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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

Correspondence theorem (group theory), Rectangle, Parallelepiped, Covering space, Unit interval, Polygon mesh, Affine transformation, Piecewise, Equivalence class, Intersection (set theory), Convex polytope, Line segment, Diameter, Disjoint sets, Bijection, Hyperplane, Y-intercept, Convex hull, Summation, Dot product, Parallelogram, Without loss of generality, Special case, Function composition, Lexicographical order, Right half-plane, Graph partition, Polytope, Geometry, Integer, Big O notation, Hyperbola, Topology, Translational symmetry, Extreme point, Right angle, Tensor product, Homeomorphism, Unit square, Parity (mathematics), Fiber bundle, Rational number, Cantor set, Coprime integers, Subset, Comparison theorem, Absolute value, Natural number, Parameter, Sanity check, Rhombus, Metric space, Theorem, Cartesian coordinate system, Compact space, Automorphism, Equation, Rotational symmetry, Calculation, Renormalization, Pixelation, Symbolic dynamics, Mathematical induction, Pairwise, Tessellation, Diagram (category theory), Degeneracy (mathematics), Fundamental domain, Outer billiard, Lattice path