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}
}
  • J. Kyncl, Zuzana Patáková
  • Published 2017
  • Mathematics, Computer Science
  • Electron. J. Comb.
  • 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

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