Discrete & Computational Geometry

, Volume 46, Issue 3, pp 599–609 | Cite as

On the Nonexistence of k-reptile Tetrahedra

Article

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 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.

Keywords

Reptile Tiling Self-similarity Geometry of a simplex 

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Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Department of Applied Mathematics and Institute of Theoretical Computer Science (ITI)Charles UniversityPraha 1Czech Republic
  2. 2.Department of Applied MathematicsCharles UniversityPraha 1Czech Republic

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