On 01.04.2010 at 12:20 in S6, there is the following noon lecture:
On the nonexistence of k-reptile tetrahedra
A d-dimensional simplex S is called a k-reptile if it can be tiled without overlaps by simplices S_1,S_2,...,S_k that are all congruent and similar to S. For d=2, k-reptile simplices (triangles) exist for many values of k and they have been completely characterized by Snover, Waiveris, and Williams. On the other hand, for d >= 3, only one construction of k-reptile simplices is known, the Hill simplices, and it provides only k of the form m^d, m=2,3,...
We prove that for d=3, k-reptile simplices (tetrahedra) exist only for k=m^3. This partially confirms a conjecture of Hertel, asserting that the only k-reptile tetrahedra are the Hill tetrahedra.
Our research has been motivated by the problem of probabilistic packet marking in theoretical computer science, introduced by Adler in 2002.
Modified: 19. 10. 2010