Geometry and Symmetry

by Paul B. Yale

Overview

This book is an introduction to the geometry of Euclidean, affine, and projective spaces with special emphasis on the important groups of symmetries of these spaces. The two major objectives of the text are to introduce the main ideas of affine and projective spaces and to develop facility in handling transformations and groups of transformations. Since there are many good texts on affine and projective planes, the author has concentrated on the n-dimensional cases.
Designed to be used in advanced undergraduate mathematics or physics courses, the book focuses on "practical geometry," emphasizing topics and techniques of maximal use in all areas of mathematics. These topics include:
Algebraic and Combinatoric Preliminaries
Isometries and Similarities
An Introduction to Crystallography
Fields and Vector Spaces
Affine Spaces
Projective Spaces
Special features include a spiral approach to symmetry; a review of the algebraic prerequisites; proofs which do not appear in other texts, such as the Polya-Burnside theorem; an extensive bibliography; and a large collection of exercises together with suggestions for term-paper topics. In addition, special emphasis is placed on the geometric significance of cosets and conjugates in a group.

Product Details

ISBN-13:	9780486169323
Publisher:	Dover Publications
Publication date:	04/07/2014
Sold by:	Barnes & Noble
Format:	eBook
Pages:	288
File size:	8 MB

Table of Contents

Chapter 1	Algebraic and Combinatoric Preliminaries
1.1	Basic notions about sets and groups	1
1.2	The algebra of permutations and examples of groups	8
1.3	Homomorphisms and permutation representations of groups	16
1.4	Automorphisms of a group	24
1.5	Cosets and orbits, Lagrange's theorem	29
1.6	The Polya-Burnside theorem	37
	Bibliography and suggestions for further reading	43
Chapter 2	Isometries and Similarities: An Intuitive Approach
2.1	Isometries and similarities	46
2.2	Involutions in S	52
2.3	The classification of isometries	57
2.4	Geometric implications of conjugacy in S	65
2.5	An exercise: Isometries in the plane	67
2.6	Dilatations and spiral similarities	68
2.7	Automorphisms of E and S	72
2.8	Homomorphisms of E and S	76
	Bibliography and suggestions for further reading	83
	Review problems for Chapters 1 and 2	84
Chapter 3	An Introduction to Crystallography
3.1	Discrete groups of isometries	86
3.2	Finite groups of isometries	89
3.3	Lattices and lattice groups	99
3.4	Crystallographic point groups	103
3.5	The seven crystal systems	108
3.6	Crystallographic space groups	113
3.7	Generalizations	117
	Bibliography and suggestions for further reading	118
Chapter 4	Fields and Vector Spaces: A Quick Review
4.1	Fields	120
4.2	Vector spaces, subspaces, echelon form	125
4.3	Linear transformations	132
4.4	Coordinate mappings, matrices for linear transformations	138
4.5	Similar matrices and commutative diagrams	142
4.6	Applications of matrix similarity	149
4.7	Symmetries of V[subscript n](F), GL[subscript n](F)	154
	Bibliography and suggestions for further reading	157
Chapter 5	Affine Spaces
5.1	Axioms for affine spaces	159
5.2	Affine subspaces	161
5.3	Parallel and skew subspaces	167
5.4	Affine coordinates	169
5.5	Affine symmetries I: Dilatations	171
5.6	Affine symmetries II: Affine transformations	174
	Term-paper topics	180
5.7	The analytic representation of affine transformations	180
5.8	Affine symmetries III: Collineations	184
5.9	Volume in real affine spaces	188
5.10	Lattices in real affine spaces	191
5.11	Collineations in real affine spaces	196
	Bibliography and suggestions for further reading	199
Chapter 6	Projective Spaces
6.1	Extended affine spaces and collapsed vector spaces	201
6.2	Projective subspaces	206
6.3	Projective planes	210
6.4	Homogeneous coordinates	214
6.5	Projective symmetries I: Perspectivities	224
6.6	Projective symmetries II: Projective transformations	234
6.7	Projective symmetries III: Collineations	244
6.8	Dual spaces and the principle of duality	252
6.9	Correlations and semi-bilinear forms	257
6.10	Quadrics and polarities	262
6.11	Real projective spaces	271
6.12	Projective spaces over noncommutative fields	275
	Bibliography and suggestions for further reading	278
	Term-paper topics	280
	Index	281
	Index of notation	286