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.

References

[a1]	P. Erdös, P.M. Gruber, J. Hammer, "Lattice points" , Longman (1989)
[a2]	P.M. Gruber, C.G. Lekkerkerker, "Geometry of numbers" , North-Holland (1987) pp. Sect. (iv) (Updated reprint)
[a3]	B. Grünbaum, G.C. Shephard, "Tilings and patterns" , Freeman (1986)
[a4]	J.H. Conway, N.J.A. Sloane, "Sphere packing, lattices and groups" , Springer (1988)