5 Lattices and their Voronoï and Delone cells Article Swipe

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Boris Zhilinskii , Michel Leduc , Michel Le Bellac ·

YOU? · · 2020 · Open Access · · DOI: https://doi.org/10.1051/978-2-7598-1952-2.c006

Lattices and their Voronoï and Delone cellsIn this section we study lattices from the point of view of their tilings by polytopes. Tilings by polytopes: some basic conceptsDefinition: polytope A polytope P is a compact body with a nonempty interior whose boundary ∂P is the union of a finite number of facets, where each facet is the (n -1)-dimensional intersection of P with a hyperplane.Two-dimensional polytopes are called polygons; three-dimensional polytopes are called polyhedra.Definition: k-face (of a polytope) For k = 0, . . ., n-2, a k-dimensional face (or k-face, for short) of a polytope is an intersection of at least (nk) facets that is not contained in the interior of a j-face for any j > k.Thus a 0-face of a polytope is a point that lies in the intersection of at least n facets but not in the interior of any 1-face, 2-face, etc.As a customary, we use the terms vertex and edge, respectively for the 0-dimensional and 1-dimensional faces of tiles, and facets for faces of dimension n -1.In the tilings we will study, the tiles will be convex polytopes in E n .Remember that the polytope P is convex if P contains the line segments joining any two points in P or on its boundary.Definition: tiling A tiling T of E n is a partition of E n into a countable number of closed cells with non-overlapping interiors:(5.1)The words tiling and tessellation are used interchangeably; similarly, tiles are often called cells.Definition: prototile set A prototile set P for a tiling T is a set of polytopes such that every tile of T is an isometric copy of an element of P.

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Type: book-chapter
Language: en
Landing Page: https://doi.org/10.1051/978-2-7598-1952-2.c006
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Authors: Boris Zhilinskii, Michel Leduc, Michel Le Bellac

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