1st Edition

Foundations of Three-Dimensional Euclidean Geometry

By I. Vaisman Copyright 1980

    This book presents to the reader a modern axiomatic construction of three-dimensional Euclidean geometry in a rigorous and accessible form. It is helpful for high school teachers who are interested in the modernization of the teaching of geometry.

    Chapter 0. Introduction
    1. The Axiomatic Method and It’s Utilization in Euclidean Geometry
    a. The axiomatic method
    b. Axiomatics of Euclidean geometry
    2. Useful Notions from other Mathematical Theories
    a. Binary relations
    b. Groups
    c. Fields and linear spaces
    d. Topological spaces
    2. Useful Notions from Other Mathematical Theories
    a. Binary relations
    b. Groups
    c. Fields and linear spaces
    d. Topological spaces
    Chapter 1. Affine Spaces
    1. Incidence Axioms and Their Consequences
    2. The Axiom of the Parallels and Its Influence on the Incidence Properties
    a. Affine spaces
    b. Projective spaces
    c. Desargues theorems
    3. The Fundamental Algebraic Structures of an Affine Space
    a. Vectors of an affine space
    b. The vector sum
    c. Scalars of an affine space
    d. Scalar algebraic operations
    e. Properties of the scalar field
    4. Coordinates in Affine Spaces
    a. The linear space structure
    b. Frames and coordinates
    c. Affine spaces over a field
    5. Affine Transformations
    a. Characteristic properties of affine transformations
    b. Special affine transformations
    c. The structure of the affine group
    d. Determination of affine transformations
    Chapter 2 Ordered Spaces
    1. The Order Axioms and Their First Consequences
    a. Linear Order Properties
    b. Plane and Spatial Order Properties
    2. Polygons and Polyhedra
    a. Convex Polygons
    b. The Jordan separation theorem
    c. Problems on polyhedral
    3. Ordered Affine Spaces
    a. Equivalent order axioms
    b. Determination of the ordered affine spaces
    c. Orientation of ordered affine spaces
    4. Continuity Axioms
    a. The Dedekind continuity axiom
    b. Continuously ordered spaces
    c. The axioms of Archimedes and Cantor
    Chapter 3. Euclidean Spaces
    1. The congruence Axioms and Their Relations with the Incidence and Order Axioms
    a. A preliminary discussion of congruence
    b. Elementary properties of congruence of segments
    c. The Euclidean group of a line
    d. Plane congruence properties
    e. Miscellaneous congruence properties
    2. Euclidean Spaces
    a. Congruence and the parallel axiom
    b. The scalar field of and Euclidean space
    c. Euclidean structures and quadratic forms
    d. Real Euclidean space
    e. Absolute geometry
    3. A short History of the Parallel Axiom
    a. Historical discussion
    b. Mathematical discussion
    4. The Independence of the Parallel Axiom
    Hints for Solving the Problems


    Izu Vaisman