Computing maximal copies of polytopes contained in a polytope

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.

💡 Research Summary

The paper addresses the geometric optimization problem of fitting the largest possible copy of a polytope P inside another polytope Q, where the copy P′ must be similar to P (i.e., obtained by a uniform scaling and a rigid motion). Historically, Kepler (1619) and Croft (1980) studied this for regular polytopes, but Croft’s analysis left six of the twenty non‑trivial inclusions of the five Platonic solids unresolved.

The authors formulate the problem as a quadratically constrained quadratic program (QCQP). Let Q be defined by half‑spaces H₁,…,Hₘ and let w₁,…,wₙ be the vertices of P. Variables are the scaling factor s (the square of the dilation) and the coordinates vᵢ of the vertices of the unknown copy P′. The constraints are: (i) each vᵢ lies in every half‑space Hₖ (linear inequalities), and (ii) for every pair i<j the squared distance ‖vᵢ−vⱼ‖² equals s·‖wᵢ−wⱼ‖² (quadratic equalities). Maximizing s yields the largest similar copy.

To reduce the size of the problem, the authors select an affine basis of P (p+1 points) and express all other vertices as linear combinations of this basis. Consequently the number of variables drops to (p+1)·q+1, independent of the number of vertices of P. Moreover, only the distances between vectors of the affine basis need to be constrained, leading to ½·(q+1)(q+2) quadratic equations instead of O(n²).

The QCQP is solved numerically with SCIP, a global non‑linear mixed‑integer programming solver that uses branch‑and‑bound techniques. SCIP provides a solution (ŝ, v̂) with a prescribed numerical precision (e.g., 10⁻¹⁰). Because numerical solvers can miss a global optimum or produce spurious local maxima, the authors state three assumptions: (1) the computed solution lies near a unique local maximum, (2) this local maximum is globally optimal, and (3) the vertex‑face incidences of the approximate solution match those of the true optimum.

When these assumptions hold, the authors refine the numerical solution to very high precision using multidimensional Newton iteration. They then apply integer‑relation algorithms (LLL) to each coordinate to guess an exact algebraic number (root of a low‑degree polynomial with small coefficients). Finally, they verify the guessed algebraic solution by substituting it back into the original quadratic system, thus obtaining an exact, provably optimal configuration.

The methodology is applied to all 20 non‑trivial inclusions of the five Platonic solids (tetrahedron T, cube C, octahedron O, dodecahedron D, icosahedron I). The solver, with the improved formulation, solves each case within minutes on a single Xeon core, using less than 8 GB of RAM. The authors reproduce all known results from Croft, correct three typographical errors in his table, and provide exact symbolic expressions (involving square roots and the golden ratio φ) for the six previously unsolved cases. Highlights include:

• Dodecahedron ⊂ Icosahedron – concentric placement; five vertices of a dodecahedral face lie on five edges incident to a common icosahedral vertex, the opposite face mirrors this configuration. The optimal edge length is (15 − √5)/22 ≈ 0.58018.
• Icosahedron ⊂ Dodecahedron – each icosahedral vertex lies in the interior of a distinct dodecahedral face; the scaling factor is (1 + 2φ)/2 ≈ 1.30902.
• Cube ⊂ Icosahedron – concentric; two opposite cube vertices lie on two adjacent icosahedral edges that are not co‑planar; the optimal edge length is (5 + 7√5)/22 ≈ 0.93874.
• Dodecahedron ⊂ Octahedron – a pair of opposite dodecahedral edges are placed in a plane spanned by four octahedral vertices; each octahedral face contains either one edge or one vertex of the dodecahedron. The optimal edge length is (25√2 − 9√10)/22 ≈ 0.31340.
• Tetrahedron ⊂ Icosahedron – one tetrahedral vertex coincides with an icosahedral vertex, another lies on an icosahedral edge, and the remaining two lie inside faces; the optimal scaling matches that of the icosahedron‑in‑dodecahedron case.
• Tetrahedron ⊂ Dodecahedron – a similar configuration with appropriate vertex‑face contacts.

For each inclusion the paper supplies a table with decimal approximations and exact symbolic formulas, as well as a description of the optimal vertex‑face incidences.

The authors conclude that the combination of (i) a general QCQP formulation, (ii) variable and constraint reduction via affine bases, and (iii) high‑precision numerical solving followed by algebraic reconstruction, provides a robust pipeline for solving maximal‑inclusion problems for arbitrary polytopes. The successful completion of the Platonic solid cases demonstrates that computer‑assisted methods can achieve results traditionally obtained by intricate geometric reasoning. Future work may explore extensions to higher dimensions, non‑regular polytopes, and situations where the optimal configuration is not algebraic, requiring alternative certification techniques.