Wolfram Language

Exploring Periodic Tilings

The "PeriodicTiling" entity domain contains more than 50 tilings that fill the plane periodically.

These include the three tilings by regular polygons.

While the regular pentagon does not tile the plane, there are exactly 15 distinct periodic tilings using identical (but nonregular) pentagons that do.

Another interesting class of tilings is the eight semiregular tilings consisting of two or more convex regular polygons such that the same polygons in the same order surround each polygon vertex.

The first three of these are related to polyhedra bearing similar-sounding names.

To see how periodic tilings can be built up from primitive parts, take the primitive unit of a particular pentagonal plane tiling.

Break up the unit into its six constituent polygons.

Now retrieve the translation vectors leading to a periodic tiling.

Finally, beginning with the original primitive unit, perform translations using the two translation vectors to build up the tiling.

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