The Tri-Quarter Framework: Radial Dual Triangular Lattice Graphs with Exact Bijective Dualities and Equivariant Encodings via the Inversive Hexagonal Dihedral Symmetry Group 𝕋 24

The Tri-Quarter Framework: Radial Dual Triangular Lattice Graphs with Exact Bijective Dualities and Equivariant Encodings via the Inversive Hexagonal Dihedral Symmetry Group 𝕋 24 | Authorea try { document.documentElement.classList.add('js'); } catch (e) { } var _gaq = _gaq || []; _gaq.push(['_setAccount', 'G-8VDV14Y67G']); _gaq.push(['_trackPageview']); (function() { var ga = document.createElement('script'); ga.type = 'text/javascript'; ga.async = true; ga.src = ('https:' == document.location.protocol ? 'https://ssl' : 'http://www') + '.google-analytics.com/ga.js'; var s = document.getElementsByTagName('script')[0]; s.parentNode.insertBefore(ga, s); })(); Skip to main content Preprints Collections Wiley Open Research IET Open Research Ecological Society of Japan All Collections About About Authorea FAQs Contact Us Quick Search anywhere Search for preprint articles, keywords, etc. Search Search ADVANCED SEARCH SCROLL This is a preprint and has not been peer reviewed. Data may be preliminary. 29 September 2025 V2 Latest version Share on The Tri-Quarter Framework: Radial Dual Triangular Lattice Graphs with Exact Bijective Dualities and Equivariant Encodings via the Inversive Hexagonal Dihedral Symmetry Group 𝕋 24 Author : Nathan O. Schmidt 0009-0006-4027-6046 [email protected] Authors Info & Affiliations https://doi.org/10.22541/au.175883133.31924293/v2 553 views 287 downloads Contents Abstract Supplementary Material Information & Authors Metrics & Citations View Options References Figures Tables Media Share Abstract The Tri-Quarter framework unleashes a radial dual triangular lattice graph with unified complex-Cartesian-polar coordinates, structured orientation phase pair assignments for directional labeling, and topological zones to build exact bijective mappings without approximations. By establishing combinatorial duality for radial separation, Escher reflective duality for zone swapping, and bijective self-duality for reversible transformations, the discretized framework leverages the lattice graph's order-6 rotational symmetry to natively support angular sectors, modular decompositions, equivariant encodings, and trihexagonal six-coloring for conflict-free parallel algorithms. At this discretized framework's core is the Tri-Quarter Inversive Hexagonal Dihedral Symmetry Group 𝕋 24 — the order-24 semidirect product D 6 ⋊ ℤ 2 — which exploits rotational, reflective, and inversive symmetries to unlock these bijective transformations with exact precision. We provide formal proofs of these dualities, along with numerous step-by-step examples, and demonstrate practical efficiency through benchmarked simulations to achieve ~2x speedups with inversion-based path mirroring via bijections and up to ~6x reductions in symmetry-reduced clustering via rotational orbits. This work advances scalable computations on symmetric structures, with applications in computational geometry, graph traversals, tiling, robotics path planning, multi-agent coordination, lattice-based cryptography, image processing, and signal processing. This work aims to solidify a mathematical and computational foundation for both classical and non-classical computing paradigms — targeting future integrations in complex emergent systems that harness intricate "superposition-like" symmetries to advance symmetry-aware algorithms and data structures across diverse computing architectures. Supplementary Material File (tri-quarter_radialduallatticegraph_nathanoschmidt_2025-09-26.pdf) Download 1000.92 KB File (tri-quarter_triangularlattice_nathanoschmidt_2025-09-24.pdf) Download 929.98 KB Information & Authors Information Version history V1 Version 1 25 September 2025 V2 Version 2 29 September 2025 Copyright This work is licensed under a Creative Commons Attribution 4.0 International License Keywords bijective dualities computer science computing and processing data structures and algorithms (cs.ds) equivariant encodings graph symmetries graphs and networks lattices python code scientific computing Authors Affiliations Nathan O. Schmidt 0009-0006-4027-6046 [email protected] View all articles by this author Metrics & Citations Metrics Article Usage 553 views 287 downloads .FvxKWukQNSOunydq8rnd { width: 100px; } Citations Download citation Nathan O. Schmidt. The Tri-Quarter Framework: Radial Dual Triangular Lattice Graphs with Exact Bijective Dualities and Equivariant Encodings via the Inversive Hexagonal Dihedral Symmetry Group 𝕋 24. Authorea . 29 September 2025. DOI: https://doi.org/10.22541/au.175883133.31924293/v2 If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download. For more information or tips please see 'Downloading to a citation manager' in the Help menu . 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