Flexagons Inside Out PDF
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Flexagons Inside Out leslie philip (Les) pook was born in Middlesex, England in 1935. He obtained a BSc in metallurgy from the University of London in 1956. He started his career at Hawker Siddeley Aviation Ltd, Coventry, in 1956. In 1963 he moved to the National Engineering Laboratory, East Kilbride, Glasgow. In 1969, while at the National Engineering Laboratory he obtained a PhD in mechanical engineering from the University of Strathclyde. Dr Pook moved to University College London in 1990. He retired formally in 1998 but remained affiliated to University College London as a Visiting Professor. He is a Fellow of the Institution of MechanicalEngineersandaFellowoftheInstituteofMaterials,Minerals and Mining. Dr Pook has wide experience of both research and practical engineer- ing problems involving metal fatigue and fracture mechanics. In these fields he has published four books and over a hundred papers. Present professional interests include the fatigue behaviour of cracks in complex stress fields, finite element analysis, and language editing of translated technical material. His leisure activities have for many years included recreationalmathematics,horology,inwhichhehaspublishedonepaper, and gardening. He regards his extensive DIY activities, especially plumb- ing, as a chore. Les married his wife, Ann, in 1960. They have a daughter, Stephanie, and a son, Adrian. Flexagons Inside Out Les Pook UniversityCollegeLondon Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo Cambridge University Press The Edinburgh Building, Cambridge , United Kingdom Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521819701 © Cambridge University Press 2003 This book is in copyright. Subject to statutory exception and to the provision of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published in print format 2003 - ---- eBook (NetLibrary) - --- eBook (NetLibrary) - ---- hardback - --- hardback - ---- paperback - --- paperback Cambridge University Press has no responsibility for the persistence or accuracy of s for external or third-party internet websites referred to in this book, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate. Contents Preface page ix List of notation xi Introduction 1 1 Making and flexing flexagons 5 1.1 The trihexaflexagon 5 1.2 The pinch flex 6 1.3 A simple square flexagon 7 1.4 General assembly instructions 8 1.5 Decoration of faces 13 2 Early history of flexagons 14 2.1 The discovery of flexagons 14 2.2 The Flexagon Committee 15 2.3 Revival of interest 16 3 Geometry of flexagons 18 3.1 Flexagon characteristics 19 3.2 Appearance of main positions 20 3.3 Flexagons as linkages 21 3.4 Mapping flexagons 24 3.5 Flexagon figures and symbols 29 4 Hexaflexagons 31 4.1 Degrees of freedom of hexaflexagons 32 4.2 The trihexaflexagon revisited 33 4.3 Multicycle hexaflexagons 36 4.4 Hexaflexagon design 40 4.5 Hexaflexagon face arrangements 42 5 Hexaflexagon variations 47 5.1 Triangle flexagons 47 5.2 The V-flex 49 v vi Contents 5.3 Origami with hexaflexagons 52 6 Square flexagons 53 6.1 Single cycle square flexagons 54 6.2 Reconstruction from a flexagon figure 58 6.3 Square flexagons with main position links 59 6.4 Distinct types of square flexagon with main position links 69 6.5 Square flexagons with box position links 70 6.6 Square flexagon face arrangements 76 6.7 The interleaf flex 78 7 Introduction to convex polygon flexagons 82 7.1 The convex polygon family 83 7.2 Degrees of freedom of intermediate positions 84 7.3 The digon flexagon 87 7.4 Triangle flexagons revisited 90 7.5 Square flexagons revisited 93 8 Typical convex polygon flexagons 96 8.1 Characteristics of typical convex polygon flexagons 97 8.2 Pentagon flexagons 98 8.3 Hexagon flexagons 106 8.4 Octagon flexagons 110 9 Ring flexagons 119 9.1 Star flexagons 120 9.2 The ring flexagon family 121 9.3 Compound flexagons 123 9.4 Compound square flexagons 126 9.5 Compound pentagon flexagons 128 9.6 Compound hexagon flexagons 132 10 Distorted polygon flexagons 137 10.1 Distorted polygons 137 10.2 Some distorted polygon flexagons 140 10.3 Right angle triangle flexagons 149 11 Flexahedra 154 11.1 Four dimensions 154 11.2 Prism flexahedra 155 Contents vii 11.3 Pyramid flexahedra 157 11.4 Cube flexahedra 158 References 164 Flexagon index 167 Subject index 169 Preface Mathematics, rightly viewed, possesses not only truth, but supreme beauty – abeautycoldandaustere,likethatofsculpture. BertrandRussell,MysticismandLogic. Flexagons are hinged polygons that have the intriguing property of dis- playing 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. The aesthetic appeal of flexagons is in their dynamic behaviour rather than the static appeal of, say, polyhedra. One of the attractions of flexagons is that it is possible to explore their dynamic behaviour exper- imentally as well as theoretically. Flexagons may be appreciated at three differentlevels:firstlyastoysorpuzzles,secondlyasarecreationalmath- ematics topic, and finally as the subject of serious mathematical study. Well made models of simple flexagons can be bought, but they are not widely available. Ifirstbecameinterestedinflexagonsinthelate1960sthroughreading two of Martin Gardner’s books. At that time I made numerous paper models of flexagons, and carried out some theoretical analysis. I did try writing up some of the material for publication, but editors showed a markedlackofenthusiasmandIlostinterest.In1997,whilebrowsingin alibraryIcameacrossapaperonflexagons,andthisrevivedmyinterest. Fortunately, my collection of flexagon models and associated notes had survived three house moves. I carried out a literature search as part of my renewed investigations. Thisshowedthatthereisonlyalimitedamountofpublishedinformation on flexagons. Furthermore, this information is scattered and some items are difficult to locate. There may well be significant items which I have missed. The only books I found devoted entirely to flexagons are at the toy or puzzle level. These usually contain cut out nets for paper models of flexagons, but making these up destroys the book. At the recreational mathematics level I found several books containing sections or chapters on flexagons, but the information included tends to be fragmentary, and ix x Preface is sometimes difficult to appreciate fully without some prior knowledge of flexagons. At this level I did not find any books devoted entirely to flexagons. Thefirstpaperattheseriousmathematicslevelwaspublishedin1957 and a comprehensive report on flexagons was issued in 1962. I only re- cently obtained a copy of this report and it turned out to include quite a lot of material which I had rediscovered for myself. At this level the subsequent literature is sparse. The main features of flexagons have been understood for half a century, but a long standing, and still not fully resolved, problem is how best to describe their structure and dynamic behaviour. For this reason publications on flexagons at the serious math- ematics level tend either to be not very informative or to need close reading by a competent mathematician in order to fully appreciate their content. Contemplationofmycollectionofpublicationsonflexagons,together withtheresultsofmyownearlyandrecentinvestigations,showedthatI had enough information to write a book on flexagons at the recreational mathematics level. Helpful comments made by anonymous reviewers of the first draft of the book encouraged me to proceed. This book is the result. It is assumed that the reader has some knowledge of elementary geometry. No previous knowledge of flexagons is assumed, so the book is suitable as an introduction to flexagons at the toy or puzzle level. In general, detailed proofs are long and tedious so they are not included. Wherethereisuncertaintyovertheaccuracyofaconclusionthisismade clear in the text. There is an infinite number of possible types of flexagon so no book on flexagons can be comprehensive. The material included is very much a personal choice. It is arranged roughly in order of increasing difficulty rather than in a strictly logical order which would be appropriate for a formal textbook. The terminology used is mostly based on that used by previous authors but to keep the text concise it was found that some new terms were needed. Specialised terms are listed in the index so that definitions can be easily located. Afeatureofthebookisacollectionofnets,withassemblyinstructions, for a wide range of paper models of flexagons. They are printed full size and laid out so that they can be photocopied. Some of the flexagons are difficult either to make or to manipulate, and this is noted in captions. Theflexagonshavebeenchosentocomplementthetext.Someofthenets were specially designed using methods described in the book.
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