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A packing in <img align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071050/p07105024.png" /> which is also a  covering in <img align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071050/p07105025.png" /> (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 <img align="absmiddle"  border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071050/p07105026.png" /> without gaps or overlaps. The sets are called tiles. If all sets are  congruent, they are the copies of a prototile.
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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 <img align="absmiddle"  border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071050/p07105027.png" /><img align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071050/p07105028.png" />, where <img  align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071050/p07105029.png" /> 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 <img  align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071050/p07105031.png" />-partitions, see  [[#References|[a1]]] and [[Voronoi lattice types|Voronoi lattice  types]].
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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====
 
====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>
 
<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>

Latest revision as of 21:44, 11 April 2014

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)
How to Cite This Entry:
Tiling. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Tiling&oldid=31580