New Models for Quasicrystals
Construction methods for non-periodic tilings of the plane can be extended to space. Above all with the help of the substitution principle one can generate three-dimensional non-periodic space fillings and thus provide a model for those strange solids which physicists call quasicrystals.
The last episode of this section was about how artists use the special properties of the so-called Penrose parquet for their purposes. Its original building blocks are two "golden triangles": isosceles triangles with the aspect ratio of the golden ratio (τ ≈ 1.618), a wide one with a long base and short sides and a tall one with the sides τ times as long as the base. Two specimens of a golden triangle, placed bases together, form a thick and thin rhombus, respectively, and these are the cobblestones with which one can lay what is in principle an infinite parquet. However, not randomly! Special "layout rules", illustrated by a pattern on the stones or a deformation of their edges, mean that the parquet is non-periodic. This means that, unlike in classic bathroom tiling, there is no parallel shift that would cause the entire parquet floor to cover itself. Rather, there's a five-fold symmetry to the whole pattern-sort of.
A very powerful theoretical tool for understanding non-periodic tilings is the so-called substitution. Each stone is broken down into a number of smaller specimens from the same assortment of stones and the parquet fragment obtained is enlarged so that each stone has the original size again. These two steps can be repeated as often as you like, with the effect that any small part of a parquet floor - a single stone is enough - gradually grows into ever larger areas and, in the limit, covers the entire infinite plane.
From the point of view of solid state physicists, the whole beautiful theory of Penrose floors is just a preliminary exercise for the same thing in three dimensions instead of two…
