Periodic Tilings

Imagine you pick up a copy of a tiling, shift it in some direction (without rotating it), and when you put copy down again it matches up exactly with the original. In this case, this shift is called a translation and we say that this translation is a symmetry of the tiling.

A tiling is said to be periodic if there exist, among the symmetries of the tiling, at least two translations in non-parallel directions.

Although the tiling above admits a translation as a symmetry, it is not a periodic tiling because all the translations in its symmetry group are in the same direction – it does not admit two translations in nonparallel directions.

The tiling below is an example of a periodic tiling; the green arrows indicate two non-parallel translations that are symmetries of the tiling.

Another example of a periodic tiling:

A tiling that is not periodic (i.e. does not have two translations in non-parallel directions) is said to be nonperiodic.

The monohedral tiling below is an example of a nonperiodic tiling. The central "star" occurs nowhere else in the tiling, and so no translations are possible.

Here is a nonperiodic dihedral tiling: