Isomorphic 4D Extensions of the Regular Dodecahedron

This record presents a minimal, coordinate-based framework for extending a verified three-dimensional regular dodecahedron into four-dimensional Euclidean space. Starting from an explicit 3D reference model with established regularity, the paper introduces a canonical embedding into R4\mathbb{R}^4R4 and examines layered duplication and correspondence relations across an additional coordinate dimension. The approach preserves all intrinsic three-dimensional geometry exactly, including distances, adjacency relations, and face structure. Four-dimensional structure is introduced explicitly and incrementally through coordinate extension and vertex-wise correspondence, allowing intrinsic geometry to be analysed independently from extended relations. Visualisation is treated strictly as representational. Projection methods are used to illustrate four-dimensional relations in three dimensions and are carefully distinguished from the underlying geometric structure. No attempt is made to define four-dimensional faces or cells, nor to construct or approximate regular four-dimensional polytopes. Where the 120-cell is referenced, it serves only as contextual orientation within known four-dimensional geometry. The constructions presented here do not derive, approximate, or claim equivalence with that polytope. This work makes no physical, biological, cognitive, or metaphysical claims. The fourth coordinate is treated purely as an ambient geometric dimension. The paper is intended as a disciplined study in geometric embedding and representation, suitable for computational, educational, or visualisation contexts where higher-dimensional extension of a fixed 3D structure is 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