Table of Contents
1. Euclidean Geometry.
Overview and History.
Constructions, Congruence, and Parallels: Euclid's Approach to Geometry.
Biography: Archimedes.
A Critique of Euclid-Modern Axiomatics.
Biography: David Hilbert.
Axiomatic Systems, Models, and Metamathematics.
Similar Figures.
Three-Dimensional Geometry.
Projects for Chapter 1.
Suggested Readings.
Suggested Media.
2. Analytic Geometry
Overview and History.
Biography: Rene Descartes.
Conics and Locus Problems.
Biography: Pierre Fermat.
Further Topics in Analytic Geometry.
Curves in Computer-Aided Design.
Higher Dimensional Analytic Geometry.
Biography: Gaspard Monge.
Projects for Chapter 2.
Suggested Readings.
Suggested Media.
3. Non-Euclidean Geometries.
Overview and History.
Biography: Nikolai Lobachevsky and Janos Bolyai.
Biography: Carl Friedrich Gauss.
Biography: Georg Fredrich Bernhard Riemann.
Properties of Lines and Omega Triangles.
Saccheri Quadrilaterals and Triangles.
Area and Hyperbolic Designs.
Spherical and Single Elliptic Geometries.
Projects for Chapter 3.
Suggested Readings.
Suggested Media.
4. Transformational Geometry.
Overview and History.
Isometries.
Biography: Flex Klein.
Algebraic Representation of Transformations.
Similarities and Affine Transformation.
Biography: Sophus Lie.
Transformations in Higher Dimensions: Computer-Aided Design.
Inversions and the Complex Plane.
Biography: Augustus Mobius.
Projects for Chapter 4.
Suggested Readings.
Suggested Media.
5. Symmetry.
Overview and History.
Finite Plane SymmetryGroups.
Symmetry in the Plane.
Symmetries in Higher Dimensions.
Symmetry in Science.
Biography: Marjorie Senechal.
Fractals.
Biography: Benoit Mandelbrot.
Projects for Chapter 5.
Suggested Readings.
Suggested Media.
6. Projective Geometry.
Overview and History.
Axiomatic Projective Geometry.
Biography: Jean Victor Poncelet.
Analytic Projective Geometry.
Projective Transformations.
Subgeometries.
Biography: Arthur Cayley.
Projective Space.
Projects for Chapter 6.
Suggested Readings.
Suggested Media.
7. Finite Geometries.
Overview and History.
Biography: Leonhard Euler.
Affine and Projective Planes.
Design Theory.
Biography: Sir Ronald A. Fisher.
Finite Analytic Geometry.
Projects for Chapter 7.
Suggested Readings.
Appendices.
Definitions, Postulates, Common Notions, and Propositions from Book I of Euclid's Elements.
Hilbert's Axioms for Euclidean Plane Geometry.
Selected Answers.
Index.
