Tiling

A packing in $\mathbf R^4$ which is also a covering in $\mathbf R^4$ (cf. Covering and packing; Covering (of a set)) is called a tiling or tesselation. In other words: A tiling is a countable family of closed sets which cover $\mathbf R^4$ without gaps or overlaps. The sets are called tiles. If all sets are congruent, they are the copies of a prototile.

In the geometry of numbers, lattice tilings are of interest; there are tilings of the form $M+a$, $a\in\Lambda$, where $\Lambda$ is a lattice of points. For an exhaustive account of planar tilings see [a3]. Higher-dimensional results and, in particular, relations to crystallography are treated in [a2], [a1]. Classical types of tilings are Dirichlet–Voronoi tilings and Delone triangulations or $L$-partitions, see [a1] and Voronoi lattice types.

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