This book is devoted to the theory of geometries which are locally
Euclidean, in the sense that in small regions they are identical to
the geometry of the Euclidean plane or Euclidean 3-space. Starting
from the simplest examples, we proceed to develop a general theory
of such geometries, based on their relation with discrete groups of
motions of the Euclidean plane or 3-space; we also consider the
relation between discrete groups of motions and crystallography.
The description of locally Euclidean geometries of one type shows
that these geometries are themselves naturally represented as the
points of a new geometry. The systematic study of this new geometry
leads us to 2-dimensional Lobachevsky geometry (also called
non-Euclidean or hyperbolic geometry) which, following the logic of
our study, is constructed starting from the properties of its group
of motions. Thus in this book we would like to introduce the reader
to a theory of geometries which are different from the usual
Euclidean geometry of the plane and 3-space, in terms of examples
which are accessible to a concrete and intuitive study. The basic
method of study is the use of groups of motions, both discrete
groups and the groups of motions of geometries. The book does not
presuppose on the part of the reader any preliminary knowledge
outside the limits of a school geometry course.
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