#### 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 [43] 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.