Making Mathematics with Needlework
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i i making mathematics with needlework i i i i i i i i making mathematics with needlework ten papers and ten projects edited by SARAH-MARIE BELCASTRO CAROLYN YACKEL A K Peters, Ltd. Wellesley, Massachusetts i i CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2007 by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S. Government works Version Date: 20140130 International Standard Book Number-13: 978-1-4398-6513-2 (eBook - PDF) This book contains information obtained from authentic and highly regarded sources. Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use. 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Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com i i To my grandmothers, who were excellent needleworkers but didn’t know they were mathematicians. —Carolyn Yackel To my mother, who knits, and my father, who sews—both needleworkers who ought to know by now that they’re mathematicians. —sarah-marie belcastro i i i i CONTENTS i i i i Acknowledgments xi Introduction: An Overview of Mathematics and Fiber Arts 1 1 Welcome!.................................................... 1 2 AShortHistory................................................. 1 3 FiberArtsandMathematicsEducation.................................... 2 4 CurrentandFutureResearch......................................... 4 5 Conclusion.................................................... 7 1 Quilted Mobius¨ Band 13 1 Overview.................................................... 13 2 Mathematics.................................................. 17 3 TeachingIdeas................................................. 21 4HowtoMakeMobiusQuilts..........................................¨ 24 2 Picking Up Stitches and Diophantine Equations 31 1 Overview.................................................... 31 2 Mathematics.................................................. 34 3 TeachingIdeas................................................. 36 4 HowtoMakeaBi-DirectionalHat....................................... 36 3 The Sierpinski Variations: Self-Similar Crochet 41 1 Overview.................................................... 41 2 Mathematics.................................................. 41 3 TeachingIdeas................................................. 48 4 HowtoMakeaSierpinskiShawl........................................ 50 4 Only Two Knit Stitches Can Create a Torus 55 1 Overview.................................................... 55 2 Mathematics.................................................. 56 3 TeachingIdeas................................................. 62 4 HowtoMakeaTorus.............................................. 65 5 SymmetryPatternsinCross-Stitch 71 1 Overview.................................................... 71 2 Mathematics.................................................. 75 3 TeachingIdeas................................................. 81 4 HowtoMaketheSymmetriesSampler.................................... 83 CONTENTS vii i i i i 6 Socks with Algebraic Structure 91 1 Overview.................................................... 91 2 Mathematics.................................................. 93 3 TeachingIdeas................................................. 98 4 HowtoMakeAlgebraicSocks......................................... 100 7 Fortunatus’s Purse 105 1 Overview.................................................... 105 2 Mathematics.................................................. 106 3 TeachingIdeas................................................. 112 4 HowtoMakeFortunatus’sPurse....................................... 114 8 (K)Not Cables, Braids 119 1 Overview.................................................... 119 2 Mathematics.................................................. 120 3 TeachingIdeas................................................. 125 4 How to Make a Pillow of Braid Equivalence . 127 9 The Graph Theory of Blackwork Embroidery 137 1 Overview.................................................... 137 2 Mathematics.................................................. 141 3 TeachingIdeas................................................. 148 4 HowtoEmbroideraHolbeinianGraph.................................... 149 10 Stop Those Pants! 155 1 Overview.................................................... 155 2 Mathematics.................................................. 158 3 TeachingIdeas................................................. 166 4 HowtoMakeHyperbolicPants........................................ 169 About the Contributors 179 Credits 181 Index 183 viii CONTENTS i i i i i i i i ACKNOWLEDGMENTS i i i i As with any book, there are a large number of people larly grateful to those who provided us with measure- who contributed to this volume’s completion. Obviously ments for the hyperbolic pants, including Julie Brand we are grateful to the chapter authors, for without them Asbornsen (who measured 13 children), Tim and Elaine there would literally have been no book! Also in this cat- Comar, Dana Rowland, Erika King via Ruth Haas, Cheri egory are Sharon Frechette of the College of the Holy Kraske (who measured a one-day-old grandchild), Alice Cross for proposing the chapter on cabling and Daina Peters, Kate Queeney, Elizabeth Wilmer, and Dan Zook. Taimina of Cornell University for writing articles and pro- The community of fiber artists with a significant in- ducing objects that inspired us to create the chapter terest in mathematics and mathematicians who think on hyperbolic pairs of pants. (Keep an eye out for her about fiber arts would be poorer without people like forthcoming book on Exploring Non-Euclidean Geome- try Through Crochet.) We thank the American Mathe- Lara Neel of the Math4Knitters blog and podcast; matical Society for approving our request to hold the Brenda Dayne, who wrote the “Geek Chic” article in In- Special Session in Mathematics and Mathematics Educa- terweave Knits that seemed to take the world by storm tion in Fiber Arts, which took place at the January 2005 (and who now does the popular podcast Cast On); Joint Mathematics Meetings in Atlanta, Georgia. We also thank the session speakers, many of whom are chapter Debbie New, who was a microbiologist before she be- authors. We are especially grateful to Alice Peters for came famous for her (frequently mathematical) knit- agreeing to publish the book and for helping us to fig- ting; ureouthowtodoit,toMandyLoveforkeepinguson track in Fall 2006, Tom Hull for assisting sarah-marie with Lesley Perg of “thomasina’s Guide to Geeky Knitting,” diagrams and talking through some of the mathemat- who maintains a web index, broken out by discipline, ics, Sean Kinlin and Tom Hull for listening to sarah-marie of scientifically-related knitting; despair over the book, Erna “Mom” Yackel for serving as a limitless resource and sounding board for Carolyn Ruth Stewart of the Impulse of Delight website who with respect to mathematics education, and Creighton is a physician, a knitter, and a jeweler who knits and Rosental for feeding Carolyn and keeping her sane. crochets with wire (and creates Mobius¨ bands, trefoil Thanks to our project testers, Joan “Mom” Belcas- knots, hyperbolic orchids...); tro, Ann Black, Jane Carder, Charlotte Davis, Emily Mark Shoulson, Claire Irving, and Nate Berglund, who Gavin, Ashley Hatfield, Zia Marek-Loftus (who tested knit and crochet a variety of embeddings of topolog- four projects), Kim Plofker, Jeri Riggs, Em Schoch, Sue ical objects; Sierra, and Aunt Ida Thornton. Thanks also to Smith mathematics students Constance Baltera, Natasha Re- Hinke Osinga, who created the crocheted Lorenz sendes, Liz Sullivan, and Angela Turner, who partici- manifold; pated in the Fortunatus’s Hat photo shoot but aren’t pic- tured in the book (and Portia Parker and Madison Stu- Miss Lime and Miss Violet (of Lime & Violet podcast art, who are) and to additional models Arzachel, Frank fame) for liking algebra and spacial relations, respec- “Daddy” and