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July 26, 2026Open Access

Coxeter Groups Unify Polytopes, Tessellations, and Molecular Symmetries — E8 Intelligence Research

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Authors

ACAndrew Stewart Caldin

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Overview

This research demonstrates the algebraic classification of polytopes and symmetries, highlighting their interconnectedness.

Key Points

  • The aim is to classify regular polytopes and their reflection symmetries using Coxeter groups, providing a unifying framework.
  • Developed a Coxeter group presentation to explore algebraic classifications.
  • Utilized Dynkin diagrams to represent specific relationships and configurations.
  • Analyzed properties of crystallographic Coxeter groups and their implications for molecular symmetries.
  • Established that Coxeter groups unify polytopes, tessellations, and molecular symmetries.
  • Identified that H₃ and H₄ groups show unique relationships involving the golden ratio and dihedral angles.
  • Demonstrated Coxeter numbers connect to E₈ lattice and quasicrystalline symmetries.

Cite This Study

Andrew Stewart Caldin (2026) studied this question.

synapsesocial.com/papers/6a65a825d3aea3239cd78a80https://doi.org/10.5281/zenodo.21525037
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  1. 1Coxeter Groups as Algebraic Skeleton for Symmetry-Breaking in Cryptography and Algorithms — E8 Intelligence Research2026
  2. 2Coxeter Groups, Dynkin Diagrams, and Euler Characteristic in Regular Polytopes — E8 Intelligence Research2026
  3. 3Icosahedral Symmetry: Golden Ratio in Coxeter Group H₃ and Platonic Solids — E8 Intelligence Research2026
  4. 4Coxeter Groups Bridge Root Symmetries and Quantum Topological Invariants — E8 Intelligence Research2026
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