On the Nonexistence of k-reptile Tetrahedra
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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≥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.
KeywordsReptile Tiling Self-similarity Geometry of a simplex
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