Zuzana Safernova: On the nonexistence of k-reptile simplices in R^3 and R^4

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 mutually 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> 2, only one construction of k-reptile simplices was known until recently, the Hill simplices, and it provides only k of the form m^d, m=2, 3, ...

Anwei Liu and Barry Joe introduced in their paper an 8-reptile tetrahedron, which is not Hill (as noticed and pointed to us by Herman Haverkort). This excludes a conjecture of Hertel, asserting that the only k-reptile tetrahedra are the Hill tetrahedra. We prove that for d=3, k-reptile simplices (tetrahedra) exist only for k=m^3. We also prove that for d=4, k-reptile simplices can exist only for k=m^2. For d=4 it remains to find out whether there really exist m^2-reptile simplices for m non-square and whether all m^4-reptile simplices are indeed Hill simplices. In a talk we give a simplified proof of the first claim and sketch the second proof.

The first part is joint work with Jiri Matousek, the second one with Honza Kyncl.