This table shows the 11 convex uniform tilings (regular and semiregular) of the Euclidean plane, and their dual tilings.
There are three regular, and eight semiregular, tilings in the plane. The semiregular tilings form new tilings from their duals, each made from one type of irregular face.
Uniform tilings are listed by their vertex configuration, the sequence of faces that exist on each vertex. For example 4.8.8 means one square and two octagons on a vertex.
These 11 uniform tilings have 32 different uniform colorings. A uniform coloring allows identical sided polygons at a vertex to be colored differently, while still maintaining vertex-uniformity and transformational congruence between vertices. (Note: Some of the tiling images shown below are not color uniform)
In addition to the 11 convex uniform tilings, there are also 14 nonconvex tilings, using star polygons, and reverse orientation vertex configurations.
Dual tilings are listed by their face configuration, the number of faces at each vertex of a face. For example V4.8.8 means isosceles triangle tiles with one corner with 4 triangles, and two corners containing 8 triangles.
In the 1987 book, Tilings and Patterns, Branko Grünbaum calls the vertex uniform tilings Archimedean in parallel to the Archimedean solids, and the dual tilings Laves tilings in honor of crystalographer Fritz Laves. John Conway calls the duals as Catalan tilings, in parallel to the Catalan solid polyhedra.
Contents |
Convex uniform tilings of the Euclidean plane
The [4,4] group family
| Uniform tilings (Platonic and Archimedean) |
Vertex figure Wythoff symbol(s) Symmetry group |
Dual uniform tilings (Called Laves or Catalan tilings) |
|---|---|---|
 Square tiling (quadrille) |
 4.4.4.4 (or 44) 4 | 2 4 *442      |
self-dual (quadrille) |
 Truncated square tiling (truncated quadrille) |
 4.8.8 2 | 4 4 4 4 2 | *442 |
 Tetrakis square tiling (kisquadrille) |
 Snub square tiling (snub quadrille) |
 3.3.4.3.4 | 4 4 2 4*2, 442, and xx  |
 Cairo pentagonal tiling (4-fold pentille) |
The [6,3] group family
| Platonic and Archimedean tilings | Vertex figure Wythoff symbol(s) Symmetry group |
Dual Laves tilings |
|---|---|---|
 Hexagonal tiling (hextille) |
 6.6.6 (or 63) 3 | 6 2 2 6 | 3 3 3 3 | *632    |
 Triangular tiling (deltile) |
 Trihexagonal tiling (hexadeltille) |
 3.6.3.6 (or (3.6)2) 2 | 6 3 3 3 | 3 *632 and *333  |
 Rhombille tiling (rhombille) |
 Truncated hexagonal tiling (truncated hextille) |
 3.12.12 2 3 | p6m (*632) |
 Triakis triangular tiling (kisdeltile) |
Triangular tiling (deltile) |
 3.3.3.3.3.3 (or 36) 6 | 3 2 3 | 3 3 | 3 3 3 *632 and *333   |
Hexagonal tiling (hextile) |
 Rhombitrihexagonal tiling (rhombihexadeltille) |
 3.4.6.4 3 | 6 2 *632 |
 Deltoidal trihexagonal tiling (tetrille) |
 Truncated trihexagonal tiling (truncated hexadeltille) |
 4.6.12 or *632 2 6 3 | *632 |
 Bisected hexagonal tiling (kisrhombille) |
 Snub hexagonal tiling (snub hexatille) |
 3.3.3.3.6 | 6 3 2 632 |
 Floret pentagonal tiling (6-fold pentille) |
Non-Wythoffian uniform tiling
| Platonic and Archimedean tilings | Vertex figure Wythoff symbol(s) Symmetry group |
Dual Laves tilings |
|---|---|---|
 Elongated triangular tiling (isosnub quadrille) |
 3.3.3.4.4 2 | 2 (2 2) cmm none |
 Prismatic pentagonal tiling (iso(4-)pentille) |
See also
- Convex uniform honeycomb - The 28 uniform 3-dimensional tessellations, a parallel construction to the convex uniform Euclidean plane tilings.
- Uniform tilings in hyperbolic plane
References
- Grünbaum, Branko; Shephard, G. C. (1987). Tilings and Patterns. W. H. Freeman and Company. ISBN 0-7167-1193-1.
- H.S.M. Coxeter, M.S. Longuet-Higgins, J.C.P. Miller, Uniform polyhedra, Phil. Trans. 1954, 246 A, 401-50.
- Williams, Robert (1979). The Geometrical Foundation of Natural Structure: A Source Book of Design. Dover Publications, Inc. ISBN 0-486-23729-X. (Section 2-3 Circle packings, plane tessellations, and networks, p 34-40)
- John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 [1] (Chapter 19, Archimedean tilings, table 19.1, Chapter 21, Naming Archimedean and Catalan polyhedra and tilings, p288 table)
External links
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