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Outer billiards is a basic dynamical system defined relative to a convex shape in the plane. B. H. Neumann introduced this system in the 1950's, and J. Moser popularized it as a toy model for celestial mechanics. All along, the so-called Moser-Neumann question has been one of the central problems in the field. This question asks whether or not one can have an outer billiards system with an unbounded orbit. The Moser-Neumann question is an idealized version of the question of whether, because of small disturbances in its orbit, the Earth can break out of its orbit and fly away from the Sun. In Outer Billiards on Kites, Richard Schwartz presents his affirmative solution to the Moser-Neumann problem. He shows that an outer billiards system can have an unbounded orbit when defined relative to any irrational kite. A kite is a quadrilateral having a diagonal that is a line of bilateral symmetry. The kite is irrational if the other diagonal divides the quadrilateral into two triangles whose areas are not rationally related. In addition to solving the basic problem, Schwartz relates outer billiards on kites to such topics as Diophantine approximation, the modular group, self-similar sets, polytope exchange maps, profinite completions of the integers, and solenoids--connections that together allow for a fairly complete analysis of the dynamical system.

Hyperbolic spaces. --- Singularities (Mathematics) --- Transformations (Mathematics) --- Geometry, Plane. --- Plane geometry --- Algorithms --- Differential invariants --- Geometry, Differential --- Geometry, Algebraic --- Hyperbolic complex manifolds --- Manifolds, Hyperbolic complex --- Spaces, Hyperbolic --- Geometry, Non-Euclidean --- Abelian group. --- Automorphism. --- Big O notation. --- Bijection. --- Binary number. --- Bisection. --- Borel set. --- C0. --- Calculation. --- Cantor set. --- Cartesian coordinate system. --- Combination. --- Compass-and-straightedge construction. --- Congruence subgroup. --- Conjecture. --- Conjugacy class. --- Continuity equation. --- Convex lattice polytope. --- Convex polytope. --- Coprime integers. --- Counterexample. --- Cyclic group. --- Diameter. --- Diophantine approximation. --- Diophantine equation. --- Disjoint sets. --- Disjoint union. --- Division by zero. --- Embedding. --- Equation. --- Equivalence class. --- Ergodic theory. --- Ergodicity. --- Factorial. --- Fiber bundle. --- Fibonacci number. --- Fundamental domain. --- Gauss map. --- Geometry. --- Half-integer. --- Homeomorphism. --- Hyperbolic geometry. --- Hyperplane. --- Ideal triangle. --- Intersection (set theory). --- Interval exchange transformation. --- Inverse function. --- Inverse limit. --- Isometry group. --- Lattice (group). --- Limit set. --- Line segment. --- Linear algebra. --- Linear function. --- Line–line intersection. --- Main diagonal. --- Modular group. --- Monotonic function. --- Multiple (mathematics). --- Orthant. --- Outer billiard. --- Parallelogram. --- Parameter. --- Partial derivative. --- Penrose tiling. --- Permutation. --- Piecewise. --- Polygon. --- Polyhedron. --- Polytope. --- Product topology. --- Projective geometry. --- Rectangle. --- Renormalization. --- Rhombus. --- Right angle. --- Rotational symmetry. --- Sanity check. --- Scientific notation. --- Semicircle. --- Sign (mathematics). --- Special case. --- Square root of 2. --- Subsequence. --- Summation. --- Symbolic dynamics. --- Symmetry group. --- Tangent. --- Tetrahedron. --- Theorem. --- Toy model. --- Translational symmetry. --- Trapezoid. --- Triangle group. --- Triangle inequality. --- Two-dimensional space. --- Upper and lower bounds. --- Upper half-plane. --- Without loss of generality. --- Yair Minsky.

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A mathematical journey through the most fascinating problems of extremes and how to solve them. What is the best way to photograph a speeding bullet? How can lost hikers find their way out of a forest? Why does light move through glass in the least amount of time possible? When Least Is Best combines the mathematical history of extrema with contemporary examples to answer these intriguing questions and more. Paul Nahin shows how life often works at the extremes--with values becoming as small (or as large) as possible--and he considers how mathematicians over the centuries, including Descartes, Fermat, and Kepler, have grappled with these problems of minima and maxima. Throughout, Nahin examines entertaining conundrums, such as how to build the shortest bridge possible between two towns, how to vary speed during a race, and how to make the perfect basketball shot. Moving from medieval writings and modern calculus to the field of optimization, the engaging and witty explorations of When Least Is Best will delight math enthusiasts everywhere.

Mathematics --- Maxima and minima. --- MATHEMATICS / History & Philosophy. --- History. --- Minima --- Math --- Science --- AP Calculus. --- Addition. --- Almost surely. --- American Mathematical Monthly. --- Arc (geometry). --- Calculation. --- Cambridge University Press. --- Cartesian coordinate system. --- Catenary. --- Central angle. --- Chain rule. --- Change of variables. --- Circumference. --- Clockwise. --- Convex function. --- Coordinate system. --- Curve. --- Cycloid. --- Cylinder (geometry). --- Derivative. --- Diameter. --- Differential calculus. --- Differential equation. --- Dimension. --- Dynamic programming. --- Elementary function. --- Equation. --- Equilateral triangle. --- Euler–Lagrange equation. --- Fermat's principle. --- Fluxion. --- Geometry. --- Honeycomb conjecture. --- Hyperbolic function. --- Hypotenuse. --- Illustration. --- Inequality of arithmetic and geometric means. --- Instant. --- Integer. --- Isoperimetric problem. --- Iteration. --- Jensen's inequality. --- Johann Bernoulli. --- Kinetic energy. --- Length. --- Line (geometry). --- Line segment. --- Linear programming. --- Logarithm. --- Mathematical maturity. --- Mathematical problem. --- Mathematician. --- Mathematics. --- Newton's method. --- Notation. --- Parabola. --- Parametric equation. --- Partial derivative. --- Perimeter. --- Philosopher. --- Physicist. --- Pierre de Fermat. --- Polygon. --- Polynomial. --- Potential energy. --- Princeton University Press. --- Projectile. --- Pumping station. --- Pythagorean theorem. --- Quadratic equation. --- Quadratic formula. --- Quantity. --- Ray (optics). --- Real number. --- Rectangle. --- Refraction. --- Refractive index. --- Regiomontanus. --- Requirement. --- Result. --- Right angle. --- Right triangle. --- Science. --- Scientific notation. --- Second derivative. --- Semicircle. --- Sign (mathematics). --- Simple algebra. --- Simplex algorithm. --- Snell's law. --- Special case. --- Square root. --- Summation. --- Surface area. --- Tangent. --- Trigonometric functions. --- Variable (mathematics). --- Vertex angle. --- Writing.

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Everybody knows that mathematics is indispensable to physics--imagine where we'd be today if Einstein and Newton didn't have the math to back up their ideas. But how many people realize that physics can be used to produce many astonishing and strikingly elegant solutions in mathematics? Mark Levi shows how in this delightful book, treating readers to a host of entertaining problems and mind-bending puzzlers that will amuse and inspire their inner physicist.Levi turns math and physics upside down, revealing how physics can simplify proofs and lead to quicker solutions and new theorems, and how physical solutions can illustrate why results are true in ways lengthy mathematical calculations never can. Did you know it's possible to derive the Pythagorean theorem by spinning a fish tank filled with water? Or that soap film holds the key to determining the cheapest container for a given volume? Or that the line of best fit for a data set can be found using a mechanical contraption made from a rod and springs? Levi demonstrates how to use physical intuition to solve these and other fascinating math problems. More than half the problems can be tackled by anyone with precalculus and basic geometry, while the more challenging problems require some calculus. This one-of-a-kind book explains physics and math concepts where needed, and includes an informative appendix of physical principles.The Mathematical Mechanic will appeal to anyone interested in the little-known connections between mathematics and physics and how both endeavors relate to the world around us.

Mathematical physics. --- Problem solving. --- MATHEMATICS / General. --- Methodology --- Psychology --- Decision making --- Executive functions (Neuropsychology) --- Physical mathematics --- Physics --- Mathematics --- Addition. --- Analytic function. --- Angular acceleration. --- Angular velocity. --- Axle. --- Calculation. --- Capacitor. --- Cartesian coordinate system. --- Cauchy's integral formula. --- Center of mass (relativistic). --- Center of mass. --- Centroid. --- Ceva's theorem. --- Clockwise. --- Complex analysis. --- Complex number. --- Conservation of energy. --- Convex curve. --- Curvature. --- Curve. --- Cylinder (geometry). --- Derivative. --- Diameter. --- Differential geometry. --- Dimension. --- Division by zero. --- Dot product. --- Eigenvalues and eigenvectors. --- Electric current. --- Equation. --- Euler's formula. --- Euler–Lagrange equation. --- Fermat's principle. --- Friction. --- Fundamental theorem of calculus. --- Gaussian curvature. --- Generating function. --- Geodesic curvature. --- Geometry. --- Gravity. --- Green's theorem. --- Heat flux. --- Hinge. --- Hooke's law. --- Horizontal plane. --- Hypotenuse. --- Inductance. --- Instant. --- Kinetic energy. --- Line integral. --- Linear map. --- Mathematics. --- Mechanics. --- Moment of inertia. --- Newton's laws of motion. --- Normal (geometry). --- Ohm's law. --- Optics. --- Partial derivative. --- Potential energy. --- Proportionality (mathematics). --- Pythagorean theorem. --- Quadratic function. --- Quantity. --- Rectangle. --- Resistor. --- Right angle. --- Right triangle. --- Second law of thermodynamics. --- Semicircle. --- Series and parallel circuits. --- Sign (mathematics). --- Slinky. --- Snell's law. --- Soap bubble. --- Soap film. --- Special case. --- Spring (device). --- Stiffness. --- Summation. --- Surface area. --- Surface tension. --- Tangent space. --- Tangent. --- Telescope. --- Theorem. --- Thought experiment. --- Tractrix. --- Trapezoid. --- Trigonometric functions. --- Two-dimensional gas. --- Uncertainty principle. --- Unit circle. --- Unit vector. --- Vacuum. --- Variable (mathematics). --- Vector field. --- Voltage drop. --- Voltage. --- Wavefront.