Explore Nonperiodic Tilings
The "NonperiodicTiling" entity domain contains more than 15 tilings that fill the plane only nonperiodically.
A number of these tilings correspond to so-called rep-tiles, i.e. tiles that can be dissected into smaller copies of themselves.
Perhaps the best-known nonperiodic tiling is the kites and darts tiling.
Using Wolfram|Alpha itself, you can visualize the way in which the tiling is built up.
You can also explore nonperiodic tilings directly. Consider the substitution diagram of the L-tetromino rep-tiling.
Pick out the vertices on the left- and right-hand sides of the substitution.
Now use FindGeometricTransform to find geometric transformations needed to reorient each L-tetromino to its position on the right-hand side of the substitution diagram.
Finally, iterate the transformation and randomly color the resulting tetrominoes.