Geometries and GroupsThis 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. |
Contents
I | 1 |
II | 4 |
III | 11 |
V | 15 |
VI | 19 |
VII | 24 |
VIII | 30 |
IX | 36 |
XXVIII | 139 |
XXIX | 149 |
XXXI | 153 |
XXXII | 160 |
XXXIII | 166 |
XXXIV | 170 |
XXXV | 184 |
XXXVI | 185 |
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Common terms and phrases
2-dimensional locally Euclidean angle axis belongs centre circle Cn or Dn complex numbers composite consider constructed contained coordinates covering crystal crystallographic groups defined denote disc D(A disc of radius discrete groups distance between points endpoint equal equivalent points Euclidean plane example Exercise fixed point follows fundamental domain geometry corresponding given glide reflection glueing group of motions groups of Type hence identity inequality integer intersection Klein bottle lattice Lemma length line segment Lobachevsky geometry Lobachevsky plane locally Cn locally Euclidean geometries modular figure modular group motion F motions of 3-space notion of equivalence obtained obviously pair of vectors parallel perpendicular plane geometry points equivalent properties prove quadratic residues reader real line rotation satisfies second kind semicircle set of points similar sphere spherical geometry spherical neighbourhood square strip superposition Suppose symmetry group torus triangle Type III.b uniformly discontinuous group upper half-plane z1 and z2
References to this book
Oxford Users' Guide to Mathematics Eberhard Zeidler,W. Hackbusch,Hans Rudolf Schwarz No preview available - 2004 |