Nonexistence of 2-Reptile Simplices

摘要

A simplex S is called an m-reptile if it can be tiled without overlaps by simplices S_1,S_2,... ,S_m that are all congruent and similar to S. The only m-reptile d-simplices that seem to be known for d ≥ 3 have m = k~d, 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.