Tiling By Regular Polygons - Archimedean, Uniform or Semiregular Tilings

Archimedean, Uniform or Semiregular Tilings

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

If the requirement of flag-transitivity is relaxed to one of vertex-transitivity, while the condition that the tiling is edge-to-edge is kept, there are eight additional tilings possible, known as Archimedean, uniform or semiregular tilings. Note that there are two mirror image (enantiomorphic or chiral) forms of 34.6 (snub hexagonal) tiling, both of which are shown in the following table. All other regular and semiregular tilings are achiral.


34.6
Snub hexagonal tiling

34.6
Snub hexagonal tiling reflection

3.6.3.6
Trihexagonal tiling

33.42
Elongated triangular tiling

32.4.3.4
Snub square tiling

3.4.6.4
Rhombitrihexagonal tiling

4.82
Truncated square tiling

3.122
Truncated hexagonal tiling

4.6.12
Truncated trihexagonal tiling

Gr├╝nbaum 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. Though these yield the same set of tilings in the plane, in other spaces there are Archimedean tilings which are not uniform.

Read more about this topic:  Tiling By Regular Polygons

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