On the geometry of Penrose tilings: projection and substitution
The Penrose tiling is a planar tiling with two rhombic tiles, orginally
designed by Roger Penrose in the early 1970s to illustrate the fact that
local properties of tiles (matching rules) can enforce global
aperiodicity. Penrose demonstrated this by a renormalization argument
that involves a substitution rule. Some ten years later, Nico de Bruijn
showed that Penrose's tiling can also be viewed as the projection of a
slice of a 5-dimensional lattice. Penrose tilings have received much
attention in the context of quasicrystals (that were discovered to exist
in 1984). Despite many other examples of quasicrystaline tilings have
been discovered, few shared the remarkable combination of properties of
the Penrose tiling.
The main result of my joint work with Edmund Harriss (Arkansas) concerns
a comprehensive characterisation of all tilings of "Penrose type", i.e.
tilings of $R^n$ that, like Penrose's original example, have matching
rules, substitutions rules, and can be constructed by De Bruijn's
projection method. A generalisation of Rauzy-Veech renormalisation for
interval-exchange transformations lies at the foundation of this result.
Our results are constructive and reveal an infinity of novel examples of
tilings of Penrose type.