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Polygons, Polyominoes and Polycubes

  • Book
  • © 2009

Overview

Editors:
  1. Anthony J. Guttman

    1. Department of Mathematics and Statistics, The University of Melbourne, Victoria, Australia

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  • The only book devoted to polygons
  • Presents a class of ultra-fast counting algorithms
  • New experimental mathematics techniques to conjecture exact solutions
  • Powerful mathematical tools to solve polygon problems

Part of the book series: Lecture Notes in Physics (LNP, volume 775)

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Table of contents (16 chapters)

  1. Front Matter

    Pages i-xix
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  2. History and Introduction to Polygon Models and Polyominoes

    • Anthony J Guttmann
    Pages 1-21

  3. Lattice Polygons and Related Objects

    • Stuart G Whittington
    Pages 23-41

  4. Exactly Solved Models

    • Mireille Bousquet-Mélou, Richard Brak
    Pages 43-78

  5. Why Are So Many Problems Unsolved?

    • Anthony J Guttmann
    Pages 79-91

  6. The Anisotropic Generating Function of Self-Avoiding Polygons is not D-Finite

    • Andrew Rechnitzer
    Pages 93-115

  7. Polygons and the Lace Expansion

    • Nathan Clisby, Gordon Slade
    Pages 117-142

  8. Exact Enumerations

    • Ian G. Enting, Iwan Jensen
    Pages 143-179

  9. Series Analysis

    • Anthony J. Guttmann, Iwan Jensen
    Pages 181-202

  10. Monte Carlo Methods for Lattice Polygons

    • E. J. Janse van Rensburg
    Pages 203-233

  11. Effect of Confinement: Polygons in Strips, Slabs and Rectangles

    • Anthony J Guttmann, Iwan Jensen
    Pages 235-246

  12. Limit Distributions and Scaling Functions

    • Christoph Richard
    Pages 247-299

  13. Interacting Lattice Polygons

    • Aleks L Owczarek, Stuart G Whittington
    Pages 301-315

  14. Fully Packed Loop Models on Finite Geometries

    • Jan de Gier
    Pages 317-346

  15. Conformal Field Theory Applied to Loop Models

    • Jesper Lykke Jacobsen
    Pages 347-424

  16. Stochastic Lowner Evolution and the Scaling Limit of Critical Models

    • Bernard Nienhuis, Wouter Kager
    Pages 425-467

  17. Appendix: Series Data and Growth Constant, Amplitude and Exponent Estimates

    • Anthony J Guttmann, Iwan Jensen
    Pages 469-482

  18. Back Matter

    Pages 483-490
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Keywords

About this book

The problem of counting the number of self-avoiding polygons on a square grid, - therbytheirperimeterortheirenclosedarea,is aproblemthatis soeasytostate that, at ?rst sight, it seems surprising that it hasn’t been solved. It is however perhaps the simplest member of a large class of such problems that have resisted all attempts at their exact solution. These are all problems that are easy to state and look as if they should be solvable. They include percolation, in its various forms, the Ising model of ferromagnetism, polyomino enumeration, Potts models and many others. These models are of intrinsic interest to mathematicians and mathematical physicists, but can also be applied to many other areas, including economics, the social sciences, the biological sciences and even to traf?c models. It is the widespread applicab- ity of these models to interesting phenomena that makes them so deserving of our attention. Here however we restrict our attention to the mathematical aspects. Here we are concerned with collecting together most of what is known about polygons, and the closely related problems of polyominoes. We describe what is known, taking care to distinguish between what has been proved, and what is c- tainlytrue,but has notbeenproved. Theearlierchaptersfocusonwhatis knownand on why the problems have not been solved, culminating in a proof of unsolvability, in a certain sense. The next chapters describe a range of numerical and theoretical methods and tools for extracting as much information about the problem as possible, in some cases permittingexactconjecturesto be made.

Editors and Affiliations

  • Department of Mathematics and Statistics, The University of Melbourne, Victoria, Australia

    Anthony J. Guttman

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