On the nonexistence of k-reptile simplices in ℝ3 ana ℝ4

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 Matoušek 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.KeywordsDihedral AngleBarycentric SubdivisionDisjoint InteriorProbabilistic PacketRational AngleThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.