Kynčl, Jan
Jan
Kynčl
Patakova, Zuzana
Zuzana
Patakova0000-0002-3975-1683
On the nonexistence of k reptile simplices in ℝ^3 and ℝ^4
International Press
2017
2018-12-11T11:48:00Z
2019-11-14T08:43:45Z
journal_article
https://research-explorer.app.ist.ac.at/record/701
https://research-explorer.app.ist.ac.at/record/701.json
10778926
544042 bytes
application/pdf
A d-dimensional simplex S is called a k-reptile (or a k-reptile simplex) if it can be tiled by k simplices with disjoint interiors that are all mutually congruent and similar to S. For d = 2, triangular k-reptiles exist for all k of the form a^2, 3a^2 or a^2+b^2 and they have been completely characterized by Snover, Waiveris, and Williams. On the other hand, the only k-reptile simplices that are known for d ≥ 3, have k = m^d, where m is a positive integer. We substantially simplify the proof by Matoušek and the second author that for d = 3, k-reptile tetrahedra can exist only for k = m^3. We then prove a weaker analogue of this result for d = 4 by showing that four-dimensional k-reptile simplices can exist only for k = m^2.