List of Euclidean uniform tilings


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.
John Conway calls these uniform duals Catalan tilings, in parallel to the Catalan solid polyhedra.
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.
In addition to the 11 convex uniform tilings, there are also 14 nonconvex tilings, using star polygons, and reverse orientation vertex configurations.

Laves tilings

In the 1987 book, Tilings and Patterns, Branko Grünbaum calls the vertex-uniform tilings Archimedean in parallel to the Archimedean solids. Their dual tilings are called Laves tilings in honor of crystallographer Fritz Laves. They're also called Shubnikov–Laves tilings after Shubnikov, Alekseĭ Vasilʹevich. John Conway called the uniform duals Catalan tilings, in parallel to the Catalan solid polyhedra.
The Laves tilings have vertices at the centers of the regular polygons, and edges connecting centers of regular polygons that share an edge. The tiles of the Laves tilings are called planigons. This includes the 3 regular tiles and 8 irregular ones. Each vertex has edges evenly spaced around it. Three dimensional analogues of the planigons are called stereohedrons.
These 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 four triangles, and two corners containing eight triangles. The orientations of the vertex planigons are consistent with the vertex diagrams in the below sections.

Convex uniform tilings of the Euclidean plane

All reflectional forms can be made by Wythoff constructions, represented by Wythoff symbols, or Coxeter-Dynkin diagrams, each operating upon one of three Schwarz triangle,, or, with symmetry represented by Coxeter groups: , , or ]. Alternated forms such as the snub can also be represented by special markups within each system. Only one uniform tiling can't be constructed by a Wythoff process, but can be made by an elongation of the triangular tiling. An orthogonal mirror construction also exists, seen as two sets of parallel mirrors making a rectangular fundamental domain. If the domain is square, this symmetry can be doubled by a diagonal mirror into the family.
Families:
Uniform tilings
Vertex figure and dual face
Wythoff symbol
Symmetry group
Coxeter diagram		Dual-uniform tilings

The 6,3 group family

Platonic and Archimedean tilingsVertex figure and dual face
Wythoff symbol
Symmetry group
Coxeter diagram		Dual Laves tilings

Non-Wythoffian uniform tiling

Platonic and Archimedean tilingsVertex figure and dual face
Wythoff symbol
Symmetry group
Coxeter diagram		Dual Laves tilings

Uniform colorings

There are a total of 32 uniform colorings of the 11 uniform tilings:
  1. Triangular tiling – 9 uniform colorings, 4 wythoffian, 5 nonwythoffian
  2. *
  3. Square tiling – 9 colorings: 7 wythoffian, 2 nonwythoffian
  4. *
  5. Hexagonal tiling – 3 colorings, all wythoffian
  6. *
  7. Trihexagonal tiling – 2 colorings, both wythoffian
  8. *
  9. Snub square tiling – 2 colorings, both alternated wythoffian
  10. *
  11. Truncated square tiling – 2 colorings, both wythoffian
  12. *
  13. Truncated hexagonal tiling – 1 coloring, wythoffian
  14. *
  15. Rhombitrihexagonal tiling – 1 coloring, wythoffian
  16. *
  17. Truncated trihexagonal tiling – 1 coloring, wythoffian
  18. *
  19. Snub hexagonal tiling – 1 coloring, alternated wythoffian
  20. *
  21. Elongated triangular tiling – 1 coloring, nonwythoffian
  22. *