**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:

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