Chapter 5: Two Dimensions and Beyond

Section 7: Systems Based on Constraints

Tiling [problems]

The constraints discussed here are similar to those encountered in covering the plane with tiles of various shapes. Of regular polygons, only squares, triangles and hexagons can be used to do this, and in these cases the tilings are always repetitive. For some time it was believed that any set of tiles that could cover the plane could be arranged to do so repetitively. But in 1964 Robert Berger demonstrated that this was not the case, and constructed a set of about 20,000 tiles that could cover the plane only in a nested fashion. Later Berger reduced the number of tiles needed to 104. Then Raphael Robinson in 1971 reduced the number tiles to six, and in 1974 Roger Penrose showed that just two tiles were necessary. Penrose's tiles can cover the plane only in a nested pattern that can be constructed from a substitution system that successively subdivides each tile, as shown on page 932. (Note that various dissections of these tiles can also be used. The edges of the particular shapes shown should strictly be distinguished in order to prevent trivial periodic arrangements.) The triangles in the construction have angles which are multiples of π/5, so that the whole tiling has an approximate 5-fold symmetry (see page 994). Repetitive tilings of the plane can only have 3-, 4- or 6-fold symmetry.

No single shape is known which has the property that it can tile the plane only non-repetitively, although one strongly suspects that one must exist. In 3D, John Conway has found a single biprism that can fill space only in a sequence of layers with an irrational rotation angle between each layer.

In addition, in no case has a simple set of tiles been found which force a pattern more complicated than a nested one. The results on page 221 in this book can be used to construct a complicated set of tiles with this property, but I suspect that a much simpler set could be found.

(See also page 1139.)

From Stephen Wolfram: A New Kind of Science [citation]