6-spheres & R, C, Q, O series...

Higher-dimensional geometry (previously "Polyshapes").

6-spheres & R, C, Q, O series...

Postby thigle » Sun Jul 31, 2005 10:40 pm

although i am not mathematician, i would like to know the following (i apologize to mathematicians for my unscholarly formulations to follow):

how come that there exist only 4 division algebras - i.e. real, complex, quaternion and octonion ? i know each succesive step through this sequence makes one property of the previous algebra vanish, so one ends up with no operators after octonions, as all 4 basic properties of Reals dissappear. but can this be somehow explained with minimum algebraic notation (though I can handle it), i.e. through imaginable geometric/spatial metaphors ? i think i read somewhere that the proof for finitness of this series has something to do with properties of spheres, especially 7d spheres (6-spheres), but i cannot find the reference anymore. :roll:
any clues anyone ? thanx
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Postby wendy » Mon Aug 01, 2005 12:12 am

I suppose that 15 is not prime would have something to do with it.

You could always look at the recent novel by Conway and Smith hight 'Quaterions and Octonions', which deals with this in some detail.
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Postby jinydu » Mon Aug 01, 2005 2:04 am

I think there was an article (maybe even more than one article) about this in the archives of plus.maths.org

However, you should keep in mind that it's not that it's impossible to formulate number systems with 3, 5, 6, 7, 9, etc "dimensions" (think vector multiplication). It's just that there's no way to define multiplication of these "numbers" in such a way as to preserve the familiar and important properties of multiplication.
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Postby wendy » Mon Aug 01, 2005 2:32 am

It's just that there's no way to define multiplication of these "numbers" in such a way as to preserve the familiar and important properties of multiplication.


Someone obviously had a lend of you. One can easily set up a trimex, or three-axial matrix that effects multiplication, ie

S(i,j,k) Vi, Wj = Xk

is the multiplication for vectors.

Some of these are quite handy, here for example is the heptagonal trimex.

I gave this one a fair pasting for a year or more.
Code: Select all

    (  1  0  0  )  ( 0  1  0 ) ( 0  0  1 )
    (  0  1  0  )  ( 1  0  1 ) ( 0  1  1 )
    (  0  0  1  )  ( 0  1  1 ) ( 1  1  1 )


Where the fall-out comes, is that no multiplication exists that equates to free isometric transformations of space. The heptagonal matrix above includes sliding down hyperbolic chords, since it has units that have a modulus greater than one.

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Postby pat » Mon Aug 01, 2005 8:32 am

wendy wrote:I suppose that 15 is not prime would have something to do with it.

You could always look at the recent novel by Conway and Smith hight 'Quaterions and Octonions', which deals with this in some detail.


That book deals with it as succinctly as I've ever seen.

And... I'm not sure about the 15 as prime thing.... but Tony Smith claims (without explanation) somewhere on his website that it has to do with 1 has no non-trivial divisors, 2 has no non-trivial divisors, 3 has no non-trivial divisors, but 4 does have non-trivial divisors. This, somehow means that dimension 2<sup>0</sup>, 2<sup>1</sup>, 2<sup>2</sup>, and 2<sup>3</sup> get division algebras.
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Postby wendy » Mon Aug 01, 2005 10:59 pm

That fifteen is composite, and that four has divisors, is pretty much the same thing.

If you look at the three non-real geometries, you will see that there is a fano-space process operating, where one has, in effect, X, Y, Z are in line if XOR(X,Y,Z) = 0.

This leads to things like 2^n-1 as 1, 3, 7, 15 as n goes 1,2,3,4.

Since 2^4-1 is the product of algebraic roots 2A1 2A2 2A4 = 1 * 3 * 5, the decomposition of 4 into divisors, and the division of 1111 binary into factors, is then pretty much the same thing.

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Postby pat » Tue Aug 02, 2005 1:30 pm

Ah, thank-you, Wendy.... that makes a lot of sense....
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Postby thigle » Sat Aug 06, 2005 9:54 pm

thank you all. it didn't clarify my confusion in these matters much, but certainly enlarged the scope of my questioning. i'll check that book by Conway/Smith. however, i would really like to get at least some understanding (=imagination for me), so I ask more: :lol:

does it have anything to do with the fact that the hyper-syrface area of n-sphere reaches maximum for n=7(.25695) ?
or with the sequence [vector/ tensor/ spinor/ twistor] ?

jinydu: thanx for reminding of algebras with 3,5,7,...dimensions. these are cool too. however, i am interested first especially in this sequence of 4 algebras that lose these 4 properties, as these seem to correspond to 4 logical positions (of Nagarjuna's or Boolean logic), these are like 4 conceptual limits. then the other algebras, without these properties, are like the non-conceptual modes of awareness beyond this 'quaternary mandala of conceptuality'. like solitonic or whathaveyou.

wendy: could you explain a bit what does it mean that 'no multiplication exists that equates to free isometric transformations of space.' ?
also, your last post is a mystery to me, almost fully. what is fano-space process ? and by non-real geometries you mean imaginary - complex(2d), quats(4d) and octonions(8d) ? and division of 1111 binary into factors? i still don't get the 15-composite/4-divisors connection, must study more :o

pat: that thing with divisors is interesting, as i can interpret it to mean that non-triviality starts at 4, so everything under this number is trivial - i.e. divisible. so triviality would equate in certain sense with divisibility. that's a good one, thanx :wink:
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Postby wendy » Sun Aug 07, 2005 11:21 pm

Fano Space

This is a finite space, derived from the fano configuration. This is a space where there are 7 points, and 7 lines, where every lies on three lines, and there is one line between any pair of points.

It can be replicated by using the XOR operator over three variables, not including 0. The alignment is that a, b, c = 0 if a .xor. b .xor. c = 0.

One can view it as a polytope, with seven hedra (2d surtopes), and 7 edges. As one goes higher, one gets 15, 31, ... sides of the polytope, rather like a simplex.

The connexion between 4 and 15, and the factorisation of 1111, is that the sides of the fanotopes are in binary, 1, 11, 111, 1111, 11111, etc.

An 'algebraic root' is my name for the factors that you get, when you divide something like a six-digit period into factors with one, two, three and six periods (ie one can have a six digit period of the type ababab, or abcabc. For any base, these numbers take the same form

A1 = 9, A2 = 11, A3 = 111, A6 = 91 in decimal
A1 = 7, A2 = 11, A3 = 111, A6 = 71 in octal
A1 = 1, A2 = 11, A3 = 111, A6 = 11 in binary

These numbers are easier to factorise than 999999, 777777, or 111111.

A four-figure period then has algebraic roots A1, A2, A4 = 9 * 11 * 101

The implication to geometries is that a 15-space ought substain a subspace of order 3 and prehaps 5. However, we don't see a 5-space, and therefore we don't have this.

Vector-space

It is certianly possible to have a multiplication system that uses three coordinates, such as the integer-system Z7 (the span of chords of the heptagon). However, the system has an infinite number of units, and does not map onto any known set of rotations in space.

Unlike the reals, complex, quarterons and octonions, these do not correspond to rotations and dilations of real space.

The closest one can get out of Z7, for example, is the mapping of a Z7 point onto three (different) points on a number line. A point exists for every combination of (x,y,z), and there is a particular lattice in this space (actually several), which behave as integers [like Z7, Z9, and subsets of things like Z13, Z14, Z18, Z19, Z21, &c.]

Fiddling around with this kind of stuff tells you that there must exist a matrix where the first column, in order, converges on the lengths of the chords of a polygon: for example,

Code: Select all

    20   36   45
    36   65   81
    45   81  101



This matrix has a determinate of 1, and successive powers approximate the chords of the heptagon, to ever increasing decimals.
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