k_nehru wrote:
Can you just explain what is the convention in the
following: H(2), C(4), R(8), 2R(8) ?
They are taken from the reference Horace gave:
http://math.ucr.edu/home/baez/week105.html by John Baez, regarding Bott peridocity in Clifford Algebra. It is well worth the time to print out and read, as it opens a big door to understanding how various geometries work within the Reciprocal System. I'm quite excited about it!
John Baez wrote:
This Week's Finds in Mathematical Physics (Week 105)
There are some spooky facts in mathematics that you'd never guess in a million years... only when someone carefully works them out do they become clear. One of them is called "Bott periodicity".
A 0-dimensional manifold is pretty dull: just a bunch of points. 1-dimensional manifolds are not much more varied: the only possibilities are the circle and the line, and things you get by taking a union of a bunch of circles and lines. 2-dimensional manifolds are more interesting, but still pretty tame: you've got your n-holed tori, your projective plane, your Klein bottle, variations on these with extra handles, and some more related things if you allow your manifold to go on forever, like the plane, or the plane with a bunch of handles added (possibly infinitely many!), and so on.... You can classify all these things. 3-dimensional manifolds are a lot more complicated: nobody knows how to classify them. 4-dimensional manifolds are a lot more complicated: you can prove that it's impossible to classify them - that's called Markov's Theorem.
Now, you probably wouldn't have guessed that a lot of things start getting simpler when you get up around dimension 5. Not everything, just some things. You still can't classify manifolds in these high dimensions, but if you make a bunch of simplifying assumptions you sort of can, in ways that don't work in lower dimensions. Weird, huh? But that's another story. Bott periodicity is different. It says that when you get up to 8 dimensions, a bunch of things are a whole lot like in 0 dimensions! And when you get up to dimension 9, a bunch of things are a lot like they were in dimension 1. And so on - a bunch of stuff keeps repeating with period 8 as you climb the ladder of dimensions.
(Actually, I have this kooky theory that perhaps part of the reason topology reaches a certain peak of complexity in dimension 4 is that the number 4 is halfway between 0 and 8, topology being simplest in dimension 0. Maybe this is even why physics likes to be in 4 dimensions! But this is a whole other crazy digression and I will restrain myself here.)
Bott periodicity takes many guises, and I already described one in "week104". Let's start with the real numbers, and then throw in n square roots of -1, say e1,...,en. Let's make them "anticommute", so
ei ej = - ej ei
when i is different from j. What we get is called the "Clifford algebra" Cn. For example, when n = 1 we get the complex numbers, which we call C. When n = 2 we get the quaternions, which we call H, for Hamilton. When n = 3 we get... the octonions?? No, not the octonions, since we always demand that multiplication be associative! We get the algebra consisting of pairs of quaternions! We call that H + H. When n = 4 we get the algebra consisting of 2x2 matrices of quaternions! We call that H(2). And it goes on, like this:
C0 = R
C1 = C
C2 = H
C3 = H + H
C4 = H(2)
C5 = C(4)
C6 = R(8)
C7 = R(8) + R(8)
C8 = R(16)
Note that by the time we get to n = 8 we just have 16x16 matrices of real numbers. And that's how it keeps going: Cn+8 is just 16x16 matrices of guys in Cn! That's Bott periodicity in its simplest form.
Actually right now I'm in Vienna, at the Schroedinger Institute, and one of the other people visiting is Andrzej Trautman, who gave a talk the other day on "Complex Structures in Physics", where he mentioned a nice way to remember the above table. Imagine the day is only 8 hours long, and draw a clock with 8 hours. Then label it like this:
(clock diagram)
k_nehru wrote:
Also I don't understand the diagram you uploaded.
The diagram is the clock chart Baez is talking about, with a couple extras added in... namely that the geometric inverses of linear/polar can be plainly seen in this periodicity, and the positioning of the various motions in the RS.
It is interesting to note that the only "Real" number in the system, C0 or "R", is UNITY. Once you have a displacement from Unity, C1, the system becomes "complex", having both a linear and rotational component, which fits right in with what we uncovered with the Euler relationships of the photon.
The sub-atomic particles, having a 2-d "magnetic" motion, place it at C2 (H) on the diagram, telling us that the time region rotations are polar, represented by quaternions -- exactly what you proposed in your earlier papers.
The atomic motion add another dimension, C3 (H+H), represented by a pair of quaternions, which again, is in perfect agreement with Larson -- the TWO, double-rotations of that atom.
C4 (H
2, 2x2 quaternion matrix) seems to have the characteristics of "prana", which you identified in other forum posts.
C5, the 4x4 complex matrix, may well be Larson's "life unit", the "real" components being our physical structure, animated via prana (the complex half).
All of the research we've done over the past 6 years is littered with similar references to this structure. I think it may be something we apprehended, but never connected with the Bott periodicity. It is definitely worth spending some time looking at.
Bruce
