Uniform tessellation
In geometry, a uniform tessellation is a vertex-transitive tessellation made from uniform polytope facets. All of its vertices are identical and there is the same combination and arrangement of faces at each vertex.
An n-dimensional uniform tessellation can be constructed on the surface of n-spheres, in n-dimensional Euclidean space, and n-dimensional hyperbolic space.
Nearly all uniform tessellations can be generated by a Wythoff construction, and represented by a Coxeter-Dynkin diagram. The terminology for the convex uniform polytopes used in uniform polyhedron, uniform polychoron, uniform polyteron, uniform polypeton, uniform tiling, and convex uniform honeycomb articles were coined by Norman Johnson.
Wythoffian tessellations can be defined by a vertex figure. For 2-dimensional tilings, they can be given by a vertex configuration listing the sequence of faces around every vertex. For example 4.4.4.4 represents a regular tessellation, a square tiling, with 4 squares around each vertex. In general an n-dimensional uniform tessellation vertex figures are define by an (n-1)-polytope with edges labeled with integers, representing the number of sides of the polygonal face at each edge radiating from the vertex.
Examples
| 2-dimensional tessellations | ||||
|---|---|---|---|---|
| Spherical | Euclidean | Hyperbolic | ||
Main article: Uniform polyhedron |
Main article: Uniform tilings in hyperbolic plane | |||
| Coxeter-Dynkin |    |
 |
 |
 |
| Picture |  Truncated icosidodecahedron. |
 Truncated trihexagonal tiling. |
 Truncated triheptagonal tiling (Poincaré disk model) |
 Truncated triapeirogonal tiling |
| Vertex figure |  |
 |
 |
|
| 3-dimensional tessellations | ||||
| 3-spherical | 3-Euclidean | 3-hyperbolic | ||
Main article: Uniform polychoron |
Main article: Convex uniform honeycomb |
Main article: Uniform honeycombs in hyperbolic space and paracompact uniform honeycomb | ||
| Coxeter diagram |   |
|
|
|
| Picture |  (Stereographic projection) 16-cell |
 cubic honeycomb |
 order-4 dodecahedral honeycomb (Beltrami-Klein model) |
 order-4 hexagonal tiling honeycomb (Poincaré disk model) |
| Vertex figure |  (Octahedron) |
 (Octahedron) |
 (Octahedron) |
(Octahedron) |
See also
- Uniform tiling
- List of uniform tilings
- Uniform tilings in hyperbolic plane
- Honeycomb (geometry)
- Wythoff construction
- Convex uniform honeycomb
- List of regular polytopes
References
- George Olshevsky, Uniform Panoploid Tetracombs, Manuscript (2006) (Complete list of 11 convex uniform tilings, 28 convex uniform honeycombs, and 143 convex uniform tetracombs)
- Branko Grünbaum, Uniform tilings of 3-space. Geombinatorics 4(1994), 49 - 56.
- Norman Johnson Uniform Polytopes, Manuscript (1991)
- Williams, Robert (1979). The Geometrical Foundation of Natural Structure: A Source Book of Design. Dover Publications, Inc. ISBN 0-486-23729-X.
- Grünbaum, Branko; Shephard, G. C. (1987). Tilings and Patterns. W. H. Freeman and Company. ISBN 0-7167-1193-1.
- H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973
- Critchlow, Keith (1970). Order in Space: A design source book. Viking Press. ISBN 0-500-34033-1.
- N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. Dissertation, University of Toronto, 1966
- A. Andreini, Sulle reti di poliedri regolari e semiregolari e sulle corrispondenti reti correlative (On the regular and semiregular nets of polyhedra and on the corresponding correlative nets), Mem. Società Italiana della Scienze, Ser.3, 14 (1905) 75–129.
External links
- Weisstein, Eric W., "Uniform tessellation", MathWorld.
- Tessellations of the Plane
- Richard Klitzing, 2D, Euclidean tesselations