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A packing in $\mathbf R^4$ which is also a  covering in $\mathbf R^4$ (cf. [[Covering  and packing|Covering and packing]]; [[Covering (of a set)|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|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|lattice of points]]. For an exhaustive account of planar tilings  see [[#References|[a3]]]. Higher-dimensional results and, in  particular, relations to crystallography are treated in  [[#References|[a2]]], [[#References|[a1]]]. Classical types of tilings  are Dirichlet–Voronoi tilings and Delone triangulations or $L$-partitions, see  [[#References|[a1]]] and [[Voronoi lattice types|Voronoi lattice  types]].

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