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.
Algebra and arithmetics: Arithmetic and combinatorics: Can a number be approximately rational? Arithmetical properties of binomial coefficients On collecting like terms, on Euler, Gauss, and MacDonald, and on missed opportunities Equations: Equations of degree three and four Equations of degree five How many roots does a polynomial have? Chebyshev polynomials Geometry of equations Geometry and topology: Envelopes and singularities: Cusps Around four vertices Segments of equal areas On plane curves Developable surfaces: Paper sheet geometry Paper Mobius band More on paper folding Straight lines: Straight lines on curved surfaces Twenty-seven lines Web geometry The Crofton formula Polyhedra: Curvature and polyhedra Non-inscribable polyhedra Can one make a tetrahedron out of a cube? Impossible tilings Rigidity of polyhedra Flexible polyhedra Two surprising topological constructions: Alexander's horned sphere Cone eversion On ellipses and ellipsoids: Billiards in ellipses and geodesics on ellipsoids The Poncelet porism and other closure theorems Gravitational attraction of ellipsoids Solutions to selected exercises Bibliography Index.