Mathematical Omnibus: Thirty Lectures on Classic Mathematics
The 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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Arithmetical Properties of Binomial Coefficients
On Collecting Like Terms on Euler Gauss and MacDonald
Equations of Degree Five
How Many Roots Does a Polynomial Have?
Geometry of Equations
The Crofton Formula
Rigidity of Polyhedra
Envelopes and Singularities
Around Four Vertices
Segments of Equal Areas
On Plane Curves
Paper Möbius Band
More on Paper Folding
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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