about tilings & how to fill areas periodically with them
It has been known for a long time that the atomic structure of some
solid materials (f.e. rocks and ice) display a highly ordered, periodic
arrangement. Materials, that meet this criteria, are called crystals.
They can be observed in nature in various shapes and symmetries and were
the subject of research for generations of scientists. But in 1982
Israelian Physicist Dan Shechtman stumbled upon an odd locking electron
diffraction pattern while conduction a routine study of an
aluminium-manganese alloy \(Al_6Mn\). On first sight this pattern seemed
to have the properties of a crystalline structure, but it lacked one
thing: periodicity. Soon after the term quasicrystal was used to
describe the observation Shechtman had made.
In the following we will try to get a grip on the mathematical
understanding of aperiodic patterns as the one Shechtman discovered. We
will start by giving the broad definition of tilings and discussing
certain types.
Informally, a tiling is the covering of a plane using geometrical
shapes, called tiles, so that the plane is filled entirely and the
tiles do not overlap. Mathematically, this can be defined as a
collection of open and disjoint sets, so that the closures of these
sets cover the whole plane. A tiling is called periodic, if a
certain translation of the plane results in an unchanged pictures.
One simple example for this is the tiling of \(\mathbb{R}^2\) using
only one regular polygons as seen in the interactive figure below
(it can be shown, that this is only possible for triangles, cubes
and hexagons).
It is obvious how conventional crystals can be thought of as
periodic tilings. But what about quasicrystals?
The fitting mathematical model for these are the so-called aperiodic
tilings, which are defined as non-periodic tilings that also do not
contain any arbitrarily large periodic regions or patches. Several
methods for generating aperiodic tilings are known, but we will
focus on the so-called Cut-and-Project-Method. This method can be
understood best by looking into a simple example, the
one-dimensional aperiodic Fibonacci tiling (see next page).
Change the number of sides