A Regular Rep Icosahedron of Order Four
Year: 2026
Type: preprint
Abstract: Can a net of a regular icosahedron be cut into smaller polygons, each of which is itself a net of a regular icosahedron? Nomi, Shiota, Kamata and Uehara explicitly recorded this existence question in 2025. We give an affirmative answer with four equal-area pieces. The outer net is a polyiamond of eighty unit equilateral triangles and folds to an icosahedron of edge length 2; each twenty-triangle piece is an edge net of one of edge length 1. Two noncongruent small-net shapes occur twice each. The search uses a half-turn of the subdivided icosahedron to find a surface cover, followed by a separate search for a compatible planar development. We present the exact construction and a standalone verifier that checks geometry, topology and every folding identification using integer arithmetic.
Author: Seamons, John
Language: English
Topic: Mathematics and Applications
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7224503125