List of uniform planar tilings
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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 as Archimedean in parallel to the Archimedean solids, and the dual tilings Laves tilings in honor of crystalographer Fritz Laves.
Contents |
[edit] Summary table of convex uniform tilings
| Uniform tiling or Archimedean tilings |
Dual tiling or Laves tilings |
|
|---|---|---|
 3.3.3.3.3.3 Triangular tiling (9 uniform colorings) |
 6.6.6 Hexagonal tiling |
|
 4.4.4.4 Square tiling (8 uniform colorings) |
4.4.4.4 Square tiling |
|
6.6.6 Hexagonal tiling (3 uniform colorings) |
3.3.3.3.3.3 Triangular tiling |
|
 3.3.3.3.6 (two chiral forms) Snub hexagonal tiling (1 uniform coloring) |
 V3.3.3.3.6 Floret pentagonal tiling |
|
 3.6.3.6 Trihexagonal tiling (2 uniform colorings) |
 V3.6.3.6 Quasiregular rhombic tiling |
|
 3.3.3.4.4 Elongated triangular tiling (2 uniform colorings) |
 V3.3.3.4.4 Prismatic pentagonal tiling |
|
 3.3.4.3.4 Snub square tiling (2 uniform colorings) |
 V3.3.4.3.4 Cairo pentagonal tiling |
|
 3.4.6.4 Small rhombitrihexagonal tiling (1 uniform coloring) |
 V3.4.6.4 Deltoidal trihexagonal tiling |
|
 4.8.8 Truncated square tiling (2 uniform colorings) |
 V4.8.8 Tetrakis square tiling |
|
 3.12.12 Truncated hexagonal tiling (1 uniform coloring) |
 V3.12.12 Triakis triangular tiling |
|
 4.6.12 Great rhombitrihexagonal tiling (1 uniform coloring) |
 V4.6.12 Hexakis triangular tiling |
[edit] Nonconvex uniform tilings
Inclusion of star polygon faces adds 14 uniform tilings. Branko Grünbaum, in Tilings and patterns, calls these hollow tilings because their interiors are undefined and they can only be drawn as polygon boundaries.
These 14 nonconvex forms, first published by H.S.M. Coxeter in the 1954 paper uniform polyhedra, given by vertex configurations are:
- 3.3.3.4/3.4/3
- 3.3.4/3.3.4/3
- 4.8/7.8/3
- 8.8/3.8/7.8/5
- 4/3.8/3.8/3
- 4.12/11.12/5
- 4.6/5.12/5
- 3/2.12.6.12
- 4.12.4/3.12/11
- 3.4/3.6.4/3
- 3.12/5.6/5.12/5
- 4.12/5.4/3.12/7
- 12.12/5.12/11.12/7
- 3/2.12/5.12/5
[edit] See also
- Convex uniform honeycomb - The 28 uniform 3-dimensional tessellations, a parallel construction to this plane set.
[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.
