Beyond the Third Dimension
by
Thomas F. Banchoff
Originally distributed by Scientific American Library, W. H. Freeman
and Co.
All electronic rights reserved, Copyright Thomas
F. Banchoff
Table of Contents
Preface
Chapter 1: Introducing Dimensions
Introduction
1. The Many Meanings of Dimension
2. Dimensions as Coordinates
3. Dimensional Progressions
4. A Revolution in Visualization Technology
Study Questions and Projects
Chapter 2: Scaling and Measurement
Introduction
1. Exponent Patterns for Basic Building Blocks
2. Volume Patterns for Pyramids
3. Fold-Out Models of Pyramids
4. The Geometry of the Binomial Theorem
5. Diagonals of Cubes in Different Dimensions
6. The Egyptian Triumph: The Volume of an Incomplete Pyramid
7. Scaling and Growth Exponents
Study Questions and Projects
Chapter 3: Slicing and Contours
Introduction
1. Slicing Basic Three-Dimensional Shapes
2. Slicing from Other Directions
3. Slicing the Hypercube
4. Slicing the Triangular Pyramid
5. Slicing Cylinders
6. Slicing Cones
7. Contour Lines and Contour Surfaces
8. Slicing Doughnuts and Bagels
Study Questions and Projects
Chapter 4: Shadows and Structures
Introduction
1. Drawing Shadows
2. Drawing Cubes and Hypercubes
3. Shadows of Hypercubes
4. Three-Dimensional Shadows of
the Hypercube
5. Counting the Edges of
Higher-Dimensional Cubes
6. Higher-Dimensional Simplexes
7. Counting the Faces of
Higher-Dimensional Cubes
8. Paleoecology and Data Visualization
Study Questions and Projects
Chapter 5: Regular Polytopes and Fold-Outs
Introduction
1. The Greek Geometry Game
2. The Search for Regular Polyhedra
3. Duals of Regular Polyhedra
4. The Search for Regular Polytopes
5. The Four-Simplex
6. The Hypercube Dual or Sixteen-Cell
7. Polytopes in Five or More Dimensions
8. The Regular 600-Cell and its Dual
9. The Self-Dual 24-Cell
10. Fold-Out Patterns in Different Dimensions
Study Questions and Projects
Chapter 6: Perspective and Animation
Introduction
1. Viewing in Perspective
2. Perspective Views of the Hypercube
3. Schlegel Diagrams of Polyhedra
4. Schlegel Polyhedra for Regular Polytopes
5. Animating the Hypercube
6. The Polyhedral Torus in the Hypercube
7. Stereographic Projection
8. Stereographic Projection from
Four-Space
Study Questions and Projects
Chapter 7: Configuration Spaces
Introduction
1. The Dimensionality of
Rehabilitation Therapy
2. Dimensionality and Dance
3. Orbits of Dynamical Systems
4. Anthropological Sites and the
Space of Circles
5. The Dimensionality of Stage Lighting
6. Configuration Spaces of
Segments and Lines
7. Wave Fronts and Focal Curves
in the Plane
8. Wave Fronts in
Three-Dimensional Space
Chapter 8: Coordinate Geometry
Introduction
1. Coordinates and Axes
2. Lengths and the Generalized
Pythagorean Theorem
3. Coordinates for the n-Simplex
4. Coordinates for Hypercube Slices
5. Coordinates for Regular Polyhedra
6. Coordinates for Regular Polytopes
7. Complex Numbers as
Two-Dimensional Numbers
8. Four-Dimensional Numbers: The
Quaternions
9. Coordinates for Circles and
Spheres
Chapter 9: Non-Euclidean Geometry and Nonorientable Surfaces
Introduction
1. The Axioms of Euclidean Plane
Geometry
2. Noncommutative Algebra
3. The Development of
Non-Euclidean Geometry
4. Three-Dimensional
Non-Euclidean Geometry
5. Higher-Dimensional Euclidean Geometry
6. Immanuel Kant and Nonorientability
7. Möbius Bands, Real
Projective Planes, and Klein Bottles
Further Readings
Acknowledgements