Excursions into Mathematics-The Millennium Edition

Preface
Contents
Foreword
Note to the Instructor
Chapter 1. Euler’s Formula for Polyhedra, and Related Topics by Donald W. Crowe
	Section 1. Introduction
		1A. Prelude and Fantasy
		1B. Euler’s Formula
			1. Exercises
			2. Theorem (Euler’s Theorem)
			3. Exercises
	Section 2. Regular Polyhedra
		Example 1
		Example 2
	Section 3. Deltahedra
	Section 4. Polyhedra without Diagonals
	Section 5. The n-Dimensional Cube and the Tower of Hanoi
		5A. The n-Cube
		5B. The Tower of Hanoi
		5C. Chinese Rings
		5D. An Application
	Section 6. The Four-Color Problem and the Five-Color Theorem
	Section 7. The Conquest of Saturn, the Scramble for Africa, and Other Problems
		7A. The Conquest of Saturn
		7B. The Scramble for Africa
		7C. Some Problems Equivalent to the Four-Color Problem
	References
Chapter 2. The Search for Perfect Numbers by Michael N. Bleicher
	Section 1. Introduction
	Section 2. Prime Numbers and Factorization
	Section 3. Euclidean Perfect Numbers
	Section 4. Primes and Their Distribution
	Section 5. Factorization Techniques
	Section 6. Non-Euclidean Perfect Numbers
	Section 7. Extensions and Generalizations
	References
Chapter 3. What is Area? by Anatole Beck
	Section 1. Introduction
	Section 2. Rectangles and Grid Figures
	Section3. Triangles
	Section 4. Polygons
	Section 5. Polygonal Regions
	Section 6. Area in General
	Section 7. Pathology
	References
Chapter 4. Some Exotic Geometries by Donald W. Crowe
	Section 1. Historical Background
	Section 2. Spherical Geometry
	Section 3. Absolute Geometry
	Section 4. More History—Saccheri, Bolyai, and Lobachevsky
	Section 5. Hyperbolic Geometry
	Section 6. New Beginnings
	Section 7. Analytic Geometry—A Reminder
	Section 8. Finite Arithmetics
	Section 9. Finite Geometries
	Section 10. Application
	Section 11. Circles and Quadratic Equations
	Section 12. Finite Affine Planes
	Section 13. Finite Projective Planes
	Section 14. Ovals in a Finite Plane
		14A. Ovals in Planes of Odd Order
		14B. Ovals in Planes of Even Order (Optional)
	Section 15. A Finite Version of Poincare’s Universe
	Definitions
	Postulates
	Common Notions
	Propositions
	References
Chapter 5. Games by Anatole Beck
	Section 1. Introduction
	Section 2. Some Tree Games
	Section 3. The Game of Hex
	Section 4. The Game of Nim
	Section 5. Games of Chance
	Section 6. Matrix Games
	Section 7. Applications of Matrix Gaines
	Section 8. Positive-sum Games
	Section 9. Sharing
	Section 10. Cooperative Games
	References
Chapter 6. What\'s in a Name? by Michael N. Bleicher
	Section 1. Introduction
	Section 2. Historical Background
	Section 3. Place Notation for the Base b
	Section 4. Some Properties of Natural Numbers Related to Notation
	Section 5. Fractions: First Comments
	Section 6. Farey Fractions
	Section 7. Egyptian Fractions
	Section 8. The Euclidean Algorithm
	Section 9. Continued Fractions
	Section 10. Decimal Fractions
	Section 11. Concluding Remarks
References
Glossary of Symbols
Index
Appendix 2000 by Anatole Beck, Michael N. Bleicher, Donald W. Crowe
	Pages Referenced
		49, 55 and 74
		93, 97, 103, 107, 108, 110, 123
		136, 138, 140, 327 and 386, 332
		338
		396, 403, 414, 424, 429, 434
		438, 451, 452, 467