Subtile is a system for computing and displaying aperiodic tilings by substitution or composition rules. It was first released in September 1995 and last updated in December 1998. It was inspired by Marjorie Senechal's Quasicrystals and Geometry (Cambridge UP, Cambridge, 1995), with support from Branko Grünbaum and G. C. Shephard, Tilings and Patterns, (WH Freeman, New York, 1987). But it's more true to say that the tilings themselves inspired the program, and that's why this new release has come about.

A tiling is a way to fit together tiles. Tiles, of course, are little pieces of ceramic which come in different shapes and often have different colors or designs on them.

Most tilings are periodic, which means that you can find the unit cell of the tiling and produce the rest of the tiling by duplicating and
moving the unit cell.

There are such things as random tilings which have no pattern at all other than the fitting together of the pieces.

Aperiodic tilings, in contrast, always seem to be on the verge of repeating themselves, but never exactly repeat themselves. As you increase the area tiled, you keep getting new patterns. But the patterns aren't random, they still have some kind of order to them.

When I first wrote subtile, there were four known ways of generating aperiodic tilings: 1) by matching edge and vertex decorations; 2) by projecting high dimensional lattices onto a plane; 3) by de Bruijn's generalized dual method; 4) by substitution rules. In 1996, Jeong and Steinhardt described a fifth method.


Branko Grünbaum and G. C. Shephard, Tilings and Patterns, WH Freeman, New York, 1987.

Marjorie Senechal, Quasicrystals and Geometry, Cambridge UP, Cambridge, 1995.

Paul J. Steinhardt, Hyeong-Chai Jeong, A simpler approach to Penrose tiling with implications for quasicrystal formation, Nature, 382: 431-433, 01 Aug 1996.