Placeholder Substructures I: The Road from NKS to Scale-Free Networks Is Paved with Zero-Divisors

Details Placeholder Substructures I: The Road from NKS to Scale-Free Networks Is Paved with Zero-Divisors Placeholder Substructures I: The Road from NKS to Scale-Free Networks Is Paved with Zero-Divisors

2007-03-26

arxiv.org/abs/math/0703745v3

Complex Systems, 17 (2007) 125-142

Mathematics

Rings and Algebras

Comments: 20 pp., 2 fig. First of 3-part "theorem/proof" exposition of 78-slide Powerpoint from Wolfram Science's NKS 2006, av

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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