Nonexistence of 2-reptile simplices

Authors:
Jiří Matoušek
Affiliations:
Department of Applied Mathematics, and Institute of Theoretical Computer Science (ITI), Charles University, Praha 1, Czech Republic
Venue:
JCDCG'04 Proceedings of the 2004 Japanese conference on Discrete and Computational Geometry
Year:
2004

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Abstract

A simplex S is called an m-reptile if it can be tiled without overlaps by simplices S1,S2,...,Sm that are all congruent and similar to S. The only m-reptile d-simplices that seem to be known for d ≥ 3 have m=kd, k ≥ 2. We prove, using eigenvalues, that there are no 2-reptile simplices of dimensions d ≥ 3. This investigation has been motivated by a probabilistic packet marking problem in theoretical computer science, introduced by Adler in 2002.