On the nonexistence of k reptile simplices in ℝ^3 and ℝ^4

J. Kynčl, Z. Patakova, The Electronic Journal of Combinatorics 24 (2017) 1–44.

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Journal Article | Published | English
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Abstract
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.
Publishing Year
Date Published
2017-07-14
Journal Title
The Electronic Journal of Combinatorics
Volume
24
Issue
3
Page
1-44
ISSN
IST-REx-ID

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Kynčl J, Patakova Z. On the nonexistence of k reptile simplices in ℝ^3 and ℝ^4. The Electronic Journal of Combinatorics. 2017;24(3):1-44.
Kynčl, J., & Patakova, Z. (2017). On the nonexistence of k reptile simplices in ℝ^3 and ℝ^4. The Electronic Journal of Combinatorics, 24(3), 1–44.
Kynčl, Jan, and Zuzana Patakova. “On the Nonexistence of k Reptile Simplices in ℝ^3 and ℝ^4.” The Electronic Journal of Combinatorics 24, no. 3 (2017): 1–44.
J. Kynčl and Z. Patakova, “On the nonexistence of k reptile simplices in ℝ^3 and ℝ^4,” The Electronic Journal of Combinatorics, vol. 24, no. 3, pp. 1–44, 2017.
Kynčl J, Patakova Z. 2017. On the nonexistence of k reptile simplices in ℝ^3 and ℝ^4. The Electronic Journal of Combinatorics. 24(3), 1–44.
Kynčl, Jan, and Zuzana Patakova. “On the Nonexistence of k Reptile Simplices in ℝ^3 and ℝ^4.” The Electronic Journal of Combinatorics, vol. 24, no. 3, International Press, 2017, pp. 1–44.
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