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The Plaid Model

(AMS-198)

Richard Evan Schwartz

PDF
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Princeton University Press img Link Publisher

Naturwissenschaften, Medizin, Informatik, Technik / Geometrie

Beschreibung

Outer billiards provides a toy model for planetary motion and exhibits intricate and mysterious behavior even for seemingly simple examples. It is a dynamical system in which a particle in the plane moves around the outside of a convex shape according to a scheme that is reminiscent of ordinary billiards. The Plaid Model, which is a self-contained sequel to Richard Schwartz’s Outer Billiards on Kites, provides a combinatorial model for orbits of outer billiards on kites.

Schwartz relates these orbits to such topics as polytope exchange transformations, renormalization, continued fractions, corner percolation, and the Truchet tile system. The combinatorial model, called “the plaid model,” has a self-similar structure that blends geometry and elementary number theory. The results were discovered through computer experimentation and it seems that the conclusions would be extremely difficult to reach through traditional mathematics.

The book includes an extensive computer program that allows readers to explore the materials interactively and each theorem is accompanied by a computer demonstration.

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Schlagwörter

Combinatorics, Parallelepiped, Mathematical induction, Sanity check, Equivalence class, Arithmetic, Compact space, Self-similarity, Extreme point, Convex polytope, Special case, Rotational symmetry, Topology, Bijection, Absolute value, Big O notation, Subset, Parallelogram, Number theory, Metric space, Right angle, Local homeomorphism, Automorphism, Homeomorphism, Inverse function, Convex hull, Piecewise linear, Affine transformation, Cartesian coordinate system, Polytope, Theorem, Calculation, Coordinate system, Formal proof, Pixelation, Without loss of generality, Summation, Fundamental domain, Symmetry group, Cantor set, Family of curves, Clockwise, Notation, Translational symmetry, Parity (mathematics), Natural number, Integer, Hyperbola, Algorithm, Linear function, Comparison theorem, Tessellation, Center of mass, Coprime integers, Diagram (category theory), Lattice path, Correspondence theorem (group theory), Concentric, Modular arithmetic, Quantity, Renormalization, Sequence, Linear algebra, Rational number, Degeneracy (mathematics), Dot product, Piecewise, Parabola, Y-intercept, Polyhedron, Parameter, Thickness (graph theory), Disjoint sets, Lexicographical order, Fiber bundle, Zero of a function, Diameter, Negation, Pairwise, Equation, Line segment, Rectangle, Unit interval, Computation, Tensor product, Vertical direction, Codimension, Symbolic dynamics, Function composition, Polygon mesh, Covering space, Graph partition, Intersection (set theory), Outer billiard, Unit square, Hyperplane, Right half-plane, Rubik's Cube, Rhombus, Geometry