The lovely tiling to the right is an example of a nonperiodic tiling.
Why? Consider the red, twelve pointed star lying in the middle of the
tiling. This star appears only at the center.
This star is not repeated throughout the tiling.
If the tiling were periodic, the star would lie in some fundamental domain for the tiling. Recall
that a fundamental domain is a piece of the tiling you can shift
around to tile the entire plane. Therefore the star would lie in
every periodic parallelogram of the lattice corresponding to the
fundamental domain. Obviously, the star cannot appears infinitely
many places in the tiling, since it appears only once in the middle of
the tiling. (The page about periodic
tilings introduces these terms and ideas, if you are feeling a