Serious fun with flexagons : a compendium and guide

Author(s)

    • Pook, Les

Bibliographic Information

Serious fun with flexagons : a compendium and guide

Les Pook

（Solid mechanics and its applications, v. 164）

Springer, c2009

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Includes bibliographical references and index

Description and Table of Contents

Description

A flexagon is a motion structure that has the appearance of a ring of hinged polygons. It can be flexed to display different pairs of faces, usually in cyclic order. Flexagons can be appreciated as toys or puzzles, as a recreational mathematics topic, and as the subject of serious mathematical study. Workable paper models of flexagons are easy to make and entertaining to manipulate. The mathematics of flexagons is complex, and how a flexagon works is not immediately obvious on examination of a paper model. Recent geometric analysis, included in the book, has improved theoretical understanding of flexagons, especially relationships between different types. This profusely illustrated book is arranged in a logical order appropriate for a textbook on the geometry of flexagons. It is written so that it can be enjoyed at both the recreational mathematics level, and at the serious mathematics level. The only prerequisite is some knowledge of elementary geometry, including properties of polygons. A feature of the book is a compendium of over 100 nets for making paper models of some of the more interesting flexagons, chosen to complement the text. These are accurately drawn and reproduced at half full size. Many of the nets have not previously been published. Instructions for assembling and manipulating the flexagons are included.

Table of Contents

  • 1 Introduction
  • 1.1 General Features
  • 1.2 Terminology
  • 1.3 Outline of Book
  • 1.4 Making Flexagons
  • 1.4.1 General Assembly Instructions
  • 1.4.2 The Two Sector First Order Fundamental Square Even Edge Flexagon
  • 2 Polygon Rings
  • 2.1 Introduction
  • 2.1.1 Multiple Polygons
  • 2.1.2 Combinations
  • 2.2 Edge Rings of Regular Polygons
  • 2.2.1 General Properties
  • 2.2.2 Regular Even Edge Rings
  • 2.2.3 Regular Odd Edge Rings
  • 2.2.4 Compound Edge Rings
  • 2.2.5 Irregular Edge Rings
  • 2.3 Edge Rings of Irregular Polygons
  • 2.3.1 General Properties
  • 2.3.2 Even Edge Rings of Silver and Bronze Triangles
  • 2.4 Vertex Rings
  • 2.4.1 General Properties
  • 2.4.2 Vertex Rings of Squares
  • 3 Fundamental Nets
  • 3.1 Introduction
  • 3.2 First Order Fundamental Edge Nets
  • 3.3 Second Order Fundamental Edge Nets
  • 3.4 Fundamental Vertex Nets
  • 3.5 Fundamental Silver and Bronze Edge Nets
  • 4 Fundamental Edge Flexagons
  • 4.1 Introduction
  • 4.1.1 Standard Face Numbering Sequences
  • 4.1.2 Truncated Flexagons
  • 4.2 First Order Fundamental Even Edge Flexagons
  • 4.2.1 General Properties
  • 4.2.2 Ring Even Edge Flexagons
  • 4.2.3 First Order Fundamental Triangle Even Edge Flexagons
  • 4.2.4 First Order Fundamental Square Even Edge Flexagons
  • 4.2.5 Detailed Analysis of Flexagons
  • 4.2.6 First Order Fundamental Pentagon Even Edge Flexagons
  • 4.2.7 First Order Fundamental Hexagon Even Edge Flexagons
  • 4.2.8 First Order Fundamental Octagon Even Edge Flexagons
  • 4.2.9 First Order Fundamental Dodecagon Even Edge Flexagons
  • 4.3 Second Order Fundamental Odd Edge Flexagons
  • 4.3.1 General Properties
  • 4.3.2 Second Order Fundamental Triangle Odd Edge Flexagons
  • 4.3.3 A Second Order Fundamental Square Odd Edge Flexagon
  • 4.3.4 A Second Order Fundamental 20-gon Odd Edge Flexagon
  • 5 Fundamental Skeletal and Point Flexagons
  • 5.1 Introduction
  • 5.2 First Order Fundamental Even Skeletal Flexagons
  • 5.2.1 General Properties
  • 5.2.2 First Order Fundamental Triangle Even Skeletal Flexagons
  • 5.2.3 A First Order Fundamental Square Even Skeletal Flexagon
  • 5.3Fundamental Point Flexagons
  • 5.3.1 General Properties and Unagons
  • 5.3.2 The Fundamental Triangle Point Flexagon
  • 5.3.3 The Fundamental Square Point Flexagon
  • 5.3.4 Fundamental Pentagon Point Flexagons
  • 5.3.5 The Fundamental Hexagon Point Flexagon
  • 5.4 Interleaved Fundamental Point Flexagons
  • 5.4.1 General Properties
  • 5.4.2 The Interleaved Fundamental Pentagon Point Flexagon
  • 5.4.3 An Interleaved Fundamental Enneagon Point Flexagon.
  • 5.5 Augmented Fundamental Point Flexagons
  • 5.5.1 General Properties
  • 5.5.2 An Augmented Fundamental Triangle Point Flexagon
  • 5.5.3 An Augmented Fundamental Square Point Flexagon
  • 5.6 Augmented Interleaved Fundamental Point Flexagons
  • 5.6.1 General Properties
  • 5.6.2 Augmented Interleaved Fundamental Triangle Point Flexagons
  • 5.6.3 An Augmented Interleaved Fundamental Square Point Flexagon
  • 6 Fundamental Compound Edge Flexagons
  • 6.1 Introduction
  • 6.2 General Properties
  • 6.3 Triangular Fundamental Compound Edge Flexagons
  • 6.3.1 Some Properties
  • 6.3.2 A Fundamental Square Compound Edge Flexagon
  • 6.3.3 A Fundamental Pentagon Compound Edge Flexagon
  • 6.4 A Square-Like Fundamental Compound Edge Flexagon
  • 6.4.1 Some Properties
  • 6.4.2 A fundamental hexagon compound edge flexagon
  • 6.5 Pentagonal Fundamental Compound Edge Flexagons
  • 6.5.1 Some Properties
  • 6.5.2 A Fundamental Square Compound Edge Flexagon
  • 6.5.3 A Fundamental Hexagon Compound Edge Flexagon
  • 6.6 A Hexagonal Fundamental Compound Edge Flexagon
  • 6.6.1 Some Properties
  • 6.6.2 A Fundamental Octagon Compound Edge Flexagon
  • 6.7 Heptagonal Fundamental Compound Edge Flexagons
  • 6.7.1 Some Properties
  • 6.7.2 A Fundamental Hexagon Compound Edge Flexagon
  • 6.7.3 A Fundamental Decagon Compound Edge Flexagon
  • 7 Irregular Cycle Flexagons
  • 7.1 Introduction
  • 7.2 Irregular Cycle Even Edge Flexagons
  • 7.2.1 General Properties
  • 7.2.2 Derivation of Nets
  • 7.2.3 The Irregular Cycle Square Even Edge Flexagon
  • 7.2.4 An Irregular Cycle Pentagon Even Edge Flexagon
  • 7.2.5 Irregular Cycle Hexagon Even Edge Flexagons

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