Periodic tiling

Remarks

Example: the tiling of $\R^2$ by the unit squares $[a, a+1] \times [b, b+1]$, $(a, b) \in \Z^2$, is periodic. Translation by $(1, 0)$ maps $[a, a+1] \times [b, b+1]$ to $[a+1, a+2] \times [b, b+1]$, so $(1, 0)$ is a period, and in the same way $(0, 1)$ is a period; these two vectors are independent.

Non-example: shift the row of squares $[a, a+1] \times [0, 1]$ by $1/2$ to the right and leave all other rows as they are. The vector $(1, 0)$ is still a period, but no vector with a nonzero second coordinate is one, because the shifted row is the only row with vertices at half-integers. So the tiling has periods in only one direction and is not periodic.

A tiling with no nonzero period at all is called non-periodic. The Penrose tilings are non-periodic.