Robert P. C. de Marrais

arXiv2008

Voyage by Catamaran: Effecting Semantic Network "Bricolage" via Infinite-Dimensional Zero-Divisor Ensembles

Continuing arguments presented [1] or announced [2][3] in "Complex Systems," zero-divisor (ZD) foundations for "scale-free" networks (evinced, in particular, in the "fractality" of the Internet) are decentralized. Spandrels, quartets of ZD-free or "hidden" box-kite-like structures (HBKs) in the 2^(N+1)- ions, are "exploded" from (and uniquely linked to) each standard box-kite in the 2^N-ions, N > 3. Any HBK houses, in a "cowbird's nest," exactly one copy of the (ZD-free) octonions, the recursive basis for all ZD ensembles. Each is a potential waystation for alien-ensemble infiltration in the large, or metaphor-like jumps in the small. Cowbirding models what evolutionary biologists [4], and structural mythologist Claude Levi-Strauss before them [5], term bricolage: the opportunistic co-opting of objects designed for one purpose to serve others unrelated to it. Such arguments entail switching focus, from the octahedral box-kite's four triangular "sails," to its trio of square "catamarans" and their box-kite-switching "twist products."

math.GM↗

arXiv2007

Placeholder Substructures II: Meta-Fractals, Made of Box-Kites, Fill Infinite-Dimensional Skies

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.

math.RA↗

arXiv2007

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

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.

math.RA↗

arXiv2007

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

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.

math.RA↗

arXiv2006

Presto! Digitization, Part I: From NKS Number Theory to "XORbitant" Semantics, by way of Cayley-Dickson Process and Zero-Divisor-based "Representations"

The objects of the great Nonlinear Revolutions - Catastrophes and Chaos in the 1960s-70s (henceforth, CT); and, small-world and scale-free Network Theory (NT), emerging quite recently - will be spliced together by a New Kind of Number Theory, focused on digitizations (i.e., binary strings). NT nodes then become feature-rich representations (nodules in a "rhizosphere") of CT contents. The "Box-Kite" formalism of zero-divisors (ZD's) - first showing in the 16-D Sedenions, then in all higher 2^N-ions derived from Imaginaries by Cayley-Dickson Process (CDP) - can model such "enriched" nodules. Its (bit-string XOR-ing vs. matrix-multiplying) operations unfold into "representations" of the objects linked in CT with "partitions of Nullity": Singularities. The route from here to fractals and Chaos, via CDP extensions to 2^N-ions for growing N, will involve us in graphics of higher-dimensional "carry-bit overflow," as manifest in the mandala-like patterns of "emanation tables" (the rough equivalent, for ZD's, of group theorists' Cayley Tables). I'll lead into this with a quote about "Hjelmslev's Net" (which I'll claim is the CDP manque') from a famous postmodern text, Deleuze and Guattari's "A Thousand Plateaus" (where "rhizosphere" imagery arose). With strong assists from the CT-based structuralism of Jean Petitot, Algirdas Greimas's "Semiotic Square" will show us how to explicitly link CT to semiotic foundations via ZD "representations," while the infinite-dimensional ZD meta-fractal or "Sky" where Box-Kites fly - first appearing in the 32-D Pathions and incorporating the higher 2^N-ions - will provide sufficient lebensraum for Levi-Strauss's "Canonical Formula" of mythwork to unfurl in. (These results serve to extend my NKS 2004 paper, available at the Wolfram Science website.)

math.RA↗

arXiv2004

Box-Kites III: Quizzical Quaternions, Mock Octonions, and Other Zero-Divisor-Suppressing "Sleeper Cell" Structures in the Sedenions and 2^n-ions

Building on two prior studies of zero-divisors (ZD's) generated by the Cayley-Dickson process, algebras we call "lariats" (Line Algebras of Real and Imaginary Axis Transforms), linkable to quantum measurement, are discovered in the Sedenions, complementing the 7 isomorphic "box-kites" (pathway systems spanning octahedral lattices) interconnecting all primitive ZD's. By switching "edge-signs," products among the diagonal line-pairs associated with each of a box-kite's 4 triangular, vertex-joined, "sails" generate not 6-cyclic ZD couplings when circuited, but 28 pairs of structures with Quaternionic multiplication tables -- provided their symbols represent the oriented diagonals as such, not point-specifiable "units" residing on them. If a box-kite's 3 "struts" (pairs of opposite vertices, the only vertex pairings which do not contain mutual ZD's) each be combined with the ZD-free Quaternion copy uniquely associated with said box-kite, 21 lariats with Octonionic multiplication, one per each box-kite strut pair, are generated. Extending this approach to "emanation tables" (box-kite analogs in higher 2^n-ions) indicates further ZD-masking "sleeper cell" structures, with renormalization's basis possibly amenable to rethinking, thanks partly to the ZDs' newfound "Trip Sync" property, inhering throughout the 2^n-ion hierarchy.

math.RA↗

arXiv2002

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

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.

math.RA↗

arXiv2000

The 42 Assessors and the Box-Kites they fly: Diagonal Axis-Pair Systems of Zero-Divisors in the Sedenions' 16 Dimensions

G. Moreno's abstract depiction of the Sedenions' normed zero-divisors, as homomorphic to the exceptional Lie group G2, is fleshed out by exploring further structures the A-D-E approach of Lie algebraic taxonomy keeps hidden. A breakdown of table equivalence among the half a trillion multiplication schemes the Sedenions allow is found; the 168 elements of PSL(2,7), defining the finite projective triangle on which the Octonions' 480 equivalent multiplication tables are frequently deployed, are shown to give the exact count of primitive unit zero-divisors in the Sedenions. (Composite zero-divisors, comprising all points of certain hyperplanes of up to 4 dimensions, are also determined.) The 168 are arranged in point-set quartets along the 42 Assessors (pairs of diagonals in planes spanned by pure imaginaries, each of which zero-divides only one such diagonal of any partner Assessor). These quartets are multiplicatively organized in systems of mutually zero-dividing trios of Assessors, a D4-suggestive 28 in number, obeying the 6-cycle crossover logic of trefoils or triple zigzags. 3 trefoils and 1 zigzag determine an octahedral vertex structure we call a box-kite -- seven of which serve to partition Sedenion space. By sequential execution of proof-driven production rules, a complete interconnected box-kite system, or Seinfeld production (German for field of being; American for 1990's television's Show About Nothing), can be unfolded from an arbitrary Octonion and any (save for two) of the Sedenions. Indications for extending the results to higher dimensions and different dynamic contexts are given in the final pages.

math.GM↗