CRITICAL TORUS !!!

[tommy1729]

Dear mathfriends (?)

In observing math , ive noticed , and im not alone, that many (proof)problems in math (especially the harder ones ) can be transformed into some kind of " critical line " problem.

The most famous example is of course Riemann's Hypothese.

But also the tanc conjecture can be rewritten as a critical line problem.

Representions theorems can be rewritten as a critical line problem.

For all clarity , i consider "critical line problems" as follows

1) a subgroup of zeros all lie on the critical line

2) none of the zeros of a subgroep lie on the critical line

3) no 3 zeros lie on a line (probably reducable to 1) or 2) so less important i guess )

Every upperbound or lowerbound conjecture is equivalent to some critical line problem

You can just literally bend functions to get a critical line , if ya already have a simple critical "function/figure/path"

SO if ya have a "critical monotonic increasing polynoom " like x^a , you can easily transform it to a line , bye transforming ( bending ) the function

And don't forget that e.g. circles are the product of 2 complex lines.

So critical problems in 2D are often easily transformable into "critical line problems in 2D"

So the "2D equivalent problems " are mainly and often simply reducable to critical lines

A bit like belonging to an NP class to make a comparison.( but for proofs instead of computations )

However !!!

some problems might not be expressible in 2D criticals , but in 3D criticals

with the concept 3D being most general !

so for instance allowing "polysigned" expressions (enter in google if you dont know what polysigned is , there is a whole website explaining the basics )

i mentioned before transforming polynoom to a line

but in 3D we have topology joining the party;

we could have a " critical torus "

or " polysigned critical torus "

or even more complicated stuff , not easily , or not at all , "transformable to a line , or even 2D disk"

we all know topology is strongly related to number theory
and metamathematics is getting popular

i hope ive made clear why i believe so strongly in this "critical theory " importance.
consider it a bit like NP=P , extended Riemann Hypothese and metamathematics.

so i am deeply intrested in the "critical torus" since it is relating so much math concepts.

and it relates them in only 2 words, nice :-)

i do not claim to be a genius or anything , but as far as i know , this has never been studied.

and as an intuitionist and constructivist , this is very appealing to me.

i could continue with more than 3 dimensions , but i think that we should start with the simplest extension ( 2D --> 3D ; holes 0 --> +1 )

so does anybody know functions that have or might have a critical torus ?

also this appears to me as a nice puzzle

would be nice to see this online , e.g. at websites and forums

this mail is not complete in the sence that the critical torus can be considered in different ways;

do the zeros lie on the surface of the torus , or in the torus ?

the most logical and intrestresting is according to me the critical surface of a torus , since other wise the condition is very loose and its more of a generalization to a strip , not a line.

there is also probably a connection with eigenvalues and cellular automata , but thats confusing me at the moment.

hope you think about it.

i find it a logical question anyway..

greetz

tommy1729

ps : id rather not have james harris replying :-)

[tommy1729]

cmon i do deserve more attention then james factoring i hope ??!

[Lee Rudolph]

tommy1729 <tomm...@gmail.com> writes:

>Dear mathfriends (?)
...

>And don't forget that e.g. circles are the product of 2 complex lines.

What???

Lee Rudolph

[jhnr...@yahoo.co.uk]

tommy1729 wrote:
>
> Dear mathfriends (?)
>
> In observing math , ive noticed , and im not alone, that many (proof)problems
> in math (especially the harder ones ) can be transformed into some kind of
> " critical line " problem.
>

> [..]

You mention circles later on, and sometimes these "critical line"
problems
relate to circles or regions of convergence, i.e. a function expressed
by,
say, a power series converges inside and possibly on the boundary
but not outside (although analytic functions expressed by continued
fractions can have fractal regions of convergence).

With the Riemann zeta function the critical line relates not to
convergence
but a kind of mirror symmetry. There's actually an identity which
relates
zeta(z) to zeta(1 - z), although that of itself doesn't imply that
roots in
the critical strip must be on the critical line - Pending a proof of
the
Riemann hypothesis, there's nothing to rule out roots occurring in
pairs either side of the critical line.

BTW the Riemann Mapping theorem states that any simply connected
region other than the entire complex plane can be conformally mapped
to any other region, such as a circle. But unless the mapping preserves
some relevant aspect or property of the original function, it's hard to
see
what use it would be to line up roots on, say, a circle.

Well hopefully the above waffle prompts a few further replies ;-P

[Timothy Golden BandTechnology.com]

tommy1729 wrote:
> Dear mathfriends (?)
>
> In observing math , ive noticed , and im not alone, that many (proof)problems in math (especially the harder ones ) can be transformed into some kind of " critical line " problem.
>
> The most famous example is of course Riemann's Hypothese.
>
> But also the tanc conjecture can be rewritten as a critical line problem.
>
> Representions theorems can be rewritten as a critical line problem.
>
>
> For all clarity , i consider "critical line problems" as follows
>
> 1) a subgroup of zeros all lie on the critical line

I'm getting this problem in odd dimensions only so far for polysign
math.
There seems to be a parity behavior going on between dimensions.
I call this critical line the identity axis. It seems to remain a line
even in high dimension. This may be an incomplete view. Perhaps it
exists for even dims and I haven't found it yet. Perhaps it is bigger
than a line in high dims but has not been discovered. So far though
there is agreement with what you are saying.

>
> 2) none of the zeros of a subgroep lie on the critical line

I pass on this one.

>
> 3) no 3 zeros lie on a line (probably reducable to 1) or 2) so less important i guess )
>

In polysign all of the zeros do lie on a line. They are all on the
identity axis, including zero. But there is still the option of some
other zero-like values out there until proven otherwise.

> Every upperbound or lowerbound conjecture is equivalent to some critical line problem
>
> You can just literally bend functions to get a critical line , if ya already have a simple critical "function/figure/path"
>
> SO if ya have a "critical monotonic increasing polynoom " like x^a , you can easily transform it to a line , bye transforming ( bending ) the function
>
> And don't forget that e.g. circles are the product of 2 complex lines.
>
> So critical problems in 2D are often easily transformable into "critical line problems in 2D"
>
> So the "2D equivalent problems " are mainly and often simply reducable to critical lines
>
> A bit like belonging to an NP class to make a comparison.( but for proofs instead of computations )
>
> However !!!
>
> some problems might not be expressible in 2D criticals , but in 3D criticals
>
> with the concept 3D being most general !
>
> so for instance allowing "polysigned" expressions (enter in google if you dont know what polysigned is , there is a whole website explaining the basics )

Thanks for the plug. Here is a link:
http://bandtechnology.com/PolySigned/PolySigned.html

>
> i mentioned before transforming polynoom to a line
>
> but in 3D we have topology joining the party;
>
> we could have a " critical torus "
>
> or " polysigned critical torus "

The polysign numbers generate spacetime as:
0D + 1D + 2D ...
where the ... are ill-behaved higher dims.
As you speak of a 'critical torus' I am forced to look at spacetime as
literally a plane combined with a line, the plane being rotational in
nature. The congruence is clear. If I were to assign a discrete shape
to this concept it would be a little prism type of thing. It can be
generated by looking at steps of -1^n in each component of the
topology. This will trace out a prism via diagonals. The structure
repeats itself :
P2 P3
-----------
- -
+ +
- *
+ -
- +
+ *
- -
... ...
This concept is extensible to higher dimensions but will go from a six
vertice structure in 3D to a 12 vertice in 6D. Below 3D is just 1D
where this thing is just a segment.
If you try to classify these things there are some possibilities of not
just left and right handed but also of offsets due to the harmonics of
for example P2 and P4. I haven't gone down this path yet but it looks
like it might be worth it sometime. Especially because of what you are
communicating.

> or even more complicated stuff , not easily , or not at all , "transformable to a line , or even 2D disk"
>
> we all know topology is strongly related to number theory
> and metamathematics is getting popular
>
> i hope ive made clear why i believe so strongly in this "critical theory " importance.
> consider it a bit like NP=P , extended Riemann Hypothese and metamathematics.
>
>
> so i am deeply intrested in the "critical torus" since it is relating so much math concepts.
>
> and it relates them in only 2 words, nice :-)
>
> i do not claim to be a genius or anything , but as far as i know , this has never been studied.
>
> and as an intuitionist and constructivist , this is very appealing to me.
>
> i could continue with more than 3 dimensions , but i think that we should start with the simplest extension ( 2D --> 3D ; holes 0 --> +1 )
>
> so does anybody know functions that have or might have a critical torus ?
>
> also this appears to me as a nice puzzle
>
> would be nice to see this online , e.g. at websites and forums
>
>
> this mail is not complete in the sence that the critical torus can be considered in different ways;
>
> do the zeros lie on the surface of the torus , or in the torus ?
>
> the most logical and intrestresting is according to me the critical surface of a torus , since other wise the condition is very loose and its more of a generalization to a strip , not a line.
>
> there is also probably a connection with eigenvalues and cellular automata , but thats confusing me at the moment.

This is probably what I am describing above. The polysign lattice is
radically different from traditional geometry in that its segments are
directed. One is not free to travel a reverse path on the lattice. To
get back to where one started one must travel equally in every
direction. Thus a loop becomes a critical path which forms signons:
http://bandtechnology.com/PolySigned/Lattice/Lattice.html
You are speaking in general terms. This matter may be confused by the
supposed topology:
0D + 1D + 2D ...
whereby each component of the topology is itself a whole dimension yet
the resultant space skips dimensions. Strict 2D is never encountered on
this topology. Yet it is a component of the topology.

-Tim

[tommy1729]

with all respect , but i believe you only partially understand.

you underestimate the importance of the critical line

its much more important then the functional equation (well perhaps not for zeta but in general )

i know about zeta function and its functional equations
heck i even conjectured the critical line before i even heard of complex numbers or riemann.
and indeed the zero's of zeta even remain on their place , even if the area around it has no convergence
(this is actually how i found the zeros , ive noticed peaks in 1/zeta to infinity where it should be 0.)

the symmetrie you mention is evident !! but why do you mention it ??? i know it. i know it for years !!

but i dont see what is the relationschip with what i said , let stand alone what the problem is ??

besides convergeance is not a main topic in my introduction ?? i dont see an important connection ?
perhaps you misunderstood that.

also it would be intresting to consider other functions then zeta , with critical lines. (like ramanujan did eg)

or other critical "stuff"

and to reply on you riemann mapping theory ;

that is quite equivalent to me saying , " the functions can be bent to a line " to quote myself more or less.

so i knew that too and believe mentioned it briefly.

btw does the riemann mapping theorem apply for changing a critical torus in polysigned 3D , to a 2D critical line ???

and further the relationship : number theory -> topology -> "critical" is very intresting.

also very notable is that EVERY conjecture CAN BE transformed in a critical problem.

and further" polysigned " could help the construction of the torus , i believe ( polysigned functional equations -> transform to complex(4d) -> 3d slice -> critical torus e.g.)

even partially recursive functions can have a critical (hypercube like functions )

but thanks for replying anyway :-)

greetz

tommy1729

[tommy1729]

thanks for replying Timothy

your role of polysigned numbers is indead mentioned bye me
but perhaps you are overestimating its role in a few ways.

maybe you should lookup the zeta Riemann conjecture and the tanc conjecture , to get a better idea of what i mean.
i know i mentioned polysigned , but its just a part , not the main thing.
its actually all over math.

my critical torus is related to all current math !!
therefore perhaps i have the impression that people only understand up to the level or part they do math
.
in higher dimensions perhaps very important.

also i consider extensions to polysigned ( actually bye thinking about fysics) lets say P3*3, meaning a P3 but with values varying in another P3.

ok this might sound strange even for you.

remember you said P3 does not need a negative for each variable ?

well suppose it did. that would give -x+x*x=0 and minus(*x)=-x+x , where both could exist , and the full equation becomes minus(-x+x*x)=-x+x*x which is actually minus 0 = 0 or simply 0 = 0 .
i would call that P3*2.
extending the variables 3signed instead of 2 , gives P3*3
which is a very powerfull and intuitive model.
( probably equivalent to a bicomplex i guess )

i will reply to your comment chronological now;
well its a pity you only got it for polymath part...
and to be honest i actually believe you have it partially wrong. my critical line is more general then yours, to say. (this reminds me of riemann's mapping theorem in the other post ..), to get the bigger picture ( which might be neccesary to understand why it is "all over math " is bye reading about the conjectures , like i mentioned above ( zeta , tanc eg) you can find them on the internet.
however i understand if ya more into the polysigned stuff.

..it is probably bigger yes .. at first thought..

"in polysign all the zeros lie on a line " ??
the problem here is they lie on "all" the lines , if im not mistaken.

following +x-x*x=0 the zeros lie on the + - * lines , so not just one line.
like i said before
my line is more general then yours.
ive noticed its pretty hard bending your 3 lines into one of mine... but since P3 is equivalent to complex it must be possible no doubt.
..a proof of (non)zero-like values cannot be so hard...

.agree at first sight mainly with the "fysics"..
although a bit messy looking i find.

greetz

tommy1729

oh ! before i forget P3 could be considered a more natural way of complex , whereas Pa*b*c....(a,b,..=2 or 3) can be considered a more natural way of matrices.

where the size of the matrix depends mainly on the number of "splitups" eg P3*3 --> 2 splitups
P3*3*2 --> 3 splitups ( so mainly number of *'s )

also have you investigated P7 ? it has nice properties , possible related to deep algebra. and perhaps P11 to geometry.

ive been thinking about Polysigned , could you addapt the concept a little more so that abs(a)abs(b)/abs(ab) = 1 always ? and multiplication remains commutativ?

i dont know if ya familiar with contour integration.

but im beginning to believe that all problems are also equivalent to some contour integration of some function defined over polysigned.
however dont get a smile yet.
this is merely equivalent if its correct at all , to my critical line.

in other words , your n-lines can be transformed to my one line , probably , mostly bye contour integration over a polysigned field ( usually of a different degree ) of a specific function , determinded bye the 2 polysigned , its beginning value , and a specific multivalued function, assuming there are infinite zeros.

it could even yield a solution to the Riemann Hypothese

further to relate polysigned to physics , you need n! rules i believe , where n is the number of signs.
that is quite a lot , and might be attacked upon bye skeptics as being "elefant fitting"
you might not understand that last at the moment , but you probably will later on.

greetz
tommy

ps this text was also ment for other people , although i meanly discussed polysigned of course

[Chip Eastham]

Okay, how many lines in the complex plane does
it take to cut a log branch?? [Two. One separates
the leaf bundles and the other forms a quality circle
around the singularity.]

But maybe the OP has something like this in mind:

X^2 + Y^2 = R^2

(x + iy)(x - iy) = r^2

See, the product of 2 complex lines!

[Yes, it's late. I'm a little punchy tonite.
Thank goodness I hit delete before I posted
this to sci.math...]

regards, chip

[Gene Ward Smith]

tommy1729 wrote:

> so for instance allowing "polysigned" expressions (enter in google if you dont know what polysigned is , there is a whole website explaining the basics )

I can'r see it has any of the basics! I'm basing that on the postings
about them here.

[Timothy Golden BandTechnology.com]

thanks for replying to my reply Tommy.
My ability to converse in the areas you are discussing is limited.
Some of the stuff that I have just read about the Zeta function is that
it is a devine marriage between summation and product. I have no idea
how they see that but it sounds good to me. You have my ear.
You say:

> but in 3D we have topology joining the party;

Why the distinction at 3D?
I am not familiar with the line transformations that you are
considering. It seems somewhat like an inverse and is neat in that time
seems to be a directed line with almost no degree of freedom. The
ability to transform back to this vanishing line would be interesting.
A dimensional reduction should be possible. In effect these are the
same things. I don't really see how the torus comes to be a part of
that structure for you but that will be explained in answer to the 3D
distinction above hopefully.

tommy1729 wrote:
> thanks for replying Timothy
>
> your role of polysigned numbers is indead mentioned bye me
> but perhaps you are overestimating its role in a few ways.
>
> maybe you should lookup the zeta Riemann conjecture and the tanc conjecture , to get a better idea of what i mean.
> i know i mentioned polysigned , but its just a part , not the main thing.
> its actually all over math.
>

The extended Riemann Hypothesis:
The first quadratic nonresidue mod p of a number is always less than
3(ln(p))^2/2 .
I find nothing on the 'tanc conjecure' except for your post so I need
more info for that one.
And I have read more on the Zeta function as well but this particular
focus is not to do with it right?
Since the polysign system is a modulo sign system it immediately looks
good. The modulo math is product related and so the ln() etc looks good
too. There is an interesting relation between polysign numbers and
Cartesian as one goes up in dimension. The two systems appear to
approach each other in terms of angle. Analysis of random numbers and
the ratio of the sum of the squares of the polysign components versus
the standard geometrical distance yields values approaching 0.5:
+990 : 0.483922
+991 : 0.48208
+992 : 0.497595
+993 : 0.515116
+994 : 0.495077
+995 : 0.498777
+996 : 0.495015
+997 : 0.508892
+998 : 0.498496
+999 : 0.488329
These are just the last of a listing of 1000 values. The last one
listed is a 999-signed value. The ratio is the Cartesian distance over
the 'polysign distance' which is simply the sum of the squares of the
polysign components in their native but reduced form.
I do not usually speak of the polysign distance generally since it is
not the standard notion of distance and only takes a constant value in
the reduced form. The source vectors for the comparison are random unit
vectors that are reduced in the polysign domain. I don't claim to
understand why these values are so well behaved. perhaps it is simply
that there are 900+ random components in each so there is a balance
being formed by the high dimension. This may have absolutely no value
to the discussion but I mention it since you may see something in it
that I do not. The notion 'polysign distance' may be ambiguous but the
relationship here is pretty clear. The ratios are not averages over
many instances. Just one single random 998-signed value generated
0.498496. Sorry if this is a long sidetrack. For efficiency sake please
delete it if not relevant.

> my critical torus is related to all current math !!
> therefore perhaps i have the impression that people only understand up to the level or part they do math
> .
> in higher dimensions perhaps very important.
>
> also i consider extensions to polysigned ( actually bye thinking about fysics) lets say P3*3, meaning a P3 but with values varying in another P3.
>
> ok this might sound strange even for you.

Yes. I have enough trouble at the bottom level. You may be right and I
am open to going there.

>
> remember you said P3 does not need a negative for each variable ?
>
> well suppose it did. that would give -x+x*x=0 and minus(*x)=-x+x , where both could exist , and the full equation becomes minus(-x+x*x)=-x+x*x which is actually minus 0 = 0 or simply 0 = 0 .
> i would call that P3*2.

Perhaps the notation P3(P2) for this space. I believe this will simply
reduce back to P3.
P3(P3) may also reduce back but that may depend on the details of your
construction.
Are you implying a Cartesian product? if so then we have to simply
grant the added dimensionality and let the parts be independent. Anyhow
if this is too much sidetrack we should ignore it for now.

> extending the variables 3signed instead of 2 , gives P3*3
> which is a very powerfull and intuitive model.
> ( probably equivalent to a bicomplex i guess )
>
> i will reply to your comment chronological now;
> well its a pity you only got it for polymath part...
> and to be honest i actually believe you have it partially wrong. my critical line is more general then yours, to say. (this reminds me of riemann's mapping theorem in the other post ..), to get the bigger picture ( which might be neccesary to understand why it is "all over math " is bye reading about the conjectures , like i mentioned above ( zeta , tanc eg) you can find them on the internet.
> however i understand if ya more into the polysigned stuff.

No, this is good. I can do some translation and keep things from
getting to polycentric. I don't mean to rob your thread of its thread.
The best means of discourse is here. ACK, SYN, ACK.
By challenging one another we develop a common language whereby
information flows. Whether one holds up to the challenges is a measure
of how much information we have to transfer, persistence aside. Your
tangential approach may be able to bring together what seem to be
divisive entities under the net you cast. If so then let's go.

>
> ..it is probably bigger yes .. at first thought..
>
> "in polysign all the zeros lie on a line " ??
> the problem here is they lie on "all" the lines , if im not mistaken.
>
> following +x-x*x=0 the zeros lie on the + - * lines , so not just one line.
> like i said before
> my line is more general then yours.
> ive noticed its pretty hard bending your 3 lines into one of mine... but since P3 is equivalent to complex it must be possible no doubt.
> ..a proof of (non)zero-like values cannot be so hard...
>
> .agree at first sight mainly with the "fysics"..
> although a bit messy looking i find.
>
> greetz
>
> tommy1729
>
> oh ! before i forget P3 could be considered a more natural way of complex , whereas Pa*b*c....(a,b,..=2 or 3) can be considered a more natural way of matrices.
>
> where the size of the matrix depends mainly on the number of "splitups" eg P3*3 --> 2 splitups
> P3*3*2 --> 3 splitups ( so mainly number of *'s )
>
> also have you investigated P7 ? it has nice properties , possible related to deep algebra. and perhaps P11 to geometry.
>
> ive been thinking about Polysigned , could you addapt the concept a little more so that abs(a)abs(b)/abs(ab) = 1 always ? and multiplication remains commutativ?
>
> i dont know if ya familiar with contour integration.
>
> but im beginning to believe that all problems are also equivalent to some contour integration of some function defined over polysigned.
> however dont get a smile yet.
> this is merely equivalent if its correct at all , to my critical line.
>
> in other words , your n-lines can be transformed to my one line , probably , mostly bye contour integration over a polysigned field ( usually of a different degree ) of a specific function , determinded bye the 2 polysigned , its beginning value , and a specific multivalued function, assuming there are infinite zeros.

I'm having trouble doing calculus in the polysign domain. Just the same
differentials seem to be clean. I haven't gotten very far with with
either yet.

>
> it could even yield a solution to the Riemann Hypothese
>
> further to relate polysigned to physics , you need n! rules i believe , where n is the number of signs.

I don't see this because I try the simplest approach. Perhaps n scales,
but you will have to instantiate these n! rules for say P3 in order for
me to get what you are saying.

> that is quite a lot , and might be attacked upon bye skeptics as being "elefant fitting"
> you might not understand that last at the moment , but you probably will later on.

Skeptical mice can stir a tribe of elephants.

>
> greetz
> tommy
>
> ps this text was also ment for other people , although i meanly discussed polysigned of course

Back to the line if you wish. I'm wondering if zero-dimensional time
will come out of your approach. The final topology will ideally be:

0D + 1D + 2D ...

where your line approach takes us through this topology. It need not
look directly like spacetime. Just flatland is OK. A product should
hopefully make itself apparent.
This topology can be formatted:
a11
a21 a22
a31 a32 a33
...
where a are magnitudes. I call this a tatrix for triangular matrix. For
instance through a3 this would be T3. You may be looking at T3 as 6D
(e.g. your zero analysis) while its rendered self is 3D. How one
interperets this conflict is an open problem.

-Tim

[Timothy Golden BandTechnology.com]

I'd be happy to help you understand them. Any shortcomings of my
website should be exposed so that I can improve it. Please be critical
by all means! That being said perhaps it is the case that the concepts
are much simpler than you assume. Then in your mind an undefined is
opined that makes one grind to a rind what could be a find.

But grind away please. I get to learn in the process as much as you. My
website does assume magnitude to be a fundamental concept that may be
defined elsewhere without the use of the real numbers. I assure you
there are no tricks or redefines. My use of the words dimension and
distance and magnitude are their usual sense. Upon considering what a
three-signed number system is I posit that you will arrive at the same
result and that therefor the construction is trivial.

-Tim

[tommy1729]

thanks for the clarification chip :-)
i mean both ways , but as you point out , espacially the second. I thought seeing a circle as 2 complex lines was classical knowledge , which needed no further explanation.

Its in almost every book that gives historical backgrounds on how math people from the past "viewed" circles. Its even the basic of some subfield of math.

no offense Rudolf , but i was just amazed.

if ya want ( i use ya often instead of you , finding it cooler :p ) you can also use riemann mapping theory to get from a line to a circle. as a nice alternative point of view.

sorry for my bad english , but chip what do you mean bye "the OP" ? dont know what that means :(

besides funny reply , im afraid you posted it afteral :p

greetz

tommy1729

[Gene Ward Smith]

Timothy Golden BandTechnology.com wrote:
> Gene Ward Smith wrote:
> > tommy1729 wrote:
> >
> > > so for instance allowing "polysigned" expressions (enter in google if you dont know what polysigned is , there is a whole website explaining the basics )
> >
> > I can'r see it has any of the basics! I'm basing that on the postings
> > about them here.
>
> I'd be happy to help you understand them. Any shortcomings of my
> website should be exposed so that I can improve it.

It seems to me the basic facts about polysigned numbers are these:

(1) The polysigned numbers, for each positive integer n, are a real
n-dimensional commutative algebra with no nilpotent elements

(2) The polysigned numbers may be described as R[Cn], where n is the
cyclic group mod n. Hence they have a basis r[0], r[1], ... , r[n-1]
where r[i]*r[j] = r[i+j], where the indicies add mod n.

(3) The polysigned numbers may also be described as the algebra of
real-valued functions on Cn, where the product is the convolution
product.

(4) If we extend the polysigned numbers to the corresponding complex
algebra, we may apply the finite Fourier transform, converting the
convolution product into a pointwise product.

(5) We may describe the above situation in terms of orthogonal
idempotent elements
e[i] = (r[0]+r[1]w^i + r[2]w^(2i) ... + r[n-1]w^(n-1))/n, where w is a
primative n-th root of unity.

(6) We may get orthogonal idempotent elements for R[Cn], ie the
polysigned numbers, by adding e[i]+e[n-i] when i is not 0 or n/2, and
using e[i] when i is 0 or n/2. Then the polysigned numbers break up
into one (if n is odd) or two (if n is even) copies of R, and (n-1)/2
(if n is odd) or (n-2)/2 (if n is even) copies of R.

These look like the basic facts to me from a purely mathematical point
of view, and I'll be happy to leave the applications to someone else.

[Timothy Golden BandTechnology.com]

Gene Ward Smith wrote:
> Timothy Golden BandTechnology.com wrote:
> > Gene Ward Smith wrote:
> > > tommy1729 wrote:
> > >
> > > > so for instance allowing "polysigned" expressions (enter in google if you dont know what polysigned is , there is a whole website explaining the basics )
> > >
> > > I can'r see it has any of the basics! I'm basing that on the postings
> > > about them here.
> >
> > I'd be happy to help you understand them. Any shortcomings of my
> > website should be exposed so that I can improve it.
>
> It seems to me the basic facts about polysigned numbers are these:
>
> (1) The polysigned numbers, for each positive integer n, are a real
> n-dimensional commutative algebra with no nilpotent elements

I disagree with 1) .
The n-signed numbers are n-1 dimensional.
This is embodied by the identity law. It requires that equal amounts in
each sign direction yields zero. The general law can be stated as:
Sum over s from 1 to n of ( s x ) = 0
where s is a sign and x is a magnitude. In P3 this yields:
- x + x * x = 0.
It's general and begets the geometry of the polysigned numbers. It is
an application of symmetry that allows the generalization of sign. The
simplex coordinate system and its nonorthogonal geometry is directly
implied by this simple equation, as is the n-1 dimensional consequence.
You have either obfuscated or obliterated this law in your rendition.
I wish you would comment on this since it has come up twice now with
you and has still not been addressed. The other time was a P3 product
such as:
( 1, 1, 1 )( 2, 3, 1 ) = 0
which is indeed zero but it is consistently zero because the first term
is zero by definition.
For instance the second term in the above equation is equivalent to:
( 1, 2, 0 )
which is called the reduced form. I would generally write such values
as:
* 1 - 2 + 0 = * 2 - 3 + 1 .
The arithmetic nature of the system is no different than the real
numbers which are the same as P2. P3 is 2D and is exactly equivalent to
the complex numbers under product and sum.
I would like to understand why the information is not passing clearly.
I'm afraid that It's the old real number mantra creeping in. Still even
if you insist on using the reals the system will work out. But the
ambiguity of such a construction is questionable. You are unwilling to
grant the notion of three signs aren't you? It sounds like a farce to
you. Well, two signs are the same farce so I'm just upping the ante.
Having overcome the farce of two signs by burning them in there is
inherently a conflict on top of a conflict with building three signs on
top of two signs. Whether it is humans using these signs or nature
using them as discrete entities tied to continuous ones is a paradigm
to ponder. We know that the complex numbers crop up regularly in
engineering,physics, and nature. Here is a construction that gets them
simply and naturally along with spacetime as well. I can see the
predicament you are in and to withhold direct judgement may be wise.
The indirect approach here is not accurate so I hope you can correct
it. I see lots of advanced thinking below that I don't understand. I am
interested in the Fourier statement. Mostly I've learned signal
analysis with it rather than general dimensional math problems.
Treating just the product as you are I would point you to Roger
Beresford's terplex algebra which should be nearly identical to the
construction you are using. But please do not confuse the two. Without
the identity relation there are no polysigned numbers.

-Tim

[Timothy Golden BandTechnology.com]

This should say the square root of the sum of the squares.
The polysigned distance is not necessarily legitimate but at high sign
the angles are becoming perpendicular so that it starts to make sense.
Also when I state that these are unit vectors that means that their
standard distance (i.e. the Cartesian form ) is unity. If we had a tape
measure in 989 dimensional space we would measure a length of one for
the first in this list. Yet the sqrt of the the sum of the squares in
polysign will yield 0.484 for this particular random unit vector, even
though the two seem nearly identical. It is a statement on the angle
discrepancy. That angle discrepancy adds up over 900+ times. So this is
not necessarily a deep thing. It is rather a sort of limit theory type
of behavior. There is no product operation involved in this behavior.
To answer why 1/2 is the puzzle and it may just be that it is a matter
of random values as an n+1 side-effect. I had guessed that the value
would approach unity so was surprised by this result. Sorry to belabor
this side-track. Just wanted to correct that missing sqrt.

-Tim

[Gene Ward Smith]

Timothy Golden BandTechnology.com wrote:
>
> I disagree with 1) .
> The n-signed numbers are n-1 dimensional.
> This is embodied by the identity law. It requires that equal amounts in
> each sign direction yields zero. The general law can be stated as:
> Sum over s from 1 to n of ( s x ) = 0
> where s is a sign and x is a magnitude. In P3 this yields:
> - x + x * x = 0.

OK; I was taking my ideas from your postings, which gave a
multiplication law. This adds another wrinkle to the definition, which
removes the value from the idempotent
(r[0] + r[1] + ... +r[n-1])/n from consideration. You now end up with
(n-1)/2 copies of the complex numbers when n is odd, and (n-2)/2
copies, plus one copy of the reals, when n is even. You can describe
this as R[Cn]/(e0), where e0 = r[0] + ... + r[n-1].

> It's general and begets the geometry of the polysigned numbers. It is
> an application of symmetry that allows the generalization of sign.

I'm not sure what you are saying, but this allows you to write every
polysign in terms of positive reals only.

The
> simplex coordinate system and its nonorthogonal geometry is directly
> implied by this simple equation, as is the n-1 dimensional consequence.
> You have either obfuscated or obliterated this law in your rendition.
> I wish you would comment on this since it has come up twice now with
> you and has still not been addressed. The other time was a P3 product
> such as:
> ( 1, 1, 1 )( 2, 3, 1 ) = 0
> which is indeed zero but it is consistently zero because the first term
> is zero by definition.

If you are hoping to have rid yourself of zero divisors by this means,
in general you haven't. The 3-polysigns are now the complex numbers,
and have no zero divisors, but the higher dimensions do have zero
divisors.

> The arithmetic nature of the system is no different than the real
> numbers which are the same as P2. P3 is 2D and is exactly equivalent to
> the complex numbers under product and sum.

Exactly. P2 is R, P3 is C, but for n>3 Pn is not a field.

> I would like to understand why the information is not passing clearly.

Mathematicians speak mathematise.

> I'm afraid that It's the old real number mantra creeping in. Still even
> if you insist on using the reals the system will work out. But the
> ambiguity of such a construction is questionable. You are unwilling to
> grant the notion of three signs aren't you?

I'm willing to grant it if you can define it in a way which makes
sense, but so far I haven't seen a definition.

[Timothy Golden BandTechnology.com]

So, in order to convince you I'll need to understand the terms under
which the definition will be acceptable. Do you accept:
http://en.wikipedia.org/wiki/Magnitude_%28mathematics%29
and
http://en.wikipedia.org/wiki/Quantity
as legitimate and fundamental references?
As I look up scalar it is a word with many definitions and I should
avoid its use in deference to magnitude, which according to the above
definition is not dependent upon the real numbers.
Aren't the real numbers simply elements with a choice of two signs and
one magnitude? If not then the mnemonic sign should be optional and
another representation more fundamental should be available that would
be preferable. Two is just a number and I raise it to three, four, etc.
Dropping it to one yields a definition of zero-dimensional numbers that
I am sure you will aprove of (snicker). They are time and allow
spacetime to exist such that the time component is not graphically
measurable.

I see you are into music. Did you see the tritare in Science News? It
was in the first June issue; a Y-string instrument with three necks.
Some Canadians made it. It's supposed to have more of a bell tone.
There is a photo of it. I'm not much of a musician but this thing looks
pretty neat. When fretted on one branch only it must have quite some
dynamics.

I suppose another way of looking at the quagmire of the polysign
definition is that the current definitions may indeed make it very
difficult to build three-signed numbers and that helps explain why
noone has done it before. Should mathematics be open to new
constructions? Must all work be derived from past work?

The book is written. One must preserve the book. Math as religion.

-Tim

[Spoonfed]

Gene Ward Smith wrote:
> Timothy Golden BandTechnology.com wrote:
> > Gene Ward Smith wrote:
> > > tommy1729 wrote:
> > >
> > > > so for instance allowing "polysigned" expressions (enter in google if you dont know what polysigned is , there is a whole website explaining the basics )
> > >
> > > I can'r see it has any of the basics! I'm basing that on the postings
> > > about them here.

He needs a bit of an update. I took most of what I know about it from
the animation, the diagram, the angle definition (Pi - ArcCos(1/(n-1))
if I remember correctly, the definitions of multiplication and
addition; particularly the additive inverse, which seems to go without
representation.

> >
> > I'd be happy to help you understand them. Any shortcomings of my
> > website should be exposed so that I can improve it.
>
> It seems to me the basic facts about polysigned numbers are these:
>
> (1) The polysigned numbers, for each positive integer n, are a real
> n-dimensional commutative algebra with no nilpotent elements
>

The n vectors start at the origin and point as far apart from each
other as they can in as many dimensions as they can span, so it creates
an n-1 dimensional space, except in the case where n=1, where it gets
wierd. I'm not familiar with the term "nilpotent."

> (2) The polysigned numbers may be described as R[Cn], where n is the
> cyclic group mod n. Hence they have a basis r[0], r[1], ... , r[n-1]
> where r[i]*r[j] = r[i+j], where the indicies add mod n.
>

Yes, I think that's right. I'd stress that r[0], r[1], ..., r[n-1] are
all unit vectors. But you'll get to that later.

> (3) The polysigned numbers may also be described as the algebra of
> real-valued functions on Cn, where the product is the convolution
> product.
>

I'm not sure. I think it may be real-valued function on Rn. I would
have to go back to abstract algebra and recall the motivation for
existence of complex numbers. It seems that P3 was already equivalent
to the complex plane, so it would be redundant to represent it by three
complex numbers.

The product of two vectors can be determined by splitting the
coordinates into a linear combination of the unit vectors r[i], ...
r[n-1], and then performing the product, which--yes, I guess it is a
convolution since every term is multiplied by every term and the final
direction is determined as you said r[i]*r[j]=r[(i+j)mod n]

> (4) If we extend the polysigned numbers to the corresponding complex
> algebra, we may apply the finite Fourier transform, converting the
> convolution product into a pointwise product.
>

That sounds interesting, and worth delving deeper.

> (5) We may describe the above situation in terms of orthogonal
> idempotent elements
> e[i] = (r[0]+r[1]w^i + r[2]w^(2i) ... + r[n-1]w^(n-1))/n, where w is a
> primative n-th root of unity.
>

That also sounds interesting.

> (6) We may get orthogonal idempotent elements for R[Cn], ie the
> polysigned numbers, by adding e[i]+e[n-i] when i is not 0 or n/2, and
> using e[i] when i is 0 or n/2. Then the polysigned numbers break up
> into one (if n is odd) or two (if n is even) copies of R, and (n-1)/2
> (if n is odd) or (n-2)/2 (if n is even) copies of R.
>
> These look like the basic facts to me from a purely mathematical point
> of view, and I'll be happy to leave the applications to someone else.

Thank you for the overview.

[Timothy Golden BandTechnology.com]

But Jon your logic applies for P2 as well. The polysigned construction
builds the real numbers so they also are not technically valid either.
The relation
- x + x = 0
causes the two-form to drop to a one-form that is the real numbers.

>
> The product of two vectors can be determined by splitting the
> coordinates into a linear combination of the unit vectors r[i], ...
> r[n-1], and then performing the product, which--yes, I guess it is a
> convolution since every term is multiplied by every term and the final
> direction is determined as you said r[i]*r[j]=r[(i+j)mod n]

I think the ultrasimple product definition looks like the following:
(s1x1)(s2x2) = s3x3
s3 = s1s2 = sum mod n( s1, s2 )
x3 = x1x2 = multiplication( x1, x2 )
This definition uses s as a sign and x as a magnitude so that sx is an
elemental polysigned number. In effect s and x being two different
domains married as one where the product operation treats their
component operations individually. elements are combinable under the
law:
s1x1( s2x2 + s3x3)
= s1s2x1x2 + s1s3x1x3 .
This is an algebraic definition and the + operation is summation or
superposition. Product is implied by lack of an operator between terms
or the terms are parenthesized. This definition is more minimal than
coordinate definitions like ( a, b, c ). The coordinate system of the
polysigned numbers does not rely upon a Cartesian product so to use it
in the definition may be a falsehood as is the usage of real numbers.
The algebraic form is sufficient. This form is also more accessible to
a wider audience.

There is a serious distinction between the polysigned numbers and their
Cartesian equivalents:
http://bandtechnology.com/PolySigned/Lattice/Lattice.html
When taken in their noncontinuous form the geometry is fundamentally
different.
The approaches you are taking will not address these differences.
Effectively this is like travelling backwards through mathematics
travelling from the continuum back to the natural number form so that
the x in sx is now discrete.
This is the level at which calculus happens and the discrepancy may
have implications there.

-Tim

[Spoonfed]

If there is the possibility that a little effort might make the article
make sense to you, use Wikipedia. If 25% or more of the words on the
page are unfamiliar to you, you might try looking elsewhere.

> As I look up scalar it is a word with many definitions and I should
> avoid its use in deference to magnitude, which according to the above
> definition is not dependent upon the real numbers.

If you are talking about an end quantity that you could easily slap a
real-valued unit on (force, distance, momentum, length, time (?),
etc...) you can call it a magnitude. You can call it a scalar whether
you can slap a unit on it or not.

> Aren't the real numbers simply elements with a choice of two signs and
> one magnitude? If not then the mnemonic sign should be optional and
> another representation more fundamental should be available that would
> be preferable.

The word "should" is not a mathematical term. You have "givens" and
you have "therefores." You can question the givens but generally not
the therefores. If you're representing anything in reference to zero,
I don't see any "choice" except which way to define as positive, and
"mnemonic" means a method to help you remember something, so I think
you may be misusing the term. Perhaps you mean convention.

<Snipping some dubious points you've made more convincingly elsewhere.>

> I suppose another way of looking at the quagmire of the polysign
> definition is that the current definitions may indeed make it very
> difficult to build three-signed numbers and that helps explain why
> noone has done it before. Should mathematics be open to new
> constructions? Must all work be derived from past work?
>
> The book is written. One must preserve the book. Math as religion.
>
> -Tim

Ehhh, no, I don't think that's what's going on. Mathematics is more of
a language. As such you can say just about anything you want with it,
except something ambiguous. The reason ambiguities come up is usually
because people have created different notations for the same thing, or
the same notation for something completely different. Mathematics is
open to new constructions, but whenever possible, you should maintain
the traditional conventions, terminology, and notation. This isn't
because one notation is more "true" than the other, but because the two
notations are like foreign languages.

I think with some modification, Gene's mathematical description could
serve as a template for a very tight definition and summary of
properties of the polysigned numbers. On your website, something like
this should go right up front, where you say what you KNOW about them.
Then applications and possible applications can be covered.

[Timothy Golden BandTechnology.com]

I'm trying to be patient and play by the rules, but when I look at real
number definitions and see six of them and not one treats magnitude as
a fundamental component then I am still stuck until this aspect gets
addressed. I have built a system that allows n-signed numbers. By
definition it should not be built from the two-signed numbers, which
are a member of the family. Now, when I look at the wiki on magnitude
mentioned above I realize that magnitude is not tied to the real
numbers by definition. That is just the most common usage of them.
Let's face it, distance comes before the real numbers come. That's
mathematical evolution. You can define a distance for the real numbers,
but that does not make the distance concept restricted to the real
numbers. Anyone with a piece of string will be able to demonstrate
distance without the use of the real numbers. This is embodied by that
wiki page. So I construct the polysign numbers from magnitude and sign
as two entities bonded together.
The real numbers are a member and therefor they can be defined this
way. Big deal. Now there are seven definitions instead of six. Who
knows how many definitions there actually are out there. Do they really
matter? Don't we understand what they are? Where is the conflict? If we
make a mistake that mistake is detectable. Where is my mistake? Two
signs and a magnitude compose the elementary real number. What is the
problem?

Perhaps the problem is that people accept that phrase 'real' as
literal. The 'real' numbers have broken symmetry. That doesn't seem
very real, yet we still represent space with them. Even though the
space does not have broken symmetry. I'm just pushing it along further
and out comes complex numbers and spacetime. So I am not creating any
more conflict than is already present. Perhaps the solution is to
rename reality. Since we have such a dim view of it currently perhaps
we can leave the mathematicians with their two pronged nipple and move
on to singality. It should be a reserved name which disallows naming of
mathematical spaces after it so that the rule makers cannot contaminate
it. Whatever it is we are subject to it regardless of our beliefs. That
should not be mistaken for a simple little number system that people
have gotten carried away with to the point of mistaking it for reality.

Wow, that's a rant!
I put a link to the P4 product analysis over on:
http://groups.google.com/group/sci.math/msg/1972adafbe1158b9

-Tim

[Gene Ward Smith]

Spoonfed wrote:

> I think with some modification, Gene's mathematical description could
> serve as a template for a very tight definition and summary of
> properties of the polysigned numbers. On your website, something like
> this should go right up front, where you say what you KNOW about them.

That strikes me as a great idea. Then, if you have some defintion of a
function assigning a "sign", give it. For real numbers, this is called
the signum function, or sign function:

http://en.wikipedia.org/wiki/Sign_function

If you don't have a definition, but just an aspiration, then keep it in
your head as something to be worked on.

> Then applications and possible applications can be covered.

At which point I'll leave you to it.

[Spoonfed]

No, sorry. I thought Gene was saying that each of the three scalars
associated with an element of P3 was a complex number. I may have
misunderstood. I wasn't saying it is redundant to have two
representations of the same concept. I was saying it is redundant to
use three complex numbers to represent a field equivalent to one
complex number.

>
> >
> > The product of two vectors can be determined by splitting the
> > coordinates into a linear combination of the unit vectors r[i], ...
> > r[n-1], and then performing the product, which--yes, I guess it is a
> > convolution since every term is multiplied by every term and the final
> > direction is determined as you said r[i]*r[j]=r[(i+j)mod n]
>
> I think the ultrasimple product definition looks like the following:
> (s1x1)(s2x2) = s3x3
> s3 = s1s2 = sum mod n( s1, s2 )
> x3 = x1x2 = multiplication( x1, x2 )
> This definition uses s as a sign and x as a magnitude so that sx is an
> elemental polysigned number. In effect s and x being two different
> domains married as one where the product operation treats their
> component operations individually. elements are combinable under the
> law:
> s1x1( s2x2 + s3x3)
> = s1s2x1x2 + s1s3x1x3 .
> This is an algebraic definition and the + operation is summation or
> superposition. Product is implied by lack of an operator between terms
> or the terms are parenthesized. This definition is more minimal than
> coordinate definitions like ( a, b, c ). The coordinate system of the
> polysigned numbers does not rely upon a Cartesian product so to use it
> in the definition may be a falsehood as is the usage of real numbers.
> The algebraic form is sufficient. This form is also more accessible to
> a wider audience.
>

I think I would advise typing it up in C++, using as simple a code
structure as possible and commenting liberally.

class Psign(){
//include properties of a Psign number including
//the number of coordinates
//the coordinates
}

function Multiply(Psign A, Psign B):Psign
{
//have a function that takes two Psign numbers, finds out the
dimension
// and returns the product of the appropriete dimension
}

function Add(Psign A,Psign B):PSign
{
//similarly
}

As far as I know these are the only parts you have clearly defined,
though we could also define

Structure CartesianCoords
{
//This structure should contain the dimension and coordinates of a
vector in n dimensions
}

function CartesianForm(PSign A):CartesianCoords
{
//A function that takes a PSign number, and returns Cartesian
Coordinates of appropriate dimension.
}

function PSignForm(CartesianCoords A):PSign
{
//Function takes a Cartesian number and returns a PSign number of
appropriate dimension
}

> There is a serious distinction between the polysigned numbers and their
> Cartesian equivalents:
> http://bandtechnology.com/PolySigned/Lattice/Lattice.html
> When taken in their noncontinuous form the geometry is fundamentally
> different.
> The approaches you are taking will not address these differences.

If the differences are there, they will show up through thorough
analysis of well-defined constructions. In particular they should show
up where generalizations of the algorithms involved lead to choices and
questions.

[Gene Ward Smith]

Spoonfed wrote:

> No, sorry. I thought Gene was saying that each of the three scalars
> associated with an element of P3 was a complex number.

Each is a real number, and since they sum to zero, define a real vector
space of dimension two, which has the structure of the complex numbers.
They are, in short, the complex numbers in funny notation.

[Timothy Golden BandTechnology.com]

I have done all of this and then some. These functions are defined for
general n. You've already seen the product code. The non-elemental sum
code is just:
for( i = 0; i < n; i++ )
x[i] = s1.x[i] + s2.x[i];
where x[] holds n magnitudes in order.
As we discuss elemental polysigned values we have to accept that when
they sum or superpose the result does not automatically combine into a
single elemental value. Instead they obey the Sum over s of (sx) = 0
law. Components of like sign can be joined together:
s1x1 + s1x2 = s1( x1 + x2)
where s is a sign and x are magnitudes.
Is that the elemental sum you are talking about in code?
I'm not quite sure what you are saying.
You know that I have code, but are you talking about demonstrating the
concepts to Gene in code here?

You said:
> > > I'm not sure. I think it may be real-valued function on Rn. I would
> > > have to go back to abstract algebra and recall the motivation for
> > > existence of complex numbers. It seems that P3 was already equivalent
> > > to the complex plane, so it would be redundant to represent it by three
> > > complex numbers.

Now I say :
It seems that P2 is already equivalent
to the real line, so it would be redundant to represent it by two
real numbers.

How can you deny my logic? The polysigned numbers define the real
numbers, not the other way around. Anyhow rather than cry it I should
sing it. It's only a matter of time before the concepts become
accepted. My gain in communicating with you guys is to understand what
the mental or mathematical blocks to the system are. Thus far there are
no mathematical blocks. What is broken? What is wrong with the new
definition of the real numbers? Just a conflict in the mind due to real
numbers having been burnt in so deep that you are incapable of going
there mentally. Your mind denies the possibility. It's impossible
right? It's a joke. Well, it is sort of funny. I can understand the
cheese factor seems very high at first whiff. But cheese is good stuff.
You try a little bit and it tastes good. You try a little more. Gee
there's another flavor, and another, and another. Now some people only
eat American cheese. That's sort of like the real numbered cheese. For
me it's P2 cheese. It's alright but I certainly wouldn't pledge
allegiance to it. It's really not that great. It claims to be great,
but when it comes down to it it's just another flavor of cheese.
Variety is more important and to constitute all the cheese you eat from
American cheese is a bad mistake.

-Tim

[!bianko]

excellent[URL=http://porn-directories.com]:)[/URL]!!

[Gene Ward Smith]

Timothy Golden BandTechnology.com wrote:

> What is wrong with the new
> definition of the real numbers?

The main thing wrong with it is that you haven't given one. Why don't
you write down explicitly your propsed definition here.

[Tom]

> Dear mathfriends (?)
>
> In observing math , ive noticed , and im not alone,
> that many (proof)problems in math (especially the
> harder ones ) can be transformed into some kind of "
> critical line " problem.
>
> The most famous example is of course Riemann's
> Hypothese.
>
> But also the tanc conjecture can be rewritten as a
> critical line problem.
>
> Representions theorems can be rewritten as a critical
> line problem.
>
>
> For all clarity , i consider "critical line problems"
> as follows
>
> 1) a subgroup of zeros all lie on the critical line
>

> 2) none of the zeros of a subgroep lie on the
> critical line
>

> 3) no 3 zeros lie on a line (probably reducable to 1)
> or 2) so less important i guess )
>

> Every upperbound or lowerbound conjecture is
> equivalent to some critical line problem
>
> You can just literally bend functions to get a
> critical line , if ya already have a simple critical
> "function/figure/path"
>
> SO if ya have a "critical monotonic increasing
> polynoom " like x^a , you can easily transform it to
> a line , bye transforming ( bending ) the function
>

> And don't forget that e.g. circles are the product of
> 2 complex lines.
>

> So critical problems in 2D are often easily
> transformable into "critical line problems in 2D"
>
> So the "2D equivalent problems " are mainly and often
> simply reducable to critical lines
>
> A bit like belonging to an NP class to make a
> comparison.( but for proofs instead of computations
> )
>
> However !!!
>
> some problems might not be expressible in 2D
> criticals , but in 3D criticals
>
> with the concept 3D being most general !
>

> so for instance allowing "polysigned" expressions
> (enter in google if you dont know what polysigned is
> , there is a whole website explaining the basics )
>

> i mentioned before transforming polynoom to a line
>

> but in 3D we have topology joining the party;
>

> we could have a " critical torus "
>
> or " polysigned critical torus "
>

Question about polysign numbers...
from the website: http://bandtechnology.com/PolySigned/PolySigned.html

" Three-signed numbers are equivalent to complex numbers.
" In three-signed math for any value x:
- x + x * x = 0. (where "*" is a new sign)

how can this be right? -1 + 1 + i doesn't equal 0
surely i is the third sign here.
and why isn't it 4 signed? +, -, +i, -i

[Timothy Golden BandTechnology.com]

Tom wrote:
> Question about polysign numbers...
> from the website: http://bandtechnology.com/PolySigned/PolySigned.html
>
> " Three-signed numbers are equivalent to complex numbers.
> " In three-signed math for any value x:
> - x + x * x = 0. (where "*" is a new sign)
>
> how can this be right? -1 + 1 + i doesn't equal 0
> surely i is the third sign here.
> and why isn't it 4 signed? +, -, +i, -i

Hi. The polysign construction is not built from the complex numbers.
The equivalence is only discovered after applying the definition of
three-signed numbers. I'll briefly cover it here but the official
tutorial is my website. The main page is mostly three-signed and so
serves as the three-signed tutorial:
http://bandtechnology.com/PolySigned/PolySigned.html
Upon getting three-signed the same rules apply for general n.
These rules define sum and product for n-signed numbers.

As I study your question is appears that you are making some
assumptions that are not valid. It appears that you are attempting to
preserve the sense of + of the two-signed numbers onto the three-signed
numbers. This will not work. The same is true of the - sign.
The idea is to generalize sign. This implies some level of symmetry
between signs such that they be 'equal' for lack of a more succinct
word. On the reals we see that:
- x + x = 0 .
where x can be a magnitude or a real number.
And this in hindsight is the proper place to implement the symmetry.
So when we say that:

- x + x * x = 0

for three-signed numbers we are implying a balancing act of these three
components.
It is no longer true that

- x + x = 0

in P3.
If that were true then we would also have:
- x * x = 0
and

+ x * x = 0

to preserve symmetry.
These will not generate a sound math system.

As we look at

- x + x * x = 0

in P3 we can instantiate general values that are not equal as well
like:
- 5 + 2 * 4 (a).
By preserving like-sign superposition so that
- 1 - 1 = - 2 ,
+ 3 + 4 = + 7 = + 1 + 2 + 4,
* 1.5 * 2.3 = * 3.8
we see that (a) can be represented as:
- 2 + 2 * 2 - 3 * 2
where the first three terms (-2+2*2) are zero and so
- 5 + 2 * 4 = - 3 * 2 .
In general any three-signed value will be reducible to two terms
whereas in the reals they are reducible to one term. Informationally
this states that three-signed numbers are two-dimensional. Now when we
ask ourselves what two-dimensional structure they fit it is not half of
the plane as you have guessed and found conflict with; it is the whole
plane. The three-signed sign vectors are the cube roots of one in the
complex plane. Simply take the expression above and mark out five units
on one of these rays and label ray '-', then 2 units out on another and
label '+', and four units out on the last denoted '*'. Perform the
vector sum and the result will be the same as graphing
- 3 * 2
on the same graph. That's because
- 1 + 1 * 1 = 0
in this coordinate system.
So you see that the signs of three-sign numbers are a different class
from two-signed numbers. Hence the need for a term such as polysigned
numbers to hold the family of n-signed systems.

The coordinate system can be extended upward and downward in sign. This
implies up and down in dimension, so four-signed numbers become
three-dimensional and you will find out that the vectors from the
center of a tetrahedron to its vertices are the proper orientations for
the unit sign vectors.

Your confusion with the 2D plane holding all higher signs is very
common so don't feel bad if you are struggling with this. It's
facinating how many people see the construction this way. Usually the
ones that see it this way initially are the ones who eventually grasp
the multidimensional nature of higher signs.

To address your four-signed argument I will simply say that in your
thinking you still have:
- 3 * 3 = 0
and
+ 4 # 4 = 0
or some configuration like these in the plane. These instances conflict
with the identity law:
- x + x * x # x = 0.
We must consider the identity law almost as a definition of zero.
Applying the planar logic would make
- 3 * 3 # 4 = # 1
whereas the proper interpretation does not allow any reduction for the
left hand side of this expression, yielding a 3D value.

Whether the components are large or small so long as they are balanced
they will remain local. The product and the sum don't care whether the
components are huge or small so in effect the reduction that we have
performed is optional to the mathematical operations.
It is a bizarre thing that the defining law which begets dimensionality
doesn't actually have to be used to do arithmetic. In effect the act of
graphing a value is performing this law. Hence in some regards and
particularly for the one-signed numbers this rendering process should
be seen as an operator.

I'll be happy to discuss the product with you. This has been a
discussion about the sum only.

-Tim

[Tom]

Thanks Tim, I get it now.
So the 3 signed numbers are like plotting points on that triangular tesselated paper you used to get at school, with one sign for each of the 3 directions that the lines go.
4 signed numbers are like a 3d tesselated tetrahedron version of this.

Cool, I've never considered this type of number/vector before...
In some way it seems simpler than cartesian vectors in that you can reach any 2d point by only moving in 3 directions rather than 4.
I wonder what would happen if people started writing the laws of physics down using polysigned numbers instead of cartesian/orthogonal dimensions for space and time... :)
Actually I wonder whether 3d rotations are any simpler to represent using polynumbers. (matrices, quaternions, axis angles all have their drawbacks).

Cheers,
Tom.

[Timothy Golden BandTechnology.com]

Tom wrote:
> Thanks Tim, I get it now.
> So the 3 signed numbers are like plotting points on that triangular tesselated paper you used to get at school, with one sign for each of the 3 directions that the lines go.
> 4 signed numbers are like a 3d tesselated tetrahedron version of this.
>
> Cool, I've never considered this type of number/vector before...
> In some way it seems simpler than cartesian vectors in that you can reach any 2d point by only moving in 3 directions rather than 4.
> I wonder what would happen if people started writing the laws of physics down using polysigned numbers instead of cartesian/orthogonal dimensions for space and time... :)
> Actually I wonder whether 3d rotations are any simpler to represent using polynumbers. (matrices, quaternions, axis angles all have their drawbacks).
>
> Cheers,
> Tom.

P4 has its own drawbacks as well, but they are different than the
quaternions. Product and sum are defined algebraically for n-signed
numbers and they obey most of the field laws. Most importantly the
associative, distributive, and commutative laws are supported in any
dimension just as they work for the reals. The catch is that even while
the arithmetic can be so well behaved the product does not obey the law
| A B | = | A || B |
beyond P3. Distances get screwed up in P4+. The product does exhibit
rotational behavior but it is coupled with this morphing as you can see
in:

http://BandTechnology.com/PolySigned/Deformation/DeformationUnitSphereP4.html
But this behavior is what allows the claim for natural spacetime
support. It is my hope to do physics with them and they suggest the
proper spacetime topology is:

0D + 1D + 2D ...

or
P1 + P2 + P3 + ...
or simply the family of polysigned numbers. I work on a fairly
classical model which would put us in a particle product space. It
requires remapping distance by
1 / ( x + 1 )
to get a geometric product that complies with the classical forces.
Under this sytem the origin becomes unity and what was infinite
distance becomes zero. Infinities disappear even for adjacent
particles. When generic points are taken relatively on the topology
above there is some evidence for charge and spin axis from pure and
simple geometry. I believe this model is invoking a 'substrate' whom we
are the arithmetic products of. This method avoids the use of a
Cartesian product.

-Tim