Beyond the Third Dimension

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


Chapter 1: Introducing Dimensions

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

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

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

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

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

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

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

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

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