On the nonexistence of $k$-reptile simplices in $\mathbb R^3$ and $\mathbb R^4$

TL;DR

This paper proves that in three and four dimensions, $k$-reptile simplices can only exist for specific values of $k$, specifically perfect cubes in 3D and perfect squares in 4D, simplifying previous proofs.

Contribution

The authors simplify the proof that $k$-reptile tetrahedra in 3D only exist for $k=m^3$ and establish a similar result for 4D, where such simplices only exist for $k=m^2$.

Findings

In 3D, $k$-reptile tetrahedra exist only for $k=m^3$.

In 4D, $k$-reptile simplices exist only for $k=m^2$.

Simplified previous proofs for the 3D case.

Abstract

A $d$-dimensional simplex $S$ is called a $k$-reptile (or a $k$-reptile simplex) if it can be tiled by $k$ simplices with disjoint interiors that are all mutually congruent and similar to $S$. For $d=2$, triangular $k$-reptiles exist for all $k$ of the form $a^2, 3a^2$ or $a^2 + b^2$ and they have been completely characterized by Snover, Waiveris, and Williams. On the other hand, the only $k$-reptile simplices that are known for $d \ge 3$, have $k = m^d$, where $m$ is a positive integer. We substantially simplify the proof by Matou\v{s}ek and the second author that for $d=3$, $k$-reptile tetrahedra can exist only for $k=m^3$. We then prove a weaker analogue of this result for $d=4$ by showing that four-dimensional $k$-reptile simplices can exist only for $k=m^2$.