Abstract
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 greater than 2, 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 greater than 2, 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.