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Crocheting adventures with hyperbolic planesHinke M. Osinga a

a University of Bristol, Bristol, UK

Online publication date: 19 February 2010

To cite this Article Osinga, Hinke M.(2010) 'Crocheting adventures with hyperbolic planes', Journal of Mathematics andthe Arts, 4: 1, 52 — 54To link to this Article: DOI: 10.1080/17513470903459526URL: http://dx.doi.org/10.1080/17513470903459526

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to explore design solutions one would never havebeen able to consider without such tools. Finally,in Chapter 7, one works with data sets as inputs for theparameters defined in programs constructing models.Data sets might be images or environmental data suchas temperature, wind or the sun’s azimuth and eleva-tion. For example, the intensity values of the pixelsin an image may specify the radii of each of a grid ofcircles, or sun position information may be used todefine louvers on different facades of a building.

Architects and designers who finish Krawczyk’sworkbook will learn a powerful set of methods forturning a simple application like AutoCAD into amore powerful modelling tool. They will have exploredthe concept of going beyond just building a model tobuilding a modeller that builds a model. That is, theywill explore designing a process of design, which iswhat Professor Mitchell alludes to in his Forewordwith the term ‘meta-design’.

As an architect whose career depends on theconcepts of parametric and algorithmic design pro-cesses, and someone who works primarily withAutoCAD and AutoLISP, I was enthusiastic to learnthat The Codewriting Workbook had been published.I have used it to help introduce my colleagues andstudents to these ways of working. As evidenced byits presence in the annual Association for ComputerAided Design (ACADIA), Smart Geometry andEducation and Research in Computer AidedArchitectural Design in Europe (eCAADe) confer-ences, parametric and algorithmic modelling aregaining more and more attention and respect.Although some may be worried that AutoCADcoupled with AutoLISP is beginning to be seen asoutdated as newer tools such as Rhinoceros 3D,Form-Z, 3ds Max and Maya emerge, I feelAutoCAD is still prominent in the field, and that onecan easily learn AutoLISP. More importantly,I see how the examples and process descriptionsfound in this book can be extended to every designer’swork.

Neil KatzSkidmore, Owings & Merrill LLP

Chicago, ILNeil.Katz@som.com� 2010 Neil Katz

Crocheting adventures with hyperbolic planes, by DainaTaimina, Wellesley, MA, A.K. Peters Ltd., 2009, 148pp., US$35.00 (hardcover), ISBN 978-1-56881-452-0.

My first impression of this book was one of sheerdelight in its excellent print quality. The overall design

and layout make it a beautiful coffee-table book.Nevertheless, this is a book about mathematics andit is not made for browsing! In fact, I found this bookchallenging to read, because I am not an expert inhyperbolic geometry. Daina Taimina takes us ona journey through the history of hyperbolic geometryby introducing the concepts in a visual and explorativeway. We are often asked to stop and think about whatwe have just read. Furthermore, she tempts her readersinto making some of the beautiful crocheted hyperbolicplanes pictured in this book in order to experienceconcepts hands-on. Therefore, this is not justa mathematics book either. Daina Taimina’s crochetedpieces are works of art that have been photographedin settings that emphasize their artistic beauty andremind us that hyperbolic shapes are familiarand occur naturally all around us. Each chapterstarts with a full-page photograph of her crochetedart placed in a natural setting, to remind us that theyare not just tools for explaining mathematical con-cepts. This book also discusses how to crochet yourown version of these beautiful objects. Any hyperbolicplane can be crocheted using the very basic crochetstitch and by repeating the simple pattern of ‘crochet nstitches, increase one’ that is perfectly suited even forsomeone who has never crocheted before.

This book tries to strike a balance betweenproviding a history of hyperbolic geometry, explaininghyperbolic geometry to a broad audience and present-ing the crocheted hyperbolic planes for tactile explora-tions, while keeping the book’s length down.This is not an easy task and one necessarily has tomake choices that cannot please everyone. In myopinion, Daina Taimina has done a wonderful job.Readers with a reasonably strong mathematical back-ground will thoroughly enjoy this book; readers withlittle knowledge of geometry may find it hard tounderstand everything, but as Bill Thurston writesin his foreword: ‘I hope this book gives you pausefor thought and changes your way of thinkingabout mathematics.’

Daina Taimina has written a very personalintroduction that summarizes what this book is aboutand how it came to fruition. She draws attention to herscholarly knowledge of the history of mathematicsand, specifically, her love for geometry. The idea ofvisualizing the hyperbolic plane through crochet grewout of her motivation and perseverance to explainhyperbolic geometry to her students. While it is notessential to ‘see’ the mathematics, a visual image of amathematical concept often provides the key togaining a more thorough understanding. Concerningthe subject of hyperbolic geometry, it has takencenturies to go beyond a mental image of a hyperbolicplane, and we owe it to Daina Taimina that we can

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now not only see hyperbolic planes, but we can touchthem and explore them without fear of breaking them.I would go as far as claiming that the tactileexperiences that crochet offers has caused a revolutionin the mathematical world. It is remarkable to readthat apparently the editor of The MathematicalIntelligencer needed to be convinced of its meritsbefore publishing her paper co-authored with DavidHenderson [2] about how to crochet a model of thehyperbolic plane! Daina Taimina and DavidHenderson claimed that the idea of using crochet tovisualize mathematical concepts was not only worthyof publication, but would be useful to others as well.Indeed, a little more than three years later, I wrote apaper with Bernd Krauskopf on crocheting the Lorenzmanifold [3]—an important surface in dynamicalsystem theory that also has negative but notconstant negative curvature—and published it inThe Mathematical Intelligencer as well; the crochetinstructions for the Lorenz manifold are much moreinvolved, but the essential features are the same as for ahyperbolic plane. Since almost five years have passed,I hope that someone will be inspired byDaina Taimina’s book to introduce the use of crochetto another area of mathematics!

The explanation of hyperbolic geometry beginswith the notion of positive and negative curvature.The reader is eased into the subject of curvature viaplanar curves and examples, such as oranges andpears, of surfaces with different curvature properties.This first chapter effectively sets the stage for therest of this book: several mathematical conceptsare explained both visually and in words, withoutbecoming too technical. However, sometimes theexplanations are quite technical and use jargon thatis not clearly explained. For example, let me quote ashort part from the section ‘The Search for a CompleteHyperbolic Surface:’ ‘Unfortunately, d depends on thelocal embedding and there is not a uniform bound forthe size of the ‘largest’ piece of the hyperbolic planethat can be isometrically embedded in 3-space.’ Thereader is left to guess what is meant by ‘embedding’, letalone ‘local embedding’ or ‘isometrical embedding’.Personally, I found the overall style and level ofexplanation quite good. If the reader is willing to stopworrying about what an embedding is, (s)he can followmost of the story at a reasonable level and still get a lotout of this book.

Already towards the end of Chapter 1 the reader isintroduced to the crochet instructions for making his/her own hyperbolic planes. The final section‘Exponential Growth’ in this chapter gives a tacitwarning that crocheting a relatively large hyperbolicplanes takes quite some time. Nevertheless, the readershould really put the book down now and start

crocheting. Chapter 2 uses the crocheted models ofhyperbolic planes as a tool to explain the concepts ofperpendicular and parallel straight lines in hyperbolicgeometry and how the sum of the angles of a triangleactually depends on the size of the triangle inhyperbolic geometry. The educational benefit fromthe tactile experience is so powerful that evidentlyevery mathematician should learn how to crochet anduse it in geometry lectures. I thoroughly enjoyed thischapter and think everybody will, even those whoremain worried about the definition of an embedding.

Chapter 3 offers a nice break from the introductionto new materials in previous chapters and discusseshow human experiences in areas as different as art/patterns, buildings/structures, navigation/stargazingand motion/machines influenced the development ofgeometry. Each strand is treated in turn by giving ahistorical account of how observations, experience anddata collection led to a scientific theory. We all knowthat mathematics originated from the applied point ofview, as a tool to solve real-world problems. Thischapter not only emphasizes the applied side ofgeometry, but also shows the breadth and depthof the subject in seemingly different areas of develop-ment. This material is quite stand-alone, and onecould easily start reading the book with this chapter,because the concepts within each strand are actuallyrather familiar. At the same time, having readthe previous chapters makes one anticipate thingsmore, and the links between the strands are more easilyrecognized because one has a better understanding ofthe difference between Euclidean (planar) andnon-Euclidean (spherical and hyperbolic) geometry.Chapter 4 gives a brief history of crochet andChapter 5 discusses the history of non-Euclideangeometry for each of the four strands introduced inChapter 3.

Chapter 6 is back on the familiar ground of usingcrochet to understand and explain mathematical con-cepts that are introduced from a historical perspective.The pseudo-sphere and a symmetric hyperbolic planeare the topics of discussion for this chapter and serve asexamples of the scope and versatility of crochetedmodels as a tool to understand geometry. Chapter 7invites the reader to play with the crocheted models.While initially it might appear that the shaping of acrocheted hyperbolic plane is entertainment of anartistic nature, the reader is quickly led back into therealm of mathematics and a wide range of mathematicaltoys pass by. It is not always easy to follow themathematical arguments here and a non-expert readercould get lost in the jargon, but the essence of usingcrochet to explore otherwise hard to visualize objects iswell maintained throughout this chapter. Chapter 8 isentirely devoted to the helicoid and the catenoid and

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great care is taken to explain what makes them special.After crocheting a helicoid it becomes an easy exerciseto turn it into a catenoid, which brilliantly illustratesthat they are, in fact, isometric. In order to achieve this,it is essential to know the gauge of your crochet stitch,but precise instructions are given to show how to do themeasurements and calculations for the pattern.

The last chapter focuses on why people are stillinterested in hyperbolic geometry and how it can beused. The breadth of applications is enlightening; notonly do they come from all branches of science, butthere are also wonderful applications in music and art.For example, Daina Taimina’s crocheted hyperbolicplanes inspired industrial designer Radu Comsa todesign the Rasta Stool [1], which is apparently verycomfortable.

This book is unique in its combination of anhistorical account of and the use of crochet as a toolfor understanding hyperbolic geometry; as a bonus, itoffers crocheted models that are genuine pieces of art.I highly recommend this book, perhaps not only as a

beautiful coffee-table book with the subtle messagethat mathematics is fun, but also because crochet is aperfect tool for testing and exploring deep mathema-tical theories.

References

[1] R. Comsa, Rasta Stool, design by Radu Comsa. Available

at http://www.raducomsa.ro/furniture/full_rs.html (accessed

12 January 2010).

[2] D.W. Henderson and D. Taimina, Crocheting

the hyperbolic plane, Math. Intelligencer 23 (2001),

pp. 17–28.

[3] H.M. Osinga and B. Krauskopf, Crocheting the

Lorenz manifold, Math. Intelligencer 26 (2004), pp. 25–37.

Hinke M. OsingaUniversity of Bristol, Bristol, UK

E-mail: H.M. Osinga@bristol.ac.uk� 2010 Hinke M. Osinga

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