2002/07/01 by Robert P. C. de Marrais, de Marrais, Robert P. C.
Mathematics · #17A99 #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Geometric and Algebraic Topology #Quantum Algebra (math.QA) #Rings and Algebras (math.RA) #math.QA #math.RA #msc:17A99
paper · pdf · doi:10.48550/arxiv.math/0207003
20 pages, incl. 4 figures, 3 textual tables. V2: Fixed boneheaded mislabeling of standard operator names on p. 4; added paragraph of simplifying conclusions at top of p. 18
openalex publication_date 2002/07/01 · arxiv created 2002/12/18 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.