On the Nonexistence of $k$-Reptile Simplices in $\mathbb R^3$ and $\mathbb R^4$

@article{Kyncl2017OnTN,
title={On the Nonexistence of \$k\$-Reptile Simplices in \$\mathbb R^3\$ and \$\mathbb R^4\$},
author={J. Kyncl and Zuzana Pat{\'a}kov{\'a}},
journal={Electron. J. Comb.},
year={2017},
volume={24},
pages={P3.1}
}

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… CONTINUE READING