Polygons, Polyominoes and Polycubes

  • Anthony J. Guttman
Part of the Lecture Notes in Physics book series (LNP, volume 775)

Table of contents

  1. Front Matter
    Pages i-xix
  2. Stuart G Whittington
    Pages 23-41
  3. Mireille Bousquet-Mélou, Richard Brak
    Pages 43-78
  4. Anthony J Guttmann
    Pages 79-91
  5. Nathan Clisby, Gordon Slade
    Pages 117-142
  6. Ian G. Enting, Iwan Jensen
    Pages 143-179
  7. Anthony J. Guttmann, Iwan Jensen
    Pages 181-202
  8. E. J. Janse van Rensburg
    Pages 203-233
  9. Anthony J Guttmann, Iwan Jensen
    Pages 235-246
  10. Christoph Richard
    Pages 247-299
  11. Aleks L Owczarek, Stuart G Whittington
    Pages 301-315
  12. Jesper Lykke Jacobsen
    Pages 347-424
  13. Bernard Nienhuis, Wouter Kager
    Pages 425-467
  14. Back Matter
    Pages 483-490

About this book

Introduction

This unique book gives a comprehensive account of new mathematical tools used to solve polygon problems.

In the 20th and 21st centuries, many problems in mathematics, theoretical physics and theoretical chemistry – and more recently in molecular biology and bio-informatics – can be expressed as counting problems, in which specified graphs, or shapes, are counted.

One very special class of shapes is that of polygons. These are closed, connected paths in space. We usually sketch them in two-dimensions, but they can exist in any dimension. The typical questions asked include "how many are there of a given perimeter?", "how big is the average polygon of given perimeter?", and corresponding questions about the area or volume enclosed. That is to say "how many enclosing a given area?" and "how large is an average polygon of given area?" Simple though these questions are to pose, they are extraordinarily difficult to answer. They are important questions because of the application of polygon, and the related problems of polyomino and polycube counting, to phenomena occurring in the natural world, and also because the study of these problems has been responsible for the development of powerful new techniques in mathematics and mathematical physics, as well as in computer science. These new techniques then find application more broadly.

The book brings together chapters from many of the major contributors in the field. An introductory chapter giving the history of the problem is followed by fourteen further chapters describing particular aspects of the problem, and applications to biology, to surface phenomena and to computer enumeration methods.

Keywords

Counting Lattice Mathematica Monte Carlo method Theoretical physics algorithms biology calculus computer computer science distribution geometry graphs mathematical physics statistics

Editors and affiliations

  • Anthony J. Guttman
    • 1
  1. 1.Department of Mathematics and StatisticsThe University of MelbourneVictoriaAustralia

Bibliographic information

  • DOI https://doi.org/10.1007/978-1-4020-9927-4
  • Copyright Information Springer Netherlands 2009
  • Publisher Name Springer, Dordrecht
  • eBook Packages Physics and Astronomy
  • Print ISBN 978-1-4020-9926-7
  • Online ISBN 978-1-4020-9927-4
  • Series Print ISSN 0075-8450
  • About this book