Platonic Solids: Euler's Formula, Dualities, and Symmetry in Crystallography — E8 Intelligence Research

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The five Platonic solids are the only convex regular polyhedra, and their dual relationships and symmetry groups fundamentally underpin crystallographic and molecular structures. Novelty: ClaimNovelty.SYNTHESIS. Consensus alignment: ConsensusAlignment.NEUTRAL.

Andrew Stewart Caldin

doi:10.5281/zenodo.22075929

August 24, 2026Open Access

Platonic Solids: Euler's Formula, Dualities, and Symmetry in Crystallography — E8 Intelligence Research

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Key Points

To analyze the mathematical relationships, dualities, and symmetry groups of the five Platonic solids and evaluate their applications in crystallography and Lie algebra root systems.
Evaluated vertex, edge, and face counts using Euler's formula (V − E + F = 2) across all five convex regular polyhedra.
Calculated rotational and full symmetry group orders (A₄×C₂, S₄×C₂, A₅×C₂) and exact dihedral angles.
Characterized exact geometric dualities (cube↔octahedron, dodecahedron↔icosahedron, and self-dual tetrahedron) along with rotational symmetry orders of 12, 24, and 60.
Identified the golden ratio (φ ≈ 1.618) in icosahedral and dodecahedral metrics, linking specific dihedral angles (e.g., 70.53°, 109.47°) to crystallographic point groups and Lie algebra root systems (A₃, B₃, C₃, H₃, H₄).

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Mathematical analysis demonstrates structural dualities and symmetry groups in Platonic solids, highlighting fundamental mappings to crystallographic structures and Lie algebras.

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Andrew Stewart Caldin (2026) studied this question.

synapsesocial.com/papers/6a8e9b92451774b83f3b458b https://doi.org/10.5281/zenodo.22075929

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Platonic Solids: Euler's Formula, Dualities, and Symmetry in Crystallography — E8 Intelligence Research

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Platonic Solids: Euler's Formula, Dualities, and Symmetry in Crystallography — E8 Intelligence Research

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