Optimizing for the Rupert property

TL;DR

This paper introduces a nonlinear optimization approach to identify Rupert polyhedra, successfully validating known cases and discovering new Rupert solids among Catalan and Johnson families.

Contribution

It develops a novel nonlinear optimization method that efficiently verifies Rupert properties and extends known classifications to include new Rupert solids.

Findings

Validated known Rupert polyhedra in seconds.

Discovered 2 new Rupert Catalan solids.

Identified 5 additional Rupert Johnson solids.

Abstract

A polyhedron is Rupert if it is possible to cut a hole in it and thread an identical polyhedron through the hole. It is known that all 5 Platonic solids, 10 of the 13 Archimedean solids, 9 of the 13 Catalan solids, and 82 of the 92 Johnson solids are Rupert. Here, a nonlinear optimization method is devised that is able to validate the previously known results in seconds. It is also used to show that 2 additional Catalan solids -- the triakis tetrahedron and the pentagonal icositetrahedron -- and 5 additional Johnson solids are Rupert.