Mathematical Omnibus: Thirty Lectures on Classic Mathematics

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.

[1]  Circles and Polygons , 2000 .

[2]  A. Sossinsky.  Knots: Mathematics with a Twist , 2002 .

[3]  Serge Tabachnikov,et al.  MORE ON PAPERFOLDING , 1999 .

[4]  A. V. Pogorelov,et al.  Hilbert's Fourth Problem , 1979 .

[5]  Semyon Grigorevich Gindikin.  Tales of physicists and mathematicians , 1988 .

[6]  Stan Wagon.  Fourteen proofs of a result about tiling a rectangle , 1987 .

[7]  Convexity and the Average Curvature of Plane Curves , 1997 .

[8]  Tiling a Square with Similar Rectangles , 2004 .

[9]  C. D. Olds.  Continued Fractions: Diophantine Equations , 1963 .

[10]  Richard A. Silverman,et al.  Hilbert's third problem , 1979 .

[11]  Serge Tabachnikov,et al.  Geometry and billiards , 2005 .

[12]  A cone eversion , 1995 .

[13]  D. Pedoe.  The Most “Elementary” Theorem of Euclidean Geometry , 1976 .

[14]  M. Dehn.  Ueber raumgleiche Polyeder , 1900 .

[15]  Sheldon Katz,et al.  Mirror symmetry and algebraic geometry , 1999 .

[16]  L. Silverman.  Poncelet's closure theorem , 2000 .

[17]  Kamal Poorrezaei,et al.  Two Proofs of Graves's Theorem , 2003, Am. Math. Mon..

[18]  Fr. Fabricius-bjerre.  On the Double Tangents of Plane Closed Curves. , 1962 .

[19]  P. Giblin,et al.  The centre symmetry set , 1999 .

[20]  Jeffrey C. Lagarias,et al.  Tiling with polyominoes and combinatorial group theory , 1990, J. Comb. Theory, Ser. A.

[21]  Combinatorics of fronts of Legendrian links and the Arnol'd 4-conjectures , 2005 .

[22]  W. Blaschke.  Einführung in die Geometrie der Waben , 1955 .

[23]  Martin Aigner,et al.  Proofs from THE BOOK , 1998 .

[24]  Décomposition des polyèdres : le point sur le troisième problème de Hilbert , 1985 .

[25]  Oded Schramm.  How to cage an egg , 1992 .

[26]  J. Sydler.  Conditions nécessaires et suffisantes pour l’équivalence des polyèdres de l’espace euclidien à trois dimensions , 1965 .

[27]  James Propp.  A pedestrian approach to a method of Conway, or, a tale of two cities , 1997 .

[28]  R. Sprague.  Über die Zerlegung von Rechtecken in lauter verschiedene Quadrate. , 1940 .

[29]  W. Thurston.  Conway's tiling groups , 1990 .

[30]  I︠u︡. I. Manin,et al.  Cubic forms; algebra, geometry, arithmetic , 1974 .

[31]  Gaston Darboux,et al.  Principes de géométrie analytique , 1919 .

[32]  Beniamino Segre,et al.  The Non-Singular Cubic Surfaces , 1942 .

[33]  G. Wanner.  The Cramer-Castillon problem and Urquhart's 'most elementary' theorem , 2006 .

[34]  L. Bianchi.  Vorlesungen über Differentialgeometrie I , 1899 .

[35]  Emmanuel Ferrand,et al.  On the Bennequin Invariant and the Geometry of Wave Fronts , 1997 .

[36]  G. Budworth.  The Knot Book , 1983 .

[37]  B. Halpern,et al.  Inverting a cylinder through isometric immersions and isometric embeddings , 1977 .

[38]  D. Freeman,et al.  The isoperimetric problem on some singular surfaces , 2005, Journal of the Australian Mathematical Society.

[39]  Idzhad Kh. Sabitov,et al.  The Volume as a Metric Invariant of Polyhedra , 1998, Discret. Comput. Geom..

[40]  V. Arnold.  Mathematical Methods of Classical Mechanics , 1974 .

[41]  M. Kapovich,et al.  On the moduli space of polygons in the Euclidean plane , 1995 .

[42]  David D. Bleecker.  Volume increasing isometric deformations of convex polyhedra , 1996 .

[43]  Serge Tabachnikov,et al.  The Four-Vertex Theorem Revisited—Two Variations on the Old Theme , 1995 .

[44]  R. Schwartz.  The Poncelet grid , 2007 .

[45]  J. P. Duncan,et al.  Folded developables , 1982, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences.

[46]  D. Hilbert,et al.  Geometry and the Imagination , 1953 .

[47]  George Polya,et al.  Induction and Analogy in Mathematics , 1954 .

[48]  N. Mishachev,et al.  Introduction to the ℎ-Principle , 2002 .

[49]  Z. L. Leibenzon.  A simple combinatorial method for proving the Jacobi identity and its generalizations , 1986 .

[50]  I. J. Schoenberg,et al.  On Cauchy's Lemma Concerning Convex Polygons , 1967, Canadian Journal of Mathematics.

[51]  P. McGinn.  Missed opportunities. , 2000, Hospitals & health networks.

[52]  J. '. Paiva,et al.  Hilbert's fourth problem in two dimensions. , 2003 .

[53]  Gerald L. Alexanderson.  The honors class: hilbert’s problems and their solvers , 2005 .

[54]  Jonathan L. King,et al.  Three Problems in Search of a Measure , 1994 .

[55]  Ted Siebuhr.  Which Way Did the Bicycle Go?: ...and Other Intriguing Mathematical Mysteries [Book Review] , 1997 .

[56]  Tilings of convex polygons , 1997 .