On φ-Rational Coordinate Constructions of the Regular Dodecahedron

This record presents a direct, reproducible coordinate construction of the regular dodecahedron in three-dimensional Euclidean space. The construction uses only the values{0,±1,±φ,±1/φ}\{0, \pm 1, \pm \varphi, \pm 1/\varphi\}{0,±1,±φ,±1/φ} (up to uniform scaling), where φ\varphiφ denotes the golden ratio. Rather than assuming regularity from symmetry arguments or quoting closed-form formulas, the paper verifies regularity explicitly through metric and combinatorial checks. From the given vertex set, the edge graph is recovered using a nearest-neighbour distance criterion. Pentagonal faces are identified as 5-cycles in the recovered graph, with planarity confirmed directly. The existence of a concentric circumsphere, midsphere, and insphere is then established by computation. The appearance of the golden ratio is treated strictly as a structural consequence of pentagonal geometry inherent to the regular dodecahedron. No optimisation, physical, biological, cognitive, or metaphysical interpretations are proposed. The scope of the work is intentionally limited to three-dimensional Euclidean geometry. All results are constructive and reproducible using elementary linear algebra and distance calculations, making the construction suitable for mathematical exposition, computational geometry, and design or modelling contexts where explicit vertex coordinates are required. Related Materials & Extended Context The following resources provide supplementary context, development history, and adjacent research. They are not required to understand or apply the framework presented in this record. Zenodo Archive — Crippin’s Theory (Finalised Volumes)https://zenodo.org/communities/atlas-codex/ NotebookLM Research Series(Developmental analysis and source synthesis)https://notebooklm.google.com/notebook/911b50f2-19a3-487d-9141-512437a668b6https://notebooklm.google.com/notebook/098f3f2a-706b-4e8d-8037-a29d7b3236aa PhilPapers Index Entryhttps://philpapers.org/rec/CRITAW Google Scholar Index (related works)https://scholar.google.com/scholar?q=Crippins+Theory Contact & AccessAuthor: Suzanne CrippinEmail: [email protected]: https://www.linkedin.com/in/atlascodex