Classes of similar tilings

Many tiings have some similarities which cannot be easily found. Here there is list of such classes which have been produced to simplify find related tilings when one is located. When a tiling is found, the class is given so that related ones can be displayed. The classes are listed below with the number in each. Click on the number to list the members of that class.

Escher's Alhambra tiling pattern(6)

This pattern is unusual since the angles ensure that all the sides are in the same ratio. The angles are already determined by the regularity of the stars. This class lists a number of small variants from the one that Escher draw while at the Alhambra. They are all from Morocco or Spain.

Large stars(11)

The class is those tilings with a star polygon with more than 18 points. Tiling with more than about 36 points are difficult to define adequately in a precise mathematical form used for the high-quality graphics available here.

Stars with 18 points(7)

The class is those tilings with a star polygon with 18 points. Note that there are two versions of the same tiling (both in brown). The first one is somewhat inferior since the irregular heptagon is not even approximately round as it should be. The second version is nearer to the original. The difference shows some of the problems in producing a good high-quality graphic.

Stars with 16 points with vertex angle not equal to 45°(16)

Very many 16-point stars have an angle of 15°, such as in Escher's Alhambra tiling pattern (see above). The other 16-point stars are listed here.

Regular polygons with more than eight sides(21)

The majority of these tiling are regular tiling whose enumeration has been completed. Only three are of Islamic origin.

Tiling whose angles are a multiple of 18°(24)

There are relatively few tilings satisfying this condition. They are a mixture of several types: Islamic, mathematical, and spiral.

Tiling with three or more proper stars(29)

By proper stars, we mean excluding diamonds in the count of the number of stars. There are four mathematical ones, and the rest are Islamic.

The last two patterns are alternative representation of the same artifact for which a photograph is available. One has straight interlacing, but the original has kinks in two places which is faithfully copied in the other alternative.

Tiling with star polygons with 14 points(10)

This number of points is comparatively rare - they are all Islamic.

Tiling with star polygons with 12 points whose vertex angle is 90°(31)

Only one mathematical tiling in this collection.

Tiling with star polygons with 12 points whose vertex angle is 60° and a hexagon(16)

This set provides a small number of related tilings.

Two-uniform tilings with regular polygons and star polygons(43)

This list is from Joseph Myers, see.

Tilings like Bourgoin, plate 118(3)

This set provides a small number of related tilings.

Monohedral tilings with symmetry group p4g(20)

Small set of tilings, mainly Islamic.

Tilings with coloured interlacing(9)

This is new to version 17 and required an enhancement to the software used to construct the graphics. The interlacing is not ignored in computing the symmetry group.

Tilings which are not constructible with straightedge and compass(53)

A regular (star) polygon is not constructible if the number of sides = 7, 9, 11, 13, 14, 18, 19, 21, 22, 23, 25, ... Hence we list them all here. The rest are almost certainly constructible.