List of k-uniform tilings

Higher k-uniform tilings are listed by their vertex figures, but are not generally uniquely identified this way.

k-uniform tilings with the same vertex figures can be further identified by their wallpaper group symmetry.

Each can be grouped by the number m of distinct vertex figures, which are also called m-Archimedean tilings.

[1] Finally, if the number of types of vertices is the same as the uniformity (m = k below), then the tiling is said to be Krotenheerdt.

In general, the uniformity is greater than or equal to the number of types of vertices (m ≥ k), as different types of vertices necessarily have different orbits, but not vice versa.

Vertex-transitivity means that for every pair of vertices there is a symmetry operation mapping the first vertex to the second.

Note that there are two mirror image (enantiomorphic or chiral) forms of 34.6 (snub hexagonal) tiling, only one of which is shown in the following table.

Grünbaum and Shephard distinguish the description of these tilings as Archimedean as referring only to the local property of the arrangement of tiles around each vertex being the same, and that as uniform as referring to the global property of vertex-transitivity.

Brian Galebach's search identified 332 5-uniform tilings, with 2 to 5 types of vertices.