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Saturday, July 14, 2007

Comments

Konstantin Kouptsov

There is an article: Complex Systems, 17 (@007) 125-142, and others are in arxiv.org

John Gonsowski

From http://www.valdostamuseum.org/hamsmith/8idempotents.html

With respect to Cellular Automata:

Michael Gibbs has been working on using Cellular Automata as neural network nodes, and Robert de Marrais has written a Box-Kites III paper (at math/0403113), leading me to think of some questions:

Could 16x16 structures such as switching yards of Box-Kites III have structures corresponding to the graded structure o the Clifford algebra Cl(8) that is the 16x16 real matrix algebra?
Since the vectors of the Cl(8) Clifford algebra are 8-dimensional and correspond to the octonions, if you take the correspondences between the 256 Wolfram CA and the Cl(8) basis elements, there is a correspondence:
CA Rule No. Octonion Basis Element
1 1
2 i
4 j
16 k
8 E
32 I
64 J
128 K
Could such a correspondence be used to construct such things as "Box-Kites" whose vertices might be regared, not just as octonions etc, but also as Cellular Automata?
Could Box-Kite type structures give useful computational structures if the vertices were considered as CA and the edge-flow-orinetations were considered as information flow in a computing system?
If such a computing system can be set up for 2^n-ionic structures for large n, then, since for 16-ions and larger you have interesting zero-divisor "sleeper-cell" substructures, could they be useful with respect to computational systems, perhaps doing things like forming loops that might let the computational system to "adjust itself" and/or "teach itself"?
Robert de Marrais commented on some of those questions, saying in part:

"... I'm finding two directions to go with box-kites next, and yes, cellular automata clearly are part of it. I did a poster session at Wolfram's [ 2004 NKS ] conference, had a long talk with him and another fellow, one Rodrigo Obando ... My poster session is currently being written up for incorporation in the conference proceedings, after which it goes out to arXiv.org -- and will have (or so I hope!) some nifty graphics for higher-dimensional cases. ... But now to the two directions, which relate to your suggestions:
(1) Boolean monotone and antitone function-pairings can be used, per Rodrigo Obando, to generate exactly all and only the complex cellular automata for a given n and r . . . and, given that for n=4 that means Dedekind's number of 168 mono- and iso- tone functions each, connections to box-kites immediately suggest themselves ... He tells me his work is leading him not merely to isolate and catalog the "complex" CA's for high n and r, but that he's finding -- when he generalizes to the n => infinity situation, that he gets violations of the continuum hypothesis ...

(2): spin networks. The key revelation (which I telescoped on the last couple pages of "Box Kites III") concerns what I call the "trip-sync property." As it turns out, this is incredibly easy to prove, for all box-kites in all dimensions. ... what is truly interesting is this: zero-divisor systems are, ironically, PRESERVERS of associative order! Specifically, each of the four "sails" on a box-kite can be represented (on an isomorphic box-kite diagram, in fact!) as a system of four interconnected Quaternion copies: write each vertex as a pairing of one uppercase and one lowercase letter (with the 'generator' of the given 2^n-ions being the divider of the two: e.g., with the Sedenions, g = the index-8 imaginary, and the pure Sedenions of index > 8 are "uppercase," with the Octonions thereby being written with "lowercase" letters). Using the standard notation in my "strut tables," the "triple-zigzag sail" has vertices (A + a), (B + b), (C + c). Since it's a triple zigzag, this means all the edge-signs are negative: hence, if one takes the diagonal "/" in the (A, a) plane, it will zero-divide the diagonal slanting like "\" in either the (B, b) or (C, c) planes. Now consider that there are 4 associative triplets here: (a, b, c); (a, B, C); (A, b, C); and, (A, B, c). Now, allow for "slippage" of the following sort: orbitings among (a, b, c) can be imagined to "slip" into one of the other 3 by keeping one of the lowercases unchanged, but allowing the other two to form "resonances" with the generator (the XOR of two uppercase is, of course, a lowercase). The trip-sync property says this: IF the "sail" is the triple zigzag, all such slippage can occur without any "flips" in orientation; however, IF the "sail" is one of the other three "trefoil" sails, then ONLY slippage with the lowercase being one of the triple zigzag's trio will preserve orientation. Importantly, this gives a way to envision "observable" and "unobservable" in a quantum mechanical manner: orientation REVERSAL will be observable, and the isomorphism of quaternion algebra to SU2 gives you (recall my graphics toward the end of the first Box-Kite paper vis a vis Catastrophe Theory?) two orthogonal circles whose centers are the "units" of two lines of diagonal idempotents (which, like the diagonals in the boxkite vertex-planes, are ALSO zero-dividers -- but only with each other!!). That is, the 4 axes in the SU2 representation are reals, the usual imaginaries, Pauli spin-matrix "mirror numbers" which square to +1, and a "commutative i" which commutes between these latter two. (This is both Cl(2) in Clifford algebra lingo, and Muses' simplest epsilon-number space.) But then, the centers of the two orthogonal circles are just the projection operators -- 1/2(1 +/- m), m the Pauli "mirror axis" unit. As systems of box-kites get very entangled in higher dimensions (in 32-D, you have systems of 7 of them forming what I call Pleiades, with some fascinating synergetic properties), spin-foams with self-organizing potential suggest themselves . . .

Now, (1) and (2) are BOTH related to my ultimate objective, which is not physics per se, but rather Levi-Strauss's canonical law of myths, and the creation of an infinite-dimensional "collage space" that can accommodate his systems of mythopoetic sign-shunting in a manner roughly reminiscent of Fourier series' infinite-dimensional backdrop for generalized harmonics. So that means I'll be busy with my hobbyhorse at least through "Box-Kites VI"!

The key notion here is that each sail can be seen as an ensemble of 5 Quaternion copies (the 4 associative triplets each are completed by the real unit, and the "sterile" zero-divisor-free triplet of generator, strut constant, and their XOR makes 5). Viewing things in closest-packing-pattern style, we have 5 interacting "unit quaternion" algebras -- with the interactions entailing (1, u), where 'u' is the shared non-real unit. Interestingly, this gives a nice way to think about the Tibetan Book of the Dead's "58 angry demons and 42 happy Buddhas," 100 in all = 5 * 16 + 2*10 = 100 distinct units in the interlinked 5-fold "unit quaternion" ensemble. So one first sees the "42 Assessors," then zooms in one one of the 7 isomorphic box-kites (which, as with all isomorphies, can be seen as identical at some higher level); then, one zooms in further on the "second box-kite" which has its struts defined by upper vs. lower case letters, and the triple zigzag analog being the "all lowercase" sail. ... I've also just purchased a domain name -- "TheoryOfZero.com" -- where I'll start building a site as soon as time permits. ... All these threads are getting ever more entangled and intriguing, aren't they? ...".

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