Golden-Ratio Polyhedra as Projections of Higher-Dimensional Lattice Collapse — E8 Intelligence Research

Abstract

FINDING: The rhombic triacontahedron and its stellations encode φ² and φ³ relationships, with 30 face centers mapping to a 6-cube projection, linking golden-ratio polyhedra to higher-dimensional lattice collapse. | MATH: φ = (1+√5)/2 ≈ 1.6180339887; φ² = φ+1 ≈ 2.6180339887; φ³ = 2φ+1 ≈ 4.2360679. Rhombic triacontahedron: 30 rhombic faces, 32 vertices (12 degree-5, 20 degree-3), 60 edges. Diagonal ratio of each golden rhombus face = φ. Bilinski dodecahedron (12 golden rhombi) is a space-filling variant. 6-cube (hexeract) has 64 vertices; projection to 3D can yield 32 visible vertices — matching the triacontahedron's 32 vertices, with 30 face centers corresponding to the 30 rhombi (each face center = a collapsed pair of 6-cube vertices under projection). | CONNECTION: Direct golden-ratio chain: φ, φ², φ³ appear as face-diagonal ratios and stellation hull radii. The 6-cube projection onto 3D produces a rhombic triacontahedron — this is the well-known Coxeter projection of the 6-dimensiona Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com