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.