Characterization of the Three-Dimensional Fivefold Translative Tiles
Abstract
This paper proves the following statement: If a convex body can form a fivefold translative tiling in $\mathbb{E}^3$, it must be a parallelotope, a hexagonal prism, a rhombic dodecahedron, an elongated dodecahedron, a truncated octahedron, a cylinder over a particular octagon, or a cylinder over a particular decagon, where the octagon and the decagon are fivefold translative tiles in $\mathbb{E}^2$. Furthermore, it presents an example of multiple tiles in $\mathbb{E}^3$ with multiplicity at most 10 which is neither a parallelohedron nor a cylinder.
- Publication:
-
arXiv e-prints
- Pub Date:
- July 2023
- DOI:
- 10.48550/arXiv.2307.07824
- arXiv:
- arXiv:2307.07824
- Bibcode:
- 2023arXiv230707824H
- Keywords:
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- Mathematics - Metric Geometry;
- 52C22;
- 52C23;
- 05B45;
- 52C17;
- 11H31