Placeholder Substructures III: A Bit-String-Driven ''Recipe Theory'' for Infinite-Dimensional Zero-Divisor Spaces

Details Placeholder Substructures III: A Bit-String-Driven ''Recipe Theory'' for Infinite-Dimensional Zero-Divisor Spaces Placeholder Substructures III: A Bit-String-Driven ''Recipe Theory'' for Infinite-Dimensional Zero-Divisor Spaces

2007-04-02

arxiv.org/abs/0704.0112v3

Mathematics

Rings and Algebras

32 pp., 1 fig. Third of 3-part "theorem/proof" exposition of 78-slide Powerpoint from NKS 2006, available at http://wolframs

Scientific paper

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.

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