Computing maximal copies of polytopes contained in a polytope

TL;DR

This paper models the problem of finding the largest homothetic copy of one polytope inside another as a quadratically constrained optimization problem, solving it numerically and algebraically for regular polyhedra.

Contribution

It extends Croft's work by providing solutions for the remaining cases of maximal inclusions of regular 3D polyhedra using a novel optimization approach.

Findings

Complete classification of maximal inclusions for all regular polyhedra.

Demonstrated numerical and algebraic methods for solving polytope inclusion problems.

Provided exact solutions for previously unresolved cases.

Abstract

Kepler (1619) and Croft (1980) have considered largest homothetic copies of one regular polytope contained in another regular polytope. For arbitrary pairs of polytopes we propose to model this as a quadratically constrained optimization problem. These problems can then be solved numerically; in case the optimal solutions are algebraic, exact optima can be recovered by solving systems of equations to very high precision and then using integer relation algorithms. Based on this approach, we complete Croft's solution to the problem concerning maximal inclusions of regular three-dimensional polyhedra by describing inclusions for the six remaining cases.