Flying Higher Than a Box-Kite: Kite-Chain Middens, Sand Mandalas, and Zero-Divisor Patterns in the 2n-ions Beyond the Sedenions

TL;DR

This paper explores the structure of zero-divisors in higher 2n-ions generated by the Cayley-Dickson process, revealing complex patterns and potential links to physics and cellular automata within algebraic systems beyond Sedenions.

Contribution

It introduces the concept of Kite-Chain Middens and sand mandalas in higher 2n-ions, extending the understanding of zero-divisor patterns and their algebraic and physical implications beyond Sedenions.

Findings

Identified infinite extensions of Box-Kite structures in higher 2n-ions.

Discovered 15 emanation tables with 168 cells each, governed by PSL(2,7).

Observed patterns of sparsity and folding indicating collapse modes between algebraic structures.

Abstract

Methods for studying zero-divisors (ZD's) in 2n-ions generated by Cayley-Dickson process beyond the Sedenions are explored. Prior work showed a ZD system in the Sedenions, based on 7 octahedral lattices ("Box-Kites"), whose 6 vertices collect and partition the "42 Assessors" (pairs of diagonals in planes spanned by pure imaginaries, one a pure Octonion, hence of subscript < 8, the other a Sedenion of subscript > 8 and not the XOR with 8 of the chosen Octonion). Potential connections to fundamental objects in physics (e.g., the curvature tensor and pair creation) are suggested. Structures found in the 32-ions ("Pathions") are elicited next. Harmonics of Box-Kites, called here "Kite-Chain Middens," are shown to extend indefinitely into higher forms of 2n-ions. All non-Midden-collected ZD diagonals in the Pathions, meanwhile, are seen belonging to a set of 15 "emanation tables," dubbed "sand mandalas." Showcasing the workings of the DMZ's (dyads making zero) among the products of each of their 14 Assessors with each other, they house 168 fillable cells each (the number of elements in the simple group PSL(2,7) governing Octonion multiplication). 7 of these emanation tables, whose "inner XOR" of their axis-pairs' indices exceed 24, indicate modes of collapsing from higher to lower 2n-ion forms, as they can be "folded up" in a 1-to-1 manner onto the 7 Sedenion Box-Kites. These same 7 also display surprising patterns of DMZ sparsity (with but 72 of 168 available cells filled), with the animation-like sequencing obtaining between these 7 "still-shots" indicating an entry-point for cellular-automata-like thinking into the foundations of number theory.