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

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 $k=m^2$.