Foundations Of Three-Dimensional Euclidean Geometry

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Overview

Foundations of Three-Dimensional Euclidean Geometry provides a modern axiomatic construction of three-dimensional geometry, in an accessible form. The method of this book is a graduated formulation of axioms, such that, by determining all the geometric spaces which satisfy the considered axioms, one may characterize the Euclidean space up to an isomorphism. A special feature of Foundations of Three-Dimensional Euclidean Geometry is the introduction of the parallel axiom at an early stage of the discussion, so that the reader can see what results may be obtained both with and without this important axiom. The many theorems, drawings, exercises, and problems richly enhance the presentation of material. Foundations of Three-Dimensional Euclidean Geometry is suitable as a textbook for a one- or two-semester course on geometry or foundations of geometry for undergraduate and beginning graduate students. Mathematics majors in M.A.T. programs will find that this exposition of a classical subject will contribute greatly to their ability to teach geometry at all levels; and logicians, philosophers, and engineers will benefit from this book's applications to their own interests.
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Product Details

Table of Contents

Chapter 0. Introduction 1
1. The Axiomatic Method and Its Utilization in Euclidean Geometry 1
a. The axiomatic method 1
b. Axiomatics of Euclidean geometry 6
2. Useful Notions from Other Mathematical Theories 9
a. Binary relations 9
b. Groups 11
c. Fields and linear spaces 13
d. Topological spaces 17
Chapter 1. Affine Spaces 21
1. Incidence Axioms and Their Consequences 21
2. The Axiom of the Parallels and Its Influence on the Incidence Properties 26
a. Affine spaces 26
b. Projective spaces 30
c. Desargues theorems 32
3. The Fundamental Algebraic Structures of an Affine Space 40
a. Vectors of an affine space 40
b. The vector sum 44
c. Scalars of an affine space 47
d. Scalar algebraic operations 51
e. Properties of the scalar field 54
4. Coordinates in Affine Spaces 59
a. The linear space structure 59
b. Frames and coordinates 62
c. Affine spaces over a field 68
5. Affine Transformations 73
a. Characteristic properties of affine transformations 73
b. Special affine transformations 76
c. The structure of the affine group 80
d. Determination of affine transformations 85
Problems 90
Chapter 2. Ordered Spaces 95
1. The Order Axioms and Their First Consequences 95
a. Linear Order Properties 95
b. Plane and Spatial Order Properties 103
2. Polygons and Polyhedra 110
a. Convex Polygons 110
b. The Jordan separation theorem 116
c. Problems on polyhedra 122
3. Ordered Affine Spaces 128
a. Equivalent order axioms 128
b. Determination of the ordered affine spaces 135
c. Orientation of ordered affine spaces 143
4. Continuity Axioms 150
a. The Dedekind continuity axiom 150
b. Continuously ordered spaces 152
c. The axioms of Archimedes and Cantor 160
Problems 167
Chapter 3. Euclidean Spaces 171
1. The Congruence Axioms and Their Relations with the Incidence and Order Axioms 171
a. A preliminary discussion of congruence 171
b. Elementary properties of congruence of segments 174
c. The Euclidean group of a line 177
d. Plane congruence properties 180
e. Miscellaneous congruence properties 186
2. Euclidean Spaces 193
a. Congruence and the parallel axiom 193
b. The scalar field of an Euclidean space 196
c. Euclidean structures and quadratic forms 202
d. Real Euclidean space 209
e. Absolute geometry 211
3. A Short History of the Parallel Axiom 219
a. Historical discussion 219
b. Mathematical discussion 225
4. The Independence of the Parallel Axiom 239
Problems 252
Hints for Solving the Problems 257
References 265
Index 267
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