This image shows an very simple way to produce a non-periodic tiling.
Most of the tilings usually encountered are periodic,
which mean that a small part of the tiling can copied and regulary arranged
to recompose the original tiling.
However, if you observe closely 20000701, you'll have trouble finding
any periodicity. The tiles looks more or less similar, but are different.
Even if you find two identical tiles, their respective neighbours won't
correspond. How can we produce such a strange tiling ?
If replace them all the same way, we should obtain something like (3).
However, it's much more interesting to choose their left-right and top-bottom
orientations randomly. This give the tiles a "similar but different" look
and since the broken lines are oriented randomly, no periodicity appears
Each tile has four side, each of which can take four different configurations
(provided the broken line has no symetry, as above). So we have 44
= 256 possibilities. However, if we consider that two tiles that differs
only by a rotation or a symetry as equal, there remains 38 different tiles
as shown below. It's much more than one can imagine seeing the original