## Mathematical Omnibus: Thirty Lectures on Classic MathematicsThe book consists of thirty lectures on diverse topics, covering much of the mathematical landscape rather than focusing on one area. The reader will learn numerous results that often belong to neither the standard undergraduate nor graduate curriculum and will discover connections between classical and contemporary ideas in algebra, combinatorics, geometry, and topology. The reader's effort will be rewarded in seeing the harmony of each subject. The common thread in the selected subjects is their illustration of the unity and beauty of mathematics. Most lectures contain exercises, and solutions or answers are given to selected exercises. A special feature of the book is an abundance of drawings (more than four hundred), artwork by an accomplished artist, and about a hundred portraits of mathematicians. Almost every lecture contains surprises for even the seasoned researcher. |

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### Contents

5 | |

Arithmetical Properties of Binomial Coefficients | 28 |

On Collecting Like Terms on Euler Gauss and MacDonald | 45 |

Equations | 65 |

Equations of Degree Five | 79 |

How Many Roots Does a Polynomial Have? | 93 |

Chebyshev Polynomials | 101 |

Geometry of Equations | 109 |

Straight Lines | 225 |

Twentyseven Lines | 239 |

Web Geometry | 253 |

The Crofton Formula | 269 |

Polyhedra | 285 |

Noninscribable Polyhedra | 301 |

Impossible Tilings | 319 |

Rigidity of Polyhedra | 335 |

Envelopes and Singularities | 125 |

Around Four Vertices | 141 |

Segments of Equal Areas | 159 |

On Plane Curves | 171 |

Developable Surfaces | 187 |

Paper Möbius Band | 203 |

More on Paper Folding | 213 |

Two Surprising Topological Constructions | 359 |

Cone Eversion | 373 |

The Poncelet Porism and Other Closure Theorems | 403 |

Solutions to Selected Exercises | 425 |

457 | |

### Other editions - View all

Mathematical Omnibus: Thirty Lectures on Classic Mathematics D. B. Fuks,Serge Tabachnikov No preview available - 2007 |

### Common terms and phrases

assume billiard boundary Chebyshev closed curve closed geodesics coefficients computation cone confocal conics Consider construction continued fraction convex coordinates coplanar corresponding cubic curve cubic equation curvature cuspidal edge cusps deformation Dehn invariant Dehn(P denote developable surface dihedral angles disc domain double points double tangent ellipse envelope equal example Exercise fold follows formula function Gauss-Bonnet Theorem geometry graph hence hexagonal horizontal hyperbola hyperboloid inside integer intersection point Lecture Lemma length loop mathematics Möbius band n-gon number of sign obtained octahedron orthogonal osculating circle osculating plane oval pairs parabola parallel translation partition permutation planar plane polygon polyhedra polyhedron polynomial of degree problem projection Prove quadratic rectangle result roots rotation ruled surfaces Section segment self-intersecting sequence shown in Figure side sign changes solutions space sphere spherical symmetric tangent lines tiling total number triangle values vector vertex vertices winding number zero