also demonstrate a remarkable rotational symmetry; every pattern that ap-
pears somewhere in the tiling also appears rotated by 36 degrees, and with
the same frequency. This “statistical symmetry” contradicts the long-held
belief that only rotations by 60, 90, 120 and 180 degrees can appear in highly
ordered structures.
The third ingredient came from physics, or you might say from materials
science. In 1982, Shechtman and coworkers [SBGC] discovered a new class
of solid, neither crystal nor amorphous, called quasicrystals. Quasicrystals
have sharp diffraction patterns, long thought to be the hallmark of a periodic
crystal, but some of these patterns have 8- or 10-fold rotational symmetry.
It didn’t take people long to realize that quasicrystals are modeled well by
aperiodic tilings, and in particular by 3-dimensional versions of the Penrose
tiling and by several other cut-and-project tilings!
The fourth ingredient came from ergodic theory and dynamical systems,
where substitution sequences had long been a subject of interest. Some of
the simplest substitutions, like the Thue-Morse substitution, were defined
over 100 years ago. However, it was only in the 1980s that people went from
substitution subshifts to substitution tilings in one dimension, and from
there to higher-dimensional substitution tilings. It didn’t hurt that the
Penrose tiling could be realized in this way. Soon people discovered other
interesting geometric “rep-tiles” and computed properties of the resulting
These trends came together in the 1990s. Tilings, including the Penrose
tiling, were used to model quasicrystals. These tilings were in turn gener-
ated in a number of ways, including local matching rules, cut-and-project
methods, and substitutions. The tilings were then studied as dynamical
systems, and their dynamical properties were related to physical properties
of the quasicrystals that they model.
Suppose you had a quasicrystal that was modeled by an aperiodic tiling.
A physicist might ask the following questions about the quasicrystal.
P1. What is the x-ray diffraction pattern of the material? This is
equivalent to the Fourier transform of the autocorrelation function of
the positions of the atoms. Sharp peaks are the hallmark of ordered
materials, such as crystals and quasicrystals.
P2. What are the possible energy levels of electrons in the material?
The locations of the atoms determine a quasiperiodic potential, and the
spectrum of the corresponding Schr¨ odinger Hamiltonian has infinitely
many gaps. What are the energies of these gaps, and what is the
density of states corresponding to each gap?
P3. Can you really tell the internal structure of the material from
diffraction data? What deformations (either local or non-local) of the
molecular structure are consistent with the combinatorics of the molec-
ular bonds? Which of these are detectable from diffraction data?
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