## Flexagons Inside OutFlexagons are hinged polygons that have the intriguing property of displaying different pairs of faces when they are flexed. Workable paper models of flexagons are easy to make and entertaining to manipulate. Flexagons have a surprisingly complex mathematical structure and just how a flexagon works is not obvious on casual examination of a paper model. Flexagons may be appreciated at three different levels. Firstly as toys or puzzles, secondly as a recreational mathematics topic and finally as the subject of serious mathematical study. This book is written for anyone interested in puzzles or recreational maths. No previous knowledge of flexagons is assumed, and the only pre-requisite is some knowledge of elementary geometry. An attractive feature of the book is a collection of nets, with assembly instructions, for a wide range of paper models of flexagons. These are printed full size and laid out so they can be photocopied. |

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### Contents

Introduction | 3 |

Early history of flexagons | 14 |

Hexaflexagons | 31 |

Hexaflexagon variations | 47 |

Introduction to convex polygon flexagons | 82 |

Typical convex polygon flexagons | 96 |

Ring flexagons | 119 |

Distorted polygon flexagons | 137 |

Flexahedra | 154 |

164 | |

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### Common terms and phrases

alternate pats angle triangle flexagon bending the leaves box position links centre compound flexagons compound ring Conrad and Hartline convex polygon flexagon cycle compound cycle square flexagon degrees of freedom dihedral angle distinct types distorted polygon flexagons dynamic behaviour enantiomorph face numbers flexagon diagram flexatube folded piles hexahexaflexagon hyperfaced cube flexahedron incomplete cycles interleaf intermediate position map irregular cycles main position 2(1 main position link McIntosh Mobius band Modified flexagon figures number of sides open ended box paper models pinch flex position 1(3 principal main position push through flex pyramid regular convex polygons regular polygons regular single cycle regular single principal right angle triangle ring flexagon rotational symmetry shown in Fig simplified map single cycle square single leaves single principal cycle square flexagon Fig star polygon subsidiary cycles subsidiary main positions three faced truncated cube Tuckerman diagram Tuckerman traverse twisted bands types of flexagon types of single typical convex polygon