Cut-and-project tilings

The famous Penrose tilings were originally described using somewhat arbitrary looking matching rules. It was later noticed by de Buijn that the P3 rhombus Penrose tiling could be constructed as a shadow of a five-dimensional cubic tiling, using the "cut-and-project" method.

The cut-and-project method to construct two-dimensional tilings consists of the following. First pick a plane in a higher-dimensional space containing a hypercubic lattice. Second keep only the points in the lattice that are close enough to the plane. (More precisely, we keep only the points that fall into the projection of a fundamental cell of the cubic tiling on the orthogonal complement of the plane.) Then project all the kept points on the plane. They form the vertices of a tiling. If the slopes defining the plane are not rational, the tiling obtained is aperiodic (i.e. never "repeats" exactly).

The P3 rhombus Penrose tiling can be obtained in this way by choosing the plane along a "maximally symmetric" direction in the five-dimensional space. The cut-and-project method applied to the maximally symmetric plane in a four-dimensional cubic lattice yields the Ammann-Beenker tiling.

In addition to a pair of direction, the cutting plane depends on an affine shift. Changing the shift doesn't change the shape of the tiles, but does change the tiling.

Here is a nice interactive applet by Grant Glouser, allowing you to play with the dimension, projection vector and shift parameter.

See also the Boyle-Steinhardt tilings project for another, aesthetically quite different generalisation of the Penrose and Ammann-Beenker tilings.

Maximally symmetric tilings

Here are the maximally symmetric tilings associated to the 4-, 5-, 6- anf 7-dimensional cubic lattice, with the affine shift set to zero.