Publication at Faculty of Mathematics and Physics |
2013
Abstract
A d-dimensional simplex S is called a k-reptile (or a k-reptile simplex) if it can be tiled without overlaps by k simplices with disjoint interiors that are all mutually congruent and similar to S. For d=2, triangular k-reptiles exist for many values of k 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 Matousek and the second author that for d=3, k-reptile tetrahedra can exist only for k=m^3.
We also 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.