Complex Systems

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

Robert P. C. de Marrais
Electronic mail address: rdemarrais@alum.mit.edu
Thothic Technology Partners,
P.O. Box 3083,
Plymouth, MA 02361

Abstract

Zero-divisors (ZDs) derived by the Cayley--Dickson process (CDP) from N-dimensional hypercomplex numbers (N a power of 2, and at least 4) can represent singularities and, as N goes to infinity, fractals and thereby, scale-free networks. Any integer less than 8 and not a power of 2 generates a metafractal or sky when it is interpreted as the strut constant (S) of an ensemble of octahedral vertex figures called box-kites (the fundamental ZD building blocks). 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.