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 $d \ge 3$, 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 $k=m^2$.
Highlights
A tiling of a closed set X in Rd is a locally finite decomposition X = i∈I Xi into closed sets with nonempty and pairwise disjoint interiors
If X has a tiling where all the tiles are congruent to a set T, we say that T tiles X, or, that X can be tiled with ( |I| copies of ) T
It is easy to see that whenever S is a d-dimensional k-reptile set, S is space-filling, that is, the space Rd can be tiled with S: using the tiling of S by its smaller copies as a pattern, one can inductively tile larger and larger similar copies of S
Summary
Except for the one-parameter family of Hill tetrahedra, two other space-filling tetrahedra described by Sommerville [41] and Goldberg [17] are k-reptiles for every k = m3 Both these tetrahedra tile the right-angled Hill tetrahedron, and their tilings are based on the barycentric subdivision of the cube. It is easy to see that the lattice tiling of Rd by barycentrically subdivided unit cubes can be obtained by cutting the space with hyperplanes xi = n/2, xi + xj = n, xi − xj = n, for every i, j ∈ [d], i = j and n ∈ Z Each tile in this tiling is congruent to the right-angled Hill simplex Hd0 defined as the convex hull of points A related question, a classification of edge-to-edge tilings of the sphere by congruent triangles, has been completely solved by Agaoka and Ueno [43]
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