@article{701,
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 ≥ 3, have k = m^d, where m is a positive integer. We substantially simplify the proof by Matoušek 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.},
author = {Kynčl, Jan and Patakova, Zuzana},
issn = {10778926},
journal = {The Electronic Journal of Combinatorics},
number = {3},
pages = {1--44},
publisher = {International Press},
title = {{On the nonexistence of k reptile simplices in ℝ^3 and ℝ^4}},
volume = {24},
year = {2017},
}