Placeholder Substructures III: A Bit-String-Driven ''Recipe Theory'' for Infinite-Dimensional Zero-Divisor Spaces
Abstract
Zero-divisors (ZDs) derived by Cayley-Dickson Process (CDP) from N-dimensional hypercomplex numbers (N a power of 2, at least 4) can represent singularities and, as N approaches infinite, fractals -- and thereby,scale-free networks. Any integer greater than 8 and not a power of 2 generates a meta-fractal or "Sky" when it is interpreted as the "strut constant" (S) of an ensemble of octahedral vertex figures called "Box-Kites" (the fundamental building blocks of ZDs). Remarkably simple bit-manipulation rules or "recipes" provide tools for transforming one fractal genus into others within the context of Wolfram's Class 4 complexity.
Keywords
Cite
@article{arxiv.0704.0112,
title = {Placeholder Substructures III: A Bit-String-Driven ''Recipe Theory'' for Infinite-Dimensional Zero-Divisor Spaces},
author = {Robert P. C. de Marrais},
journal= {arXiv preprint arXiv:0704.0112},
year = {2007}
}
Comments
32 pp., 1 fig. Third of 3-part "theorem/proof" exposition of 78-slide Powerpoint from NKS 2006, available at http://wolframscience.com/conference/2006/presentations/materials/demarrais.ppt V2: small fixes, 2 new notes. V3: Added small number of corrections (pp. 8, 15-16), one long remark (pp. 21-22), RE: 2nd type of box-kite flow pattern