§1 Quasicrystals

§ 1.1. Point symmetry.

Rotational symmetry is well-known from flowers and ornaments. In geometry it singles out regular figures. As point symmetry it governs the atomic structure of crystals and quasicrystals.

Polytopes of regular shape bounded by planes were conceived and manufactured in history long before the invention of systematic geometry, ranging from small prehistoric models to the huge pyramids of Gizeh. The Greeks were the first to consider polytopes as building blocks of matter. Platon in his dialogue Timaios [52] associated four of the five regular polytopes (the tetrahedron, octahedron, cube and icosahedron) to the four elements, which to-day we would rather associate with phases of matter. Platon noted the surface composition of these regular polytopes from regular triangles and squares and even speculated about what now we call a chemical reaction between atoms.

The five Platonic polytopes are bounded by regular polygonal faces. They can be characterized by their rotational symmetry: When we fix their center, we can look for rigid rotations which transform the initial faces into other faces. Since any such rotation can be reversed, and two of them carried out after one another give a new rotation of the faces, the finite set of all these rotations forms a group, called the point group of the polytope.
A new interpretation of regular polytopes arises by projecting them from their center to a 2-sphere, which we denote by S^{2}, surrounding the polytope. The projected faces become spherical polygons which tile the sphere without gaps or overlaps, except in edges. These two properties: no overlap, no gaps, characterize a tiling, here a tiling of S^{2}.
A finer distinction of point groups arises from compatibility with a lattice.