On the nonexistence of k-reptile tetrahedra

TL;DR

This paper proves that in three dimensions, k-reptile tetrahedra only exist for k being a perfect cube, confirming a conjecture and extending understanding of such geometric tilings beyond known Hill simplices.

Contribution

It establishes that 3D k-reptile tetrahedra only exist for k=m^3, supporting Hertel's conjecture and expanding the classification of k-reptile simplices.

Findings

k-reptile tetrahedra exist only for k=m^3

Partial confirmation of Hertel's conjecture

Extension of known classifications in 3D geometry

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 > 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=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.