- Tim
victor_me...@yahoo.co.uk
> Yes, that's correct. To the best of my knowledge my website is a
> complete rendition and all that I write here is diarrhea.
I was wondering about the nasty smell.
> It has taken
> down into areas that need more work.
How about you doing some of that (mathematical) work?
> In further defense of magnitude I would present to you the gorilla
> conjecture:
> If a gorilla can sort sticks by length with out any mathematical
> training at all then can we accept magnitude as fundamental?
Guy the gorilla sez Golden rocks!
> I'm in it for the long haul. Caught a ride on
> the freight train... Wasn't going my way, but what the hey, I'm going
> somewhat that way. Having gone too far I'll just hop off, trace the
> tracks back to the last good crossing where a road sign reads the way
> with a nice neat arrow pointing my way. I read it on the way but was a
> bit on the edge as to whether I read it right. If I read it wrong then
> the next train would be a real pain at 30 knots on the hour.
Zzzzzzzzzzzzzzzzzzzz....
> Nice to meet you on the street Victor.
Sod off, Tim the dim!
Tim BandTech.com
<victor_meldrew_...@yahoo.co.uk> wrote:
> On 5 Feb, 21:18, "Tim BandTech.com" <tttppp...@yahoo.com> wrote:
>
> > On Feb 5, 3:08 pm, "victor_meldrew_...@yahoo.co.uk"
>
> > > Just relying on a C random generator. What distribution
> > > is that meant to simulate?
>
> > It is intended to be a geometrical survey.
>
> Do you mean it's simulating a geometric distribution?
>
> > > > I do see the work that you point out. Still Victor you insist upon an
> > > > RxC representation whilst you instantiate in RxRxRxR for P4.
>
> > > I've never "instantiated in R x R x R x R" in my life :-(
>
> > Yeah, whatever. You know what I mean.
>
> Whatever what? I really don't know what you mean; you introduced
> R x R x R x R into this thread, not me, but you never
> explained its purpose.
haven't bothered to admit it yet.
Do you understand that when we declare
(x,x,x,x,x...) = 0
as in
sum over s ( s x ) = 0
- x + x * x ... @ x = 0
(All of the above meaning the same thing)
that this inherently implies the simplex balance?
Then my criticism or your usage of barycentric coordinates is false.
Still, it should be pointed out that this 4D version lays there.
Especially since Mariano's e0,e1, ... notation is pretty much there.
The point in polysign is to get to the crux without the old crap.
The truth of polysign is that sign is a general phenomenon having to
do directly with dimension. This nuance is what gets avoided yet it
seems you are part way down the rabbit hole. Now go find the cocoa
puffs. And look out for the false mimics; those round little rabbit
turds that Mariano chokes on of his own volition.
>
> As an aside, it really would help if you could post
> in a much closer approximation to the English language.
>
> > When you want isometric correspondence then we just don't know do we?
>
> We do know; at least we who are competent to have worked it out
> do. It's easy to write down the distance function on R x C
> corresponding to yours on P_4, viz. the square of your
> distance corresponds to the map (x,z) |--> (1/3)(x^2 + 2|z|^2).
> It's easy to do this for all P_n.
Hmmm. So we're in P4, so lets say I've dropped a point off for your
procedure:
y(P4) = y = - 1 + 2 # 3 .
Now I've got to get y into your x,z form. Have you inherently provided
me a transform for this? I think not since I need a spec of a plane in
P4 for your z space and a line spec in P4 for your x. Since this line
passes through the origin then one point might suffice to state the
coordinate system. Since it is not specified I should only assume that
it is # 1 + 1. Is this coherent with what you are doing? The Mario
representation would be
e0 + e2
for your x unit vector direction, scaled to unity, and not to be
confused with orthogonal unit vectors.
This should set the x component positive reference frame direction.
This is how I understand the equivalence to be layed out. If I am off
I trust you'll correct me and at least reflect this back to your own
context for a parallel communication. Thusly we might actually come to
some agreement on some peculiar thing even if not applicable for the
whole system.
We'll still need to spec the plane's reference frame. But upon
carrying all of this out what will your isometric product look like? I
myself do not withhold my own misunderstanding. You claim to
understand this and I would welcome a complete rendition here. It need
not be extraordinary. What has been done in the past is not so clean
and simple and sweet as you have portrayed it. Still you stick like a
glue to your RxC space, yet all your mumblings have not really
provided much. Because the four-signed numbers are three dimensional
then transforms exist from them to any other three dimensional system
that covers the volume of its space. Thusly you may as well state your
distance function in terms of RxRxR. Where are the benefits of the
form RxC? They are in the product aren't they? To express such
products we need to spec a transform from your RxC (x,z) to P4 (y) and
vice versa. Then upon studying the product relationship we can observe
any equivalence or difference.
Your one-liner
"map (x,z) |--> (1/3)(x^2 + 2|z|^2)."
doesn't really mean much to me without the background information that
I am asking of you. It seems to me that you are in some midstep
position with this equation which is supposed to relate to distance in
P4. As I've seen the word distance used differently here by Mariano
and also Tommy recently I am a bit miffed that there is no universal
agreement on distance. I see that we can define strange distance
functions and play with them, but the ordinary sense of distance does
not need this level of confusion. I am operating on the assumption
that you are discussing ordinary distance and especially since P4 does
carry a physical graphing capability that we could then plot such
positions out and literally measure these distances with a measuring
tape. Do you balk at this? Then perhaps we ought to step down to P3,
where such distances can be plotted in the plane of the paper.
Unfortunately without a specification of the transform you are using
statements like your one above are sitting floating in air without
much beneath them. When you say that something is obvious and so how
could it go misunderstood such as the instantiation of the P4 product
on (R,C), well, the holes in the thinking collapse it under
instantiation. I've already worked on this at
http://bandtechnology.com/PolySigned/Deformation/P4T3Comparison.html
I must assume that you are withholding information, and I do not mean
to say that this is with any ill intent. I recommend you get right
into it here as you see it and I'll try to follow along. If you have
an isometric isomorphism from P4 to (R,C) then others will take
interest in that. I'm surprised lwal hasn't spoken up yet here, but it
wouldn't surprise me if lwal does check in here and if you can do what
uit sounds like you claim then I'd point him to your post directly. It
should be a fairly complete rendition though I think. Please note:
optional long winded text at end of post was ommitted since this one
is a bit breezy up front, though without spacing out the thoughts
sufficiently wide and reinforcing them with such redundancy I doubt
that an onlooker would see much wrong with your statement. That is not
even to say that there is anything wrong with it, but that it must
carry with it some additional information, and if we are going to
start dabbling in twisting and morphing the metric of the space or
folding up its vertices then that is your bad, not mine. Whose space
will be twisted? P4, or RxC? The conveyance of your attachment to RxC
and the challenge this poses to its purity can be squashed by the
likes of you simply by putting down your information here.
- Tim
Tim BandTech.com
Hah !!!
Let the onlooker decide what you've communicated. Information station
MSA defeated.
Oh, I wrote a crappy poem for you but withheld it out of maturity. Ha.
2009020110:31:01est unposted "For future bait with MarianoSaNan"
Mariano so des ka?
Mariano sa.
Mariano so des;
Marianosa.
Mariano eche ban.
Marianosa eche san?
Marianosa nan ban.
Marianosanan...
Mariano S. A. (NAN)
Flagging it in
Flogging it out
Slogging it in
Slagging it out
Structured types are such big frights
un till the dis crete work is done.
To say there is an error language
Until there is an error language
There is an error language
The NAN is done
INF and NAN
are just a who and when
or just a when and who
but when the two are two
perhaps then real work can be done.
So Tommy makes a plea
and Marianosanan nanny ban not an eche ban
sends along a very very strong
construction endorsed by g.
To see the two as free
their lack of simplicity
Their cryptic ways
their cryptic stays
they're stuck on every thorn
waiting to be reborn
meanwhile bleeding dry.
Preserved and older forms.
Like in a book of one.
The one the eche ban.
The eche ban can never be undone.
So you say you've got a phaser
that can outdue my tazer
and I tell you mine is only set to stun.
The big drop is coming.
- T
Tim BandTech.com
OK. So you like magnitude as primitive. When we get to tacking on sign
it starts to get tacky, and this tacky feeling is what I think most
people dislike in the polysign. The recovery of a serious effort is in
the observation that sign is dimension. Finally then the seriousness
of the sign stands freely. Sign is dimension, just off by one.
Hence an argument that the human race is off by one. Thusly time goes
overlooked as a primitive arithmetic structure and occupies a
paradoxical and often talked about quality that has remained
mysterious. Then again the whole geometry can be taken as
unidirectional in that it no longer requires the direct bidirectional
feature that the cartesian space constructed. Instead we see a
balancing form of space.
- Tim
victor_me...@yahoo.co.uk
> Mariano so des ka?
> Mariano sa.
> ......
This is even worse than "galathea"'s blank verse :-(
victor_me...@yahoo.co.uk
> OK. So you like magnitude as primitive.
Do I? When I figure out what that means, I'll let you know.
> The recovery of a serious effort is in
> the observation that sign is dimension.
And what about the observation that your polysigns form a cyclic
group.
> Hence an argument that the human race is off by one.
How blessed is the human race! that it has Tim Golden to point out
the errors of its ways!
victor_me...@yahoo.co.uk
> Alright then. You are inline with the simplex reasoning
Am I? That's nice!
> Then my criticism or your usage of barycentric coordinates is false.
As I hadn't used barycentric coorinates anywhere in this thread,
this means that your criticism must be false.
> This nuance is what gets avoided yet it
> seems you are part way down the rabbit hole. Now go find the cocoa
> puffs.
You keep your cocoa puffs down your rabbit's hole? That explains much!
> I think not since I need a spec of a plane in
> P4 for your z space and a line spec in P4 for your x.
Is that a "spec" as in Grothendieck's EGA?
> Your one-liner
> "map (x,z) |--> (1/3)(x^2 + 2|z|^2)."
> doesn't really mean much to me
Couldn't you work it out? I already indicated the isomorphism
between your P_n and R^a x C^b. It's now just a question of
transporting your distance function from one to t'other.
> carry a physical graphing capability
This is sci.math, not sci.physics.
> http://bandtechnology.com/PolySigned/Deformation/P4T3Comparison.html
As an algebra over R, P_4 and R x C are isomorphic
(is "closely resembles" a Goldenballs for "is isomorphic"?).
> I must assume that you are withholding information
Oh yes, I have taught you everything you know, but I have not
taught you everything I know!
> that is your bad
My what?
Tim BandTech.com
You can see that I am really not so clever.
Still, anything to keep up the humor is wise I think.
Think of all the students bored in math class...
They know they've been eating a pile of crap...
They can feel it.
Now they can know it.
- Tim
Tim BandTech.com
<victor_meldrew_...@yahoo.co.uk> wrote:
> On 7 Feb, 16:07, "Tim BandTech.com" <tttppp...@yahoo.com> wrote:
>
> > OK. So you like magnitude as primitive.
>
> Do I? When I figure out what that means, I'll let you know.
>
> > The recovery of a serious effort is in
> > the observation that sign is dimension.
>
> And what about the observation that your polysigns form a cyclic
> group.
number.
Under my ordinary education the real number is presented as real
analysis without any attention to group theory. These old rules are
the ones that I wish to challenge. In that most modern mathematical
theory treats the reals as fundamental and that I have inserted a
clause beneath this level then I must challenge the accumulation of
thinking since that time. This goes so far as to challenge the
physicists concept of isotropic space, which is arguably a
mathematical construction. So I cast my net far and wide and so what I
have caught is an assortment of fish. This assortment is rather
familiar yet in its new context the fishiness of the polysign is
clearly there for the well educated man. I must then cast my doubt
upon the well educated man. He has overlooked a fundamental principle
for far too long. What a shame and what a sham. Now he can make good.
And then we can make better. The puzzle will remain open
unconditionally no matter how tantalizing and pristine the current
system is preached to be.
Again drewel and foam coming forth in a frothy spray of misperception
and through the haze I see it all...
- Tim
amy666
> On Feb 5, 9:21 pm, amy666 <tommy1...@hotmail.com>
> wrote:
> > mariano , how many numbers are there in 3d ?
> >
> > isomorphic to each other ?
> >
> > is your opinion :
> >
> > 1) R x C = P4
> >
> > and
> >
> > 2) R x R x R
> >
> > and no other.
> >
> > yes or no ?
>
> Those are (up to isomorphism) the only two
> reduced commutative 3-dimensional real algebras.
good then we agree upon that.
>
> The complete list of 3-dimensional commutative
> real algebras is:
>
> A1) R x R x R
> A2) R x C
> A3) R x R[x] / (x^2)
> A4) R[x, y] / (x^2, x y, y^2)
> A5) R[x] / (x^3)
>
> -- m
regards
tommy1729
Mariano Suárez-Alvarez
> mariano wrote :
>
>
>
> > On Feb 5, 9:21 pm, amy666 <tommy1...@hotmail.com>
> > wrote:
> > > mariano , how many numbers are there in 3d ?
>
> > > isomorphic to each other ?
>
> > > is your opinion :
>
> > > 1) R x C = P4
>
> > > and
>
> > > 2) R x R x R
>
> > > and no other.
>
> > > yes or no ?
>
> > Those are (up to isomorphism) the only two
> > reduced commutative 3-dimensional real algebras.
>
> good then we agree upon that.
Notice that I did not agree to anything: I simply can't,
for I have no idea what you mean by "numbers in 3d".
-- m
Tim BandTech.com
<victor_meldrew_...@yahoo.co.uk> wrote:
> On 7 Feb, 15:13, "Tim BandTech.com" <tttppp...@yahoo.com> wrote:
> > Your one-liner
> > "map (x,z) |--> (1/3)(x^2 + 2|z|^2)."
> > doesn't really mean much to me
>
> Couldn't you work it out? I already indicated the isomorphism
> between your P_n and R^a x C^b. It's now just a question of
> transporting your distance function from one to t'other.
>
My, my, my.
You are a sad lot that wants to impose an arbitrary function on top of
a product with correspondence. Perhaps this criticism can be whittled
away, but I think not fully. Still, I cannot impose a correspondence
principle upon a mathematician. Perhaps this is where we part ways and
a physicist steps in.
- Tim
Tim BandTech.com
Hi Victor. I really do not wish to give you the cold shoulder.
Rather, I pose this simple dilemma to you as a security retainer.
Upon agreement that we will be morphing your space into mine...
Since P4 was created first chronologically it must be so.
Your work will be in hindsight.
Hah! Got you there, sort of.
No, really what if you were to insist that we scale back P4's
amplitudes to match the RxC pure product? Should I not be offended?
Rather should P4 be offended?
The polysign numbers are strong enough to speak for themselves;
I am just the spin guy.
Still, the math that you accept as undone has not been done. How about
some prize money or something? I'd put up $50US if you did the same. I
seriously doubt if I will get a satisfactory purely mathematical
answer so I'd like the solution open to arbitrary precision
computation. I've already got a strong lead on that. Do you have any
idea how long the series is or if the series idea is wrong? I think it
must be right based on my computational results, but they are just
numbers. Still there is that other interesting thing of eliminating
one error at the gain of the other error. Somehow this word error here
is the trouble isn't it? Aren't you granting that your product
solution is errant and that you are fixing it up via a distance
function? Why on earth else would you need this new distance function?
Are you proscribing a space with two distance functions? One for
superposition and one for product? Let's not forget that rotational
reference frames under superposition is a fairly ordinary behavior
which could get quite contorted under your new distance function. I
really would prefer that you twist up yours. Oh well, I guess if you
argue that it could go either way then again I'll ask you if we could
just let it tweak out on your side. I don't want to see P4 go through
the wringer.
- Tim
victor_me...@yahoo.co.uk
> But the form was built back in time as the real number.
What "form"?
> In that most modern mathematical
> theory treats the reals as fundamental
Really? If that were the case, why did Weierstrass bother
with his arithmetization of analysis at all?
> This goes so far as to challenge the
> physicists concept of isotropic space,
Why don't you sod off and tell the fizzisists this?
victor_me...@yahoo.co.uk
> You are a sad lot that wants to impose an arbitrary function on top of
> a product with correspondence.
No, it's you who is the "sad lot" for imposing your arbritray
distance function on your P_n algebra.
victor_me...@yahoo.co.uk
> Upon agreement that we will be morphing your space into mine...
I don't "morph". And I don't have a "space", All I have done
is to explicate the structure of your algebra.
> Since P4 was created first chronologically it must be so.
That is a lie: direct products of fields were studied many decades
before you set up your website. The same is true of group algebras.
> Rather should P4 be offended?
Anthropomorphism :-(
> Still, the math that you accept as undone has not been done.
Goldenballs can't spell "mathematics" :-(
> How about
> some prize money or something? I'd put up $50US if you did the same.
Haven't you any hard currency? And is there any clear
statement of your challenge?
> Do you have any
> idea how long the series is or if the series idea is wrong?
What series?
> Why on earth else would you need this new distance function?
It is your distance function, not mine. See
http://bandtechnology.com/PolySigned/DistanceFunction.html .
Tim BandTech.com
Grr...
I have not imposed any arbitrary distance function.
The polysign math is built on the foundation that
sum over s ( s x ) = 0 .
From this single line the simplex geometry is evident. Again I would
strike up a conversation on ordinary distance here, but it seems you
will transgress it. Thus far the only distances that I use on my
website or here are in the sense of ordinary distance, no different
than the standard cartesian representation of distance as a sum of
squares. These distances are Euclidean in that we can take out some
dividers, transfer them, etc. This is a vector space. If I try to go
any deeper than that it will seem that I am the one wheedling so
instead I should go back to the prior question of whether you are
going to have two distance functions in your construction? Could you
then explain what is the utility of each of them? Clearly the new
distance function which is under dev is just a fixup for the product
mismatch. That is all that it is and so to call it a distance function
is a misnomer. It is more inline with a correction factor, for
wherever the RxC product mismatches the polysign product this function
will correct the error. As I understand it this is its only purpose
and so naming it a distance function is errant.
This function will be of interest but as I've alluded to here already
I believe that its complexity is going to be more than a few simple
terms. My own whiff of it from my work is that it may be a long
convergent series. In effect I should have to repeat my work on
http://bandtechnology.com/PolySigned/Deformation/P4T3Comparison.html
to reduce the error in a second iteration, then a third and so forth
until the error is whittled down.
The other way to look at this puzzle it to see that an RxC (x,z)
product such as
(x1,z1)(x2,z2) = ( x1 x2, z1 z2 )
does not have any mixing of its components across the R and C spaces,
whereas the polysign does inherently intermix all of its components
with each other. If in your correction you introduce this dimensional
mixing then why shouldn't that come back all the way into the product
definition? Then you can have one distance function (the ordinary
distance), one product, and still have a cleanly superposable basis.
Tell you what Victor. I'll cut a check for US$50 when the correction
factor is clearly expressed. This is an open bribe to anyone who can
do it first. But that's all it's going to be isn't it? A correction
factor of some weird sort? If you want to put up $50 too then that's
great. My offer is up and active as of now.
- Tim
victor_me...@yahoo.co.uk
> I have not imposed any arbitrary distance function.
You have, by insisting that your unit vectors be the vertices
of a regular simplex.
> These distances are Euclidean in that we can take out some
> dividers, transfer them, etc. This is a vector space.
Any positive definite quadratic form on a finite-dimensional
vector space over R defines a distance equivalent to the
usual Euclidean distance on R^n. To do this, you have a free
choice. You made one, but are pretending you didn't.
> If I try to go
> any deeper than that it will seem that I am the one wheedling
You are.
> Clearly the new
> distance function which is under dev
"dev"? As in Kapil Dev?
> The other way to look at this puzzle it to see that an RxC (x,z)
> product such as
> (x1,z1)(x2,z2) = ( x1 x2, z1 z2 )
> does not have any mixing of its components across the R and C spaces,
> whereas the polysign does inherently intermix all of its components
> with each other.
What the isomorphism reveals is that there are linear subspaces
in your P_n, which you hadn't noticed since your construction
hides them, which as you say, don't "mix" with each other.
> Tell you what Victor. I'll cut a check for US$50 when the correction
> factor is clearly expressed.
Is "check" American for "cheque"? And does "cutting" one reduce
its value? Of course, I would prefer a larger sum, in a negotiable
hard currency.
Tim BandTech.com
Alright, then, how about a money order for US$50?
Who has done most of the cutting in this conversation?
Hahhha ! !
Your answers are not forthcoming and you are making a backlog of
research for me.
Are you trying to stifle the polysign?
Are you trying to choke my number system to death by shoving other
numbers at its native beauty?
- Tim
victor_me...@yahoo.co.uk
> Your answers are not forthcoming and you are making a backlog of
> research for me.
Oh dear, whay don't you come back *after*
you have done this research?
> Are you trying to stifle the polysign?
All I was doing was to explain your construction in terms
of the mathematics of a century ago, and also to explain
why you do not obtain any interesting new structures
from your polysign numbers (just direct products of reals
and complexes).
victor_me...@yahoo.co.uk
> I have not imposed any arbitrary distance function.
> The polysign math is built on the foundation that
> sum over s ( s x ) = 0 .
> From this single line the simplex geometry is evident.
Is the "simplex geometry" the decision that each s1
have unit distance from the origin and that each s1
is equidistant from the rest? That certainly does not
*not* follow just from the equation sum_s s x = 0.
Tim BandTech.com
I would argue that in their simplest form without the added
complication of scaling that yes, the simplex form is fundamental.
All that we have to do is study the algebra and the dimensional nature
of the number systems is forthcoming as information. I know that this
informational form is not so clean cut to traditional mathematics so I
try not to rely on it too much. What other geometry would you suggest?
How can you refute the simplex? The superpositional behaviors are
perfectly matched.
There have been people in the past trying to claim that the four-
signed numbers map to the plane. But this is not true. One would need
to put additional constraints in the system such that in P4:
+ 1 # 1 = 0 , - 1 * 1 = 0
to make the P4 planar solution work out. These are not necessary
requirements and are not in the spirit of the construction. The point
is that the components balance to yield zero, just as the real number
does. So consider taking your argument on the real number- yes, you
could build a scaled form but that scale will get built on top of the
simple form- the simplex form. The unit vectors of the n-simplex
(there are n of them) added together come back to its center which in
this construction is the origin. This also matches the informational
dimension of the system.
This simplex interpretation does have similar bystanders, but nobody
seems to have bothered with this very simple interpretation before.
Mostly when we think of the simplex we think of the external frame as
lines connecting vertices rather than this core model. I do think that
helps explain the resistance to the geometry. When you start playing
in lattices of these things the whole thing goes kaleidoscopic on you,
so this sense of the origin and the unit vectors to vertices as
elemental is important I think to base or ground the system.
What about P1 Victor? Are you willing to consider that a simple
geometry has been overlooked? That it is zero dimensional? That it
mimics time?
"It is known that geometry assumes, as things given, both the
notion of
space and the first principles of constructions in space. She gives
definitions
of them which are merely nominal, while the true determinations appear
in
the form of axioms. The relation of these assumptions remains
consequently
in darkness; we neither perceive whether and how far their connection
is
necessary, nor a priori, whether it is possible"
- Bernhard Riemann, On the Hypotheses which lie at the Bases of
Geometry.
Translated by William Kingdon Clifford
He shortly goes on to say:
"Thus arises the problem, to discover the simplest matters of fact
from which
the measure-relations of space may be determined; a problem which from
the
nature of the case is not completely determinate, since there may be
several
systems of matters of fact which suffice to determine the measure-
relations of
space—the most important system for our present purpose being that
which
Euclid has laid down as a foundation."
From the context of Riemann I should draw a clearer boundary between
our two systems: the traditional cartesian geometry and the new
simplex geometry, these being equally consistent fabrications of
space. However, whereas one has an arithmetic product which springs
forth from its construction (polysign) the other (cartesian) has such
products laying much farther up the stream of academic accumulation,
so obscure as to be intangible to all but graduate students.
I suggest to you if Riemann had the polysign numbers in hand he'd be
working with them. Would he go on to a curved space or a scaled space?
Perhaps. But at the crux of what I see is structured spacetime. The
problems over time and its openness to interpretation lead people to
the strangest conclusions. At least now there is a bit of math that is
leading. And it leads to spacetime as unified and structured. The
geometry of electromagnetism is woven into spacetime itself. The
rotational plane P3 and the bidirectional line P2 coupled with the
unidirectional P1 will eventually yield a restatement of Maxwell's
equations whereby the electron and its spin will be inherent. This
structured spacetime will recover its isotropic character merely by
invoking the relative reference frame on top of the structured form.
This then is the conglomerated space we live in. Rotational discs with
jets spewing forth material bidirectionally are well known out in the
cosmos. That this form is quintessential is a fairly broad swath to
try and cut. Still, it is a pretty thought. I am simple minded so I
can go here, but likewise I don't have the resources to break through
all the way.
The work that you do on the traditional forms will be of value I
think, but my belief is that working in the native polysign form will
be more productive. Variational calculus does interest me and I still
don't know much about it. Likewise I've tried to get into Lagrangian
dynamics but haven't gotten very far.
Sometimes the silliest little things can be stumbling blocks- for
instance the little minus or plus symbol on the front of a number.
Sloshing a light film of foam down with a swig of coffee I say good
morning and apologize for this cacophony. Now that it is typed I can't
see anything to delete and instead could keep filling it out but it is
already too long, so you see I should not. So I won't as I said and
did not and do not and will not, except for these obnoxious few spare
words.
- Tim
victor_me...@yahoo.co.uk
> On Feb 9, 4:38 am, "victor_meldrew_...@yahoo.co.uk"
>
>
> > have unit distance from the origin and that each s1
> > is equidistant from the rest? That certainly does not
> > *not* follow just from the equation sum_s s x = 0.
>
> I would argue that in their simplest form without the added
> complication of scaling
Well there we are; in the second line of his reply Goldenballs
spouts the meaninless phrase "complication of scaling".
> How can you refute the simplex?
One can refute propositions. But a simplex is a geometric
figure not a proposition. How could one refute that?
> Mostly when we think of the simplex
The royal "we"?
> I suggest to you if Riemann had the polysign numbers in hand he'd be
> working with them.
And I suggest to you that Riemann would be calling you a bullshitter
too.
> will eventually yield a restatement of Maxwell's
> equations whereby the electron and its spin will be inherent.
> jets spewing forth material bidirectionally are well known out in the
> cosmos.
Why don't you take these insights to sci.physics and leave us in
peace?
> but my belief is that working in the native polysign form will
> be more productive.
You haven't succeeded one iota yet.
> apologize for this cacophony.
No need to apologise: just cease being cacophanous.
Tim BandTech.com
<victor_meldrew_...@yahoo.co.uk> wrote:
> On 9 Feb, 14:04, "Tim BandTech.com" <tttppp...@yahoo.com> wrote:
>
> > On Feb 9, 4:38 am, "victor_meldrew_...@yahoo.co.uk"
>
> > > Is the "simplex geometry" the decision that each s1
> > > have unit distance from the origin and that each s1
> > > is equidistant from the rest? That certainly does not
> > > *not* follow just from the equation sum_s s x = 0.
>
> > I would argue that in their simplest form without the added
> > complication of scaling
>
> Well there we are; in the second line of his reply Goldenballs
> spouts the meaninless phrase "complication of scaling".
>
> > How can you refute the simplex?
>
> One can refute propositions. But a simplex is a geometric
> figure not a proposition. How could one refute that?
Well, you are casting doubt, let's say.
I will argue that we can likewise cast doubt upon the cartesian
dimensional construction of space, much as Riemann is suggesting is
universally so.
We do see challenges of Euclid's postulates in relativity theory don't
we? Riemann's workings with spherical spaces, etc. leave us with an
opening at very low levels whereby any math can be challenged.
For instance the cartesian product is not necessary to compose the
polysign numbers.
Now, getting into the cartesian construction we see a declaration of
two real lines at perpendicular angles with a claim that when the
values on these lines are independent that we have a free 2D space. If
they were dependent upon each other such as
x = 2y
we would only see a line. While I can accept this version of 2D space
it is not necessarily the only construction of 2D space, though usage
of the very name 2D is the result of using the cartesian space. More
Euclidean would be to label this space the 'plane', but since all of
these arguments are consistent then there is no actual friction in the
constructions. I would point out that polysign simply fits as nicely
with these in P3, where an equilateral triangle whose center is the
origin and whose vertices are labelled -1, +1, and *1 sufficiently
define the plane with all of its usual behaviors. Arithmetic products
are not a necessary part of the usual vector formulation so we can
keep things simple here by overlooking the product for this argument's
sake. Given the polysign construction I can now cast doubt upon the
cartesian product as a necessary construction and state very clearly
that the real number and its behaviors were enough to get the higher
dimensional space if those behaviors were properly generalized.
I do believe that this argument is coherent and so the polysign can
cast the stronger doubt upon the cartesian than the other way around.
That said upon witnessing the break in the polysign family at
P1 P2 P3
and seeing that these components compose a spacetime with
unidirectional time I am sort of stuck taking that construction as
P1 x P2 x P3
though here each of these blocks is uniquely constructed so that they
are different numerical types. Hence the need to declare their
independence is less strong since they are already inherently
different from one another.
I did recently write something down about independence and whether it
is an observed trait or a constructed trait. At some level aren't we
forced to accept that nothing is actually independent of anything
else? From an electrostatics type of analysis this physical argument
has support. Furthermore then what right do we have to declare axes
independent? The polysign has this neat cancellation effect whereby a
change in one component does effect the whole, so for instance in P3
if we move a point position by
+ 1.1
we can accept that changes in the other components can bring the value
back to where it was by tacking on say
- 0.5 * 0.5 - 0.5 * 0.5 - 0.1 * 0.1 .
Thus the clause of independence within the cartesian construction is
broken by the very principle that builds it's real number basis:
sum over s ( s x ) = 0 .
This is stated in the context of polysign. I understand that the
cartesian construction has been effective and useful but it may also
be misleading us at some fundamental level.
I have another criticism on the cartesian construction but shall not
exercise it yet for brevity has already been offended. I don't see any
substantial information from you. If your getting tired perhaps we
should quit. It's been a good volley.
- Tim
Mariano Suárez-Alvarez
> On Feb 9, 9:49 am, "victor_meldrew_...@yahoo.co.uk"
>
>
>
> <victor_meldrew_...@yahoo.co.uk> wrote:
> > On 9 Feb, 14:04, "Tim BandTech.com" <tttppp...@yahoo.com> wrote:
>
> > > On Feb 9, 4:38 am, "victor_meldrew_...@yahoo.co.uk"
>
> > > > Is the "simplex geometry" the decision that each s1
> > > > have unit distance from the origin and that each s1
> > > > is equidistant from the rest? That certainly does not
> > > > *not* follow just from the equation sum_s s x = 0.
>
> > > I would argue that in their simplest form without the added
> > > complication of scaling
>
> > Well there we are; in the second line of his reply Goldenballs
> > spouts the meaninless phrase "complication of scaling".
>
> > > How can you refute the simplex?
>
> > One can refute propositions. But a simplex is a geometric
> > figure not a proposition. How could one refute that?
>
> Well, you are casting doubt, let's say.
> I will argue that we can likewise cast doubt upon the cartesian
> dimensional construction of space, much as Riemann is suggesting is
> universally so.
You cannot possibly have misunderstood what Victor wrote.
"Refuting a simplex" makes as much sense as "Refuting an apple":
that is, none whatsoever. Your question "How can you refute the
simplex?" is, simply, devoid of any meaning,
At most, one can refute a claim involving a simplex. But
you have yet to state an actual claim.
> We do see challenges of Euclid's postulates in relativity theory don't
> we? Riemann's workings with spherical spaces, etc. leave us with an
> opening at very low levels whereby any math can be challenged.
>
> For instance the cartesian product is not necessary to compose the
> polysign numbers.
Unless you are also planning to revolutionize set theory, there is
no way for you to even construct the set of your adored polysign
numbers without the support of cartesian products.
> [yet another amazing display of verborrhea, elided]
-- m
victor_me...@yahoo.co.uk
> For instance the cartesian product is not necessary to compose the
> polysign numbers.
Your definition of polysign numbers, using n-tuples of real
numbers, uses Cartesian products.
> these arguments are consistent then there is no actual friction in the
"friction"?
> Arithmetic products
> are not a necessary part of the usual vector formulation
So you are agreed that this polysign multiplication is unnecessary.
> From an electrostatics type of analysis this physical argument
> has support.
Take this shite to sci.physics please.
> I have another criticism on the cartesian construction but shall not
> exercise it yet for brevity has already been offended.
You certainly take a lot of words to say nothing.
> If your getting tired
If my what??
Tim BandTech.com
My gravy you guys crack me up.
You are the ones cracking here.
Not me.
You are in complete denial of whay I say because of the bifurcation
that will take place in your mind.
Such is the nature of twenty or so years of schooling on your brains.
You are hard wired for the real number aren't you?
You have proven yourselves as bots of the system.
You have proven that human social behavior is so limiting.
I am not just sorry for you.
I am even sorry for me.
Oh well.
I have no need to communicate further with either of you since the
content has come to a close from your own side.
Through avoidance mechanisms you may claim blindness in these posts,
however they are marked here for history to read.
I do not wish to push you over the edge.
That simply will not happen.
I ask that you relax your tensions and return to your previous work
and forget that polysign exists.
As if it were a mirage.
As if spacetime is nonexistent.
The true mathematician likes calling his number real for imaginative
reasons.
To sway such people to the course of the physical world is an
aesthetic they've already rejected.
Sadly the many physicists have rejected the philosophical.
Sadly many philosophers have rejected the mathematical.
Let's see, did I get around the loop yet?
Anyway the signature three are there.
These three realms shall collide as one again.
I've already coined the term:
Hyperclassical Number Theory
to mean this very merge.
As in the times of old.
When one man could practice all three.
victor_me...@yahoo.co.uk
> On Feb 9, 1:59 pm, "victor_meldrew_...@yahoo.co.uk"
>
>
> > If my what??
>
> My gravy you guys crack me up.
Can't answer the question? My what exactly?
> I ask that you relax your tensions and return to your previous work
> and forget that polysign exists.
> As if it were a mirage.
OK, but it would help me to do that if you deleted your website.
> Let's see, did I get around the loop yet?
Dunno, but you're certainly round the bend!
lwa...@lausd.net
> On Feb 6, 3:16 am, "victor_meldrew_...@yahoo.co.uk"
> > We do know; at least we who are competent to have worked it out
> > do. It's easy to write down the distance function on R x C
> > corresponding to yours on P_4, viz. the square of your
> > distance corresponds to the map (x,z) |--> (1/3)(x^2 + 2|z|^2).
> > It's easy to do this for all P_n.
> I must assume that you are withholding information, and I do not mean
> to say that this is with any ill intent. I recommend you get right
> into it here as you see it and I'll try to follow along. If you have
> an isometric isomorphism from P4 to (R,C) then others will take
> interest in that. I'm surprised lwal hasn't spoken up yet here,
Ask and ye shall receive.
There's been so much discussion about this mysterious norm
based on the vertices of a tetrahedron, which Golden insists
exists, yet no one seems to believe him. So let me look up
this norm for myself.
The vertices of a regular tetrahedron centered at origin in R^3:
d0 = (+1, +1, +1);
d1 = (-1, -1, +1);
d2 = (-1, +1, -1);
d3 = (+1, -1, -1).
Each vertex is clearly sqrt(3) units from the origin and
2sqrt(2) units from each other.
So we can define the Golden norm of a four-signed number,
written in Mariano's notation:
ae0 + be1 + ce2 + de3
by normalizing:
e0 = d0/sqrt(3), e1 = d1/sqrt(3),
e2 = d2/sqrt(3), e3 = e3/sqrt(3)
and then taking the norm in R^3 of ae0 + be1 + ce2 + de3.
But then notice that we don't obtain e1e1 = e2, or some
of the other multiplication facts, by multiplying
coordinatewise in R^3 or RxC. In particular:
e0 = (+1,+1,+1)/sqrt(3)
would be interpreted as (1/sqrt(3), (1+i)/sqrt(3)), and
certainly multiplying this by any of the other ei's fails
to give any of the ei's.
I tried rotation by 45 degrees, but to no avail.
Golden won't accept P4 as being equivalent to RxC, unless
one can find elements in RxC e0, e1, e2, e3, such that
(ei)(ej) = e(i+j) or e(i+j-4) (as previously defined)
norm(ei-ej) = constant, for all i,j distinct
So far, every isomorphism fails to be an isometry, and
vice versa.
victor_me...@yahoo.co.uk
> But then notice that we don't obtain e1e1 = e2, or some
> of the other multiplication facts, by multiplying
> coordinatewise in R^3 or RxC.
You haven't been paying attention. Consider the
space R x C where we define the distance from the origin to
(x, z) to be the square root of (x^2 + 2|z|^2)/3.
In this space, there is a regular tetrahedron,
centred at the origin, whose vertices form
a cyclic group of order 4.
Exercise for lwally and Goldenballs:
find the appropriate generalization to the n-simplex.
Tim BandTech.com
Victor do you see the difference in the information that lwal has
presented?
He has formed a clean and simple transform between the two systems.
The rotational aspects of the reference frame you choose do matter.
As lwal has demonstrated, the particular rotational frame that he took
doesn't work out.
They can be made to have a better match, but it is still not perfect.
Your simplex challenge is poorly worded.
It seems you still have not come to view the simplex as a valid
coordinate system as it is used in polysign. It perhaps should be
painted more clearly on my website. But this level of reasoning was
back down at a very different level than your criticisms here. I have
assumed that you understand the simplex coordinate system. This thread
may be worth something if that is not the case.
The simplex coordinate system can be declared independent of polysign.
It covers general dimension such that an n-1 dimensional space is
built with the n-simplex. These relations are already within the
simplex construction and perhaps it would be good to go into that
layer. I will not do that here since other exposures exist at wiki and
wolfram.
By picking n to work with, say 4 since that has been the study we've
supposedly been working on we know then that the 4-simplex is a
tetrahedron in 3D space. Now we take this picture and modify it
slightly: The frame will be removed from the tetrahedron. Now with the
four naked vertices we average their being and they meld into a point
at the center of the tetrahedron. From this one point or origin we
draw out vectors to the vertices with outward pointing arrows. These
unit vectors coupled with magnitudinal values are capable of covering
the 3D space. They are a nonorthogonal coordinate system. They are
also redundant. This redundancy on first sight appears wasteful but it
is not. This perhaps then is the best explanation for why the simplex
coordinate systeme has been skipped over. But this statement isn't
quite true since there are quadrays, synergetic coordinates, and
barycentric coordinates.
The uniquity in polysign is not the existence of this generic
coordinate system. It is in the fact that the general n-signed numbers
which form an algebra exactly match this coordinate system in n.
victor_me...@yahoo.co.uk
'kin'ell Goldenballs, even your titles are logorhhoeic now :-(
> He has formed a clean and simple transform between the two systems.
Except he hasn't, failing to translate your multiplication
into his system.
> Your simplex challenge is poorly worded.
Pot. Kettle. Black.
> This thread may be worth something if that is not the case.
This thread is totally worthless.
> The frame will be removed from the tetrahedron.
What "frame"?
> The uniquity in polysig
You misspelt "iniquity".
Tim BandTech.com
Really Victor I thought you were a very intelligent man but now I'm
coming down to the level of talking with you as if you were a child-
or rather since that can be taken as an insult I do wish to correct it
with respectfulness. I think that the proper state to ponder polysign
from is that of a child. Perhaps a child who was just introduced to
sign two numbers a few days ago. Please place yourself as such a
brilliant child that thought of it himself. Sadly in hindsight the
move is obvious and the fact that it has been overlooked by the human
race does make a large statement. It does make a large statement
especially on the mathematics community, who has claimed that they are
covering the generalizations as completely as possible. The refusal to
consider such possibility can be taken many ways, but as the
popularizer of polysign I am caught with my own judgements on the
impedance you yourself impose upon yourself and you alone. here I have
gone to the trouble of granting you tremendous attention and all that
I see is a little boy pouting in refusal as if he was told Santa
Clause is a capitalistic fraud and that his entire shopping nation is
wrong. This little boy will eventually stop pouting and when he has I
do believe he will set up shop for himself. It is the only way to make
right what has happened. Now we have the internet and it allows me to
confront you as a random sampling of what I have to deal with. As I
have opened here it leaves me guessing as to the true nature of your
own impedance. This impedance however is a matter of human social
study and so I have a very simplistic proof that the mathematician
may not divorce himself from his human form. This then sends the
mathematician back to philosophy and of course physics is in the mix
too. Our egocentric mimical format is a bizarre twist of fate. We did
not choose it yet we are subject to it. In this most simplistic way
your own free will has been coopted by the greater form. This
stability is no doubt of importance in survival so its results here
are as much an act of preservation I believe as they are of impedance.
Thus are the religions born. This is how deep the schism lays. That we
should declare math a religion might be its proper place. Your own
behaviors here show the limited form of the human and to build a
system which matches that form is the only proper design if influence
is to be had over humans.
Of course you might open your blinders and enter into the Socratic
method a layer deeper.
I honestly have no idea how intelligent a man you are given your
current behavior.
On the one hand I want to treat you as authentic and in the spirit
that I treated you at the opening of this volley. On the other hand I
know nothing about you and you did basically start spouting up where
MsaNan left off, there being somewhat merely a shadow of MsaNan's
image in your posts, though you did claim to read my site carefully...
Whatever- the interaction here is but one of many. Some onlooker can
get a better guage for the content than I myself for I have been
dramatically involved in it. I own that much of this drama has been
fabricated yet I also do feel sincere in its underpinnings. But these
are open problems. It may be that ultimately the human judgement of
acceptance operates upon a continuum and that because we have a very
large quantity of ideas flitting about through our brains that these
continuous acceptance filters are necessary. The correlations that are
strengthened are mimiced and we come to a common ground. At the very
least each word that I use here is common. I do not feel compelled to
seek mimicry of myself. I believe that every human with modest
mathematical ability will come to construct the polysign numbers on
their own, given persistence and a curious mind. This level of mimicry
I can go with. Hah- the space is open for further construction
unconditionally and forever and polysign is a shining instance in this
moment.
- Tim
victor_me...@yahoo.co.uk
> Really Victor I thought you were a very intelligent man but now I'm
> coming down to the level of talking with you as if you were a child
Down to your level?
> but as the popularizer of polysign
I see no evidence of popularity.
<rest of 37-line paragraph devoid of mathematical content snipped>
> though you did claim to read my site carefully...
Ooops! for a moment I thought you wrote "read my shite carefully" :-)
Tim BandTech.com
If I can give one small victory I'm happy to do so.
I give you the last word here.
You did come back yesterday with a challenge to the simplex geometry.
I assume that I have allayed that problem which was your focus since
no futher information has transacted here. Again as yesterday I do
wish that you will relieve yourself.
You may now relieve yourself fully here but please do not ask any
question, for then I will assume that you want to carry on. Please
make no expectation on my part in terms of the mathematics as I make
none on you. I have no desire to hold onto this communication and mean
you no harm either now or in the future, though as collateral damage
either of us is fair game. The math on which I stand is a solid
footing and its peculiar nature is fascinating beyond images of R and
C forever kaleidoscopic. I can accept this truth though on a continuum
I have argued it to be only a half truth. I would leave it there
rather than use it as a crux for some pivotal principle. Certainly
multiple representations exist and in that every representation is
capable of constructing a theory then the playing field is even and
free for all such constructions, whether they be R^a, R^bxC^c,
R^bxR^2^c, polysign, or any other system of representation.
The comparative properties of these representations or constructions
are a bit beyond simple logic.
I have been criticized here for giving these life and quality beyond
what most mathematicians will.
I don't feel that I am wrong. I am just a human who has perhaps
personified the math so as to make friends with it rather than battle
its impenetrable laws. So I have found a workaround. As they built the
wall up it was staggered at one end and so I could go there and simply
and gradually come up along its height. I just had to follow it along
long enough to come to this end. It happened that nobody had come to
this stair before in a long time. But I feel comfortable that some
have been here in the past. They have been forgotten, or perhaps we
await their electromagnetic image to arrive from spacetime. This is
how profound the codex of polysign could be. I suppose this signalling
will play out in constellation style QAM with various encryptions but
these things are just novelties. Secrecy has never been a problem.
Openness is a superior focus. From this openness paradigm I do feel
that I have obeyed by it here and throughout this thread, though I
have also injected a bit of humor, some of which is slightly crass.
You've been a good sport Victor. Cheers!
- Tim
victor_me...@yahoo.co.uk
> You did come back yesterday with a challenge to the simplex geometry.
I wasn't challenging anything.
> I do wish that you will relieve yourself.
Thanks, I'm bursting.
> will play out in constellation style QAM with various encryptions but
What??
> You've been a good sport Victor. Cheers!
Alas, I seem to have failed. I hoped to encourage you to
study some algebra, but your resistance to learning remains
implacable.
amy666
> wrote:
> > mariano wrote :
> >
> >
> >
> > > On Feb 5, 9:21 pm, amy666 <tommy1...@hotmail.com>
> > > wrote:
> > > > mariano , how many numbers are there in 3d ?
> >
> > > > isomorphic to each other ?
> >
> > > > is your opinion :
> >
> > > > 1) R x C = P4
> >
> > > > and
> >
> > > > 2) R x R x R
> >
> > > > and no other.
> >
> > > > yes or no ?
> >
> > > Those are (up to isomorphism) the only two
> > > reduced commutative 3-dimensional real algebras.
> >
> > good then we agree upon that.
>
> Notice that I did not agree to anything: I simply
> can't,
> for I have no idea what you mean by "numbers in 3d".
>
> -- m
we mean the same.