12 research outputs found
No acute tetrahedron is an 8-reptile
An -gentiling is a dissection of a shape into parts which are
all similar to the original shape. An -reptiling is an -gentiling of
which all parts are mutually congruent. This article shows that no acute
tetrahedron is an -gentile or -reptile for any , by showing that
no acute spherical diangle can be dissected into less than nine acute spherical
triangles.Comment: updated text, as in press with Discrete Mathematics, Discrete
Mathematics Available online 10 November 201
The Least-Area Tetrahedral Tile of Space
We determine the least-area unit-volume tetrahedral tile of Euclidean space,
without the constraint of Gallagher et al. that the tiling uses only
orientation-preserving images of the tile. The winner remains Sommerville's
type 4v.Comment: 72 pages, 10 figure
Surface-area-minimizing n-hedral Tiles
We provide a list of conjectured surface-area-minimizing n-hedral tiles of space for n from 4 to 14, previously known only for n equal to 5 and 6. We find the optimal orientation-preserving tetrahedral tile (n=4), and we give a nice new proof for the optimal 5-hedron (a triangular prism)
On the nonexistence of k reptile simplices in â„ť^3 and â„ť^4
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