List of uniform tilings
From Wikipedia, the free encyclopedia
This table shows the 11 convex uniform tilings 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 forms, 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.
Contents |
[edit] Convex uniform tilings of the Euclidean plane
[edit] The R3 [4,4] group family
| Platonic and Archimedean tilings | Vertex figure Wythoff symbol(s) Symmetry group |
Dual Laves tilings |
|---|---|---|
 Square tiling |
 4.4.4.4 4 | 2 4 p4m |
self-dual |
 Truncated square tiling |
 4.8.8 2 | 4 4 4 4 2 | p4m |
 Tetrakis square tiling |
 Snub square tiling |
 3.3.4.3.4 | 4 4 2 p4g |
 Cairo pentagonal tiling |
[edit] The V3 [6,3] group family
| Platonic and Archimedean tilings | Vertex figure Wythoff symbol(s) Symmetry group |
Dual Laves tilings |
|---|---|---|
 Hexagonal tiling |
 6.6.6 3 | 6 2 2 6 | 3 3 3 3 | p6m |
 Triangular tiling |
 Trihexagonal tiling |
 3.6.3.6 2 | 6 3 3 3 | 3 p6m |
 Quasiregular rhombic tiling |
 Truncated hexagonal tiling |
 3.12.12 2 3 | 6 3 3 | 3 p6m |
 Triakis triangular tiling |
Triangular tiling |
 3.3.3.3.3.3 6 | 3 2 3 | 3 3 | 3 3 3 p6m |
Hexagonal tiling |
 Small rhombitrihexagonal tiling |
 3.4.6.4 3 | 6 2 p6m |
 Deltoidal trihexagonal tiling |
 Great rhombitrihexagonal tiling |
 4.6.12 2 6 3 | p6m |
 Bisected hexagonal tiling |
 Snub hexagonal tiling |
 3.3.3.3.6 | 6 3 2 p6 |
 Floret pentagonal tiling |
[edit] Non-Wythoffian uniform tiling
| Platonic and Archimedean tilings | Vertex figure Wythoff symbol(s) Symmetry group |
Dual Laves tilings |
|---|---|---|
 Elongated triangular tiling |
 3.3.3.4.4 2 | 2 (2 2) cmm |
 Prismatic pentagonal tiling |
[edit] Expanded lists of uniform tilings
There are a number ways the list of uniform tilings can be expanded:
- Vertex figures can have retrograde faces and turn around the vertex more than once.
- Star polygons tiles can be included.
- Apeirogons, {∞}, can be used as tiling faces.
Branko Grünbaum, in the 1987 book Tilings and patterns, in section 12.3 enumerates a list of 25 uniform tilings, including the 11 convex forms, and adds 14 more he calls hollow tilings which included the first two expansions above, star polygon faces and vertex figures.
H.S.M. Coxeter et al, in the 1954 paper 'Uniform polyhedra', in Table 8: Uniform Tessellations, uses all of these and enumerates a total of 38 uniform tilings.
Finally, if a tiling made of 2 apeirogons is also counted, the total can be considered 39 uniform tilings.
The 7 new tilings with {∞} tiles, given by vertex figure and Wythoff symbol are:
- ∞.∞ (Two half-plane tiles, infinite dihedron)
- 4.4.∞ - ∞ 2 | 2 (an infinite prism)
- 3.3.3.∞ - | 2 2 ∞ (an infinite antiprism)
- 4.∞.4/3.∞ - 4/3 4 | ∞ (alternate square tiling)
- 3.∞.3.∞.3.∞ - 3/2 | 3 ∞ (alternate triangular tiling)
- 6.∞.6/5.∞ - 6/5 6 | ∞ (alternate trihexagonal tiling with only hexagons)
- ∞.3.∞.3/2 - 3/2 3 | ∞ (alternate trihexagonal tiling with only triangles)
The remaining list includes 21 tilings, 7 with {∞} tiles (apeirogons). Drawn as edge-graphs there are only 14 unique tilings, and the first is identical to the 3.4.6.4 tiling.
The 21 grouped by shared edge graphs, given by vertex figures and Wythoff symbol, are:
- Type 1
- 3/2.12.6.12 - 3/2 6 | 6
- 4.12.4/3.12/11 - 2 6 (3/2 3) |
- Type 2
- 8/3.4.8/3.∞ - 4 ∞ | 4/3
- 8/3.8.8/5.8/7 - 4/3 4 (2 ∞) |
- 8.4/3.8.∞ - 4/3 ∞ | 4
- Type 3
- 12/5.6.12/5.∞ - 6 ∞ | 6/5
- 12/5.12.12/7.12/11 - 6/5 6 (3 ∞) |
- 12.6/5.12.∞ - 6/5 ∞ | 6
- Type 4
- 12/5.3.12/5.6/5 - 3 6 | 6/5
- 12/5.4.12/7.4/3 - 2 6/5 (3/2 3) |
- 4.3/2.4.6/5 - 3/2 6 | 2
- Type 5
- 8.8/3.∞ - 4/3 4 ∞ |
- Type 6
- 12.12/5.∞ - 6/5 6 ∞ |
- Type 7
- 8.4/3.8/5 - 2 4/3 4 |
- Type 8
- 6.4/3.12/7 - 2 3 6/5 |
- Type 9
- 12.6/5.12/7 - 3 6/5 6 |
- Type 10
- 4.8/5.8/5 - 2 4 | 4/3
- Type 11
- 12/5.12/5.3/2 - 2 3 | 6/5
- Type 12
- 4.4.3/2.3/2.3/2 - non-Wythoffian
- Type 13
- 4.3/2.4.3/2.3/2 - | 2 4/3 4/3 (snub)
- Type 14
- 3.4.3.4/3.3.∞ - | 4/3 4 ∞ (snub)
[edit] Uniform tilings in hyperbolic plane
There are an infinite number of uniform tilings on the hyperbolic plane based on the (p q 2) hyperbolic regular tilings.
Shown with Poincaré disk model are two families:
[edit] The [7,3] group family
| Uniform hyperbolic tilings | Vertex figure Wythoff symbol(s) Symmetry group |
Dual tilings |
|---|---|---|
 Order-3 heptagonal tiling |
7.7.7 3 | 7 2 [7,3] |
 Order-7 triangular tiling |
 Order-3 truncated heptagonal tiling |
3.14.14 2 3 | 7 [7,3] |
 Order-7 triakis triangular tiling |
 Triheptagonal tiling |
3.7.3.7 2 | 7 3 [7,3] |
 Order-7-3 quasiregular rhombic tiling |
 Order-7 truncated triangular tiling |
7.6.6 2 7 | 3 [7,3] |
 Order-3 heptakis heptagonal tiling |
Order-7 triangular tiling |
37 7 | 3 2 [7,3] |
Order-3 heptagonal tiling |
 Small rhombitriheptagonal tiling |
3.4.7.4 3 | 7 2 [7,3] |
 Deltoidal triheptagonal tiling |
 Great rhombitriheptagonal tiling |
4.6.14 2 7 3 | [7,3] |
 Order-3 bisected heptagonal tiling |
 Order-3 snub heptagonal tiling |
3.3.3.3.7 | 7 3 2 [7,3] |
 Order-7-3 floret pentagonal tiling |
[edit] The [5,4] group family
| Uniform hyperbolic tilings | Vertex figure Wythoff symbol(s) Symmetry group |
Dual tilings |
|---|---|---|
 Order-4 pentagonal tiling |
5.5.5.5 4 | 5 2 [5,4] |
 Order-5 square tiling |
 Truncated pentagonal tiling |
4.10.10 2 4 | 5 [5,4] |
 Order-5 tetrakis square tiling |
 tetrapentagonal tiling |
4.5.4.5 2 | 5 4 [5,4] |
 Order-5-4 quasiregular rhombic tiling |
 Order-5 truncated square tiling |
8.8.5 2 5 | 4 [5,4] |
 Order-4 pentakis pentagonal tiling |
Order-5 square tiling |
45 5 | 4 2 [5,4] |
Order-4 pentagonal tiling |
 Small rhombitetrapentagonal tiling |
4.4.5.4 4 | 5 2 [5,4] |
 Deltoidal tetrapentagonal tiling |
 Great rhombitetrapentagonal tiling |
4.8.10 2 5 4 | [5,4] |
 Order-4 bisected pentagonal tiling |
 Order-4 snub pentagonal tiling |
3.3.4.3.5 | 5 4 2 [5,4] |
 Order-5-4 floret pentagonal tiling |
[edit] (4 3 3) family
| Uniform hyperbolic tilings | Vertex figure Wythoff symbol(s) Symmetry group |
Dual tilings |
|---|---|---|
 Order-4-3-3_t0 tiling |
(3.4)^3 3 | 3 4 (4 3 3) |
 Order-4-3-3_t0 dual tiling |
 Order-4-3-3_t01 tiling |
3.8.3.8 3 3 | 4 (4 3 3) |
 Order-4-3-3_t01 dual tiling |
 Order-4-3-3_t12 tiling |
3.6.4.6 4 3 | 3 (4 3 3) |
 Order-4-3-3_t12 dual tiling |
 Order-4-3-3_t2 tiling |
(3.3)4 4 | 3 3 (4 3 3) |
 Order-4-3-3_t2 dual tiling |
 Order-4-3-3_t012 tiling |
6.6.8 4 3 3 | (4 3 3) |
 Order-4-3-3_t012 dual tiling |
 Order-4-3-3_snub tiling |
3.3.3.3.3.4 | 4 3 3 (4 3 3) |
 Order-4-3-3_snub dual tiling |
[edit] See also
- Convex uniform honeycomb - The 28 uniform 3-dimensional tessellations, a parallel construction to the convex uniform Euclidean plane tilings.
[edit] External links
- Uniform Tessellations on the Euclid plane
- Tessellations of the Plane
- David Bailey's World of Tessellations
- k-uniform tilings
- n-uniform tilings
[edit] 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.
