Definitions and Fundamental Formula

A set of points in the plane is called a domain if it is open and connected. A set of points is called a region if it is the union of a domain with some, none, or all its boundary points.

By a lattice of fundamental regions in the plane we understand a sequence of congruent regions α1, α1… that satisfies the following conditions:

1. Every point P of the plane belongs to one and only one region αi;

2. Every αi; can be superposed on α0 by a motion ti that superposes on every αnan αs, that is, by a motion that takes the whole lattice onto itself.

The set of motions {ti} such that α0 = tiαi is a discrete subgroup of the

group of motions. Such groups are called crystallographic groups. There are seventeen classes of nonisomorphic crystallographic groups, but for any given group there are infinitely many possible fundamental regions. It is not our purpose to present details on these groups, which are explored, for instance, in the books of Coxeter [127] and Guggenheimer [254]. Figures 8.1 to 8.5 are examples of lattices whose fundamental regions are squares, parallelograms, hexagons, or figures of more complicated shape.

Let D0 be a figure in the plane, that is, a set of points, which can be a region bounded by a finite number of closed curves without double points, a set of rectifiable curves, a finite number of points, etc. Suppose that D0 is contained in the fundamental region α1 of a given lattice.