## A Mathematical Gift, I: The Interplay Between Topology, Functions, Geometry, and AlgebraThis book will bring the beauty and fun of learning mathematics to the classroom. It offers serious mathematics in a lively, reader-friendly style. Included are exercises and many figures that illustrate the main concepts. The first chapter presents the geometry and topology of surfaces. Among other topics, the authors discuss the Poincare-Hopf theorem on critical points of vector fields on surfaces and the Gauss-Bonnet theorem on the relation between curvature and topology (the Euler characteristics). The second chapter addresses various aspects of the concept of dimension, including the Peano curve and the Poincare approach. Also addressed are the structure of three-dimensional manifolds, in particular, provided that the three-dimensional sphere is the union of two doughnuts. This is the first of three volumes originating from a series of lectures given by the authors at Kyoto University (Japan). It is suitable for classroom use for high school mathematics teachers and undergraduate courses in science and liberal arts. |

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### Contents

Invitation to Topology Viewing Figures Globally | 1 |

Introduction | 3 |

The Euler Characteristic | 5 |

Vortices Created by Winds and the Euler Characteristic | 25 |

Curvature of a Surface and the Euler Characteristic | 43 |

The Story of Dimension | 75 |

Introduction | 77 |

Learning to Appreciate Dimension | 79 |

What is Dimension? | 89 |

ThreeDimensional Figures | 107 |

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### Common terms and phrases

ABCD barycentric subdivision called cardinality closed surface concept of dimension consider construct continuous mapping coordinate critical points cube define deformation denoted dimensional direct product discussed disk doughnut edges elliptic curving elliptic point equation Euler characteristic example four color four-dimensional space Gauss map Gauss-Bonnet theorem Gaussian curvature genus g geometry gluing hyperbolic curving imagine infinite number intersection curve intuitive Klein bottle lecture look mathematicians mathematics natural numbers Newtonian mechanics observe obtain one-dimensional line one-to-one correspondence overlap parabolic curving parabolic points particles Peano curve Poincare Poincare-Hopf theorem point at infinity point of view polygonal representation proof quantum real numbers region resulting figure saver shown in Figure shows sides sphere of radius square string superstring theory surface of constant surface of genus surface Sg tangent plane three-dimensional figures three-dimensional space three-dimensional sphere topology torsion product torus triangle unit circle unit sphere vector field velocity vertices winding number