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Three-dimensional Rep-tiles

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 Added by Ryan Blair
 Publication date 2021
  fields
and research's language is English




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A 3D rep-tile is a compact 3-manifold $X$ in $mathbb{R}^3$ that can be decomposed into finitely many pieces, each of which are similar to $X$, and all of which are congruent to each other. In this paper we classify all 3D rep-tiles up to homeomorphism. In particular, we show that a 3-manifold is homeomorphic to a 3D rep-tile if and only if it is the exterior of a connected graph in $S^3$.



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This paper proves the following statement: {it If a convex body can form a twofold translative tiling in $mathbb{E}^3$, it must be a parallelohedron.} In other words, it must be a parallelotope, a hexagonal prism, a rhombic dodecahedron, an elongated dodecahedron, or a truncated octahedron.
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