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A Mathematical Gift, I, II, III: The interplay between topology, functions, geometry, and algebra
Shigeyuki Morita, Tokyo Institute of Technology, Japan, Koji Shiga, Yokohama, Japan, Toshikazu Sunada, Tohoku University, Sendai, Japan, and Kenji Ueno, Kyoto University, Japan

Mathematical World
2005; 392 pp; softcover
ISBN-10: 0-8218-3859-8
ISBN-13: 978-0-8218-3859-4
List Price: US$83
Member Price: US$66.40
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This three-volume set addresses the interplay between topology, functions, geometry, and algebra. Bringing the beauty and fun of mathematics to the classroom, the authors offer serious mathematics in a lively, reader-friendly style. Included are exercises and many figures illustrating the main concepts. It is suitable for advanced high-school students, graduate students, and researchers.

The three-volume set includes A Mathematical Gift I, II, and III.


Advanced high-school students and undergraduates in mathematics.

Table of Contents

Part I
Invitation to topology (Viewing figures globally)
  • Introduction
  • The Euler characteristic
  • Vortices created by winds and the Euler characteristic
  • Curvature of a surface and the Euler characteristic
The story of dimension
  • Introduction
  • Learning to appreciate dimension
  • What is dimension?
  • Three-dimensional figures
  • Physics and dimension
Part II
The legacy of trigonometric functions
  • Introduction
  • Trigonometric functions and infinite series
  • Elliptic functions
Intersection of geometry and algebra
  • Introduction
  • The Poncelet closure theorem
  • The Poncelet theorem for circles
  • The Poncelet theorem in the world of complex numbers
  • Proof of the Poncelet theorem using plane geometry
  • Conclusion
Part III
  • The story of the birth of manifolds
  • The prelude to the birth of manifolds
  • The birth of manifolds
  • The story of area and volume from everyday notions to mathematical concepts
  • Transition from the notion of "size" to the concept of "area"
  • Scissors-congruent polygons
  • Scissors-congruent polyhedra
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