Interlocking of convex polyhedra: towards a geometric theory of fragmented solids

TL;DR

This paper develops a geometric theory of interlocking convex polyhedra, identifying conditions for interlocking arrangements including all platonic solids, and extends the concept to higher dimensions such as 4D cubes.

Contribution

It introduces criteria for interlocking based on cross-section transformations and generalizes the concept to higher-dimensional spaces.

Findings

All five platonic solids can form interlocked structures.

Criteria for interlocking are based on cross-section transformations.

Interlocking layers of four-dimensional cubes are described.

Abstract

We present structures comprised of identical convex polyhedra which are interlocked geometrically. These sets cannot be disassembled by removing individual polyhedra by translations and/or rotations. The shapes that permit interlocking arrangements include all five platonic solids. Criteria for interlocking based on transformations of the cross-sections of the elements in a 3D reconstruction of a layer from its middle cross-section are formulated. A generalization to higher dimensions is also given. In particular, an interlocking layer of four-dimensional cubes is described.