Three-dimensional Rep-tiles

TL;DR

This paper classifies all three-dimensional rep-tiles, showing they are exactly the exteriors of connected graphs in the 3-sphere, thus providing a complete topological characterization.

Contribution

It provides a full classification of 3D rep-tiles, linking them to the exteriors of connected graphs in $S^3$, a novel topological insight.

Findings

3D rep-tiles are exactly the exteriors of connected graphs in $S^3$.

Complete classification of 3D rep-tiles up to homeomorphism.

Abstract

A 3D rep-tile is a compact 3-manifold $X$ in $\mathbb{R}^3$ that can be decomposed into finitely many pieces, each of which are similar to $X$, and all of which are congruent to each other. In this paper we classify all 3D rep-tiles up to homeomorphism. In particular, we show that a 3-manifold is homeomorphic to a 3D rep-tile if and only if it is the exterior of a connected graph in $S^3$.