## Math in a Material World

**December 17, 2007**

Fun with fractals: Chapter 3 of the book under review in-cludes a pattern for this crocheted shawl, an iterative project that exploits the self-similarity of Sierpinksi triangles.

Book Review

Michelle Sipics

**Making Mathematics with Needlework: Ten Papers and Ten Projects**.* By Sarah-Marie Belcastro and Carolyn Yackel, eds., AK Peters, Wellesley, Massachusetts, 2007, 200 pages, $30.*

Cornell mathematics professor Daina Taimina has gained a certain celebrity over the past few years---perhaps as much for her needlework skills as for her mathematics. Some readers may know of her as the mathematician who, along with her husband (geometer and fellow Cornell professor David Henderson), designed and crocheted models of the hyperbolic plane a few years ago. Overwhelming amounts of media attention followed---from *The New York Times*, the well-known knitting magazine Interweave Knits, and Science, as well as NPR's *All Things Considered*, just for starters.

Taimina was not the first to demonstrate mathematical concepts using needlework, or even crochet, of course. Hinke Osinga of Bristol University illustrated her talk at the 2004 SIAM dynamical systems conference with a Lorenz manifold that she had designed (with Bernd Krauskopf) and crocheted; a similar flurry of media attention greeted that creation. Many others have published patterns that can be used to teach or visualize mathematical concepts. A group of mathematics and needlework enthusiasts have now produced what may be the first coherent collection of patterns and essays on this particular intersection of math and the arts: *Making Mathematics with Needlework*, edited by Mercer University professor Carolyn Yackel and Smith College professor Sarah-Marie Belcastro.

The collection consists of ten essays and projects---overviews of mathematical concepts followed by needlework projects designed to apply or demonstrate them. The book begins with a chapter on the Möbius band: simple experiments that can be done with it; an explanation of the relevant mathematics; a discussion of its origins, including Ferdinand Möbius's original uses for it; and, finally, instructions the reader can use to make a quilted Möbius band. Several other needlecrafts are represented in the other nine chapters, including knitting, crochet, and cross-stitch---as the authors put it in their introduction, "Everything except Weaving," which they say has been discussed in a large variety of publications in mathematics/fiber arts.

Belcastro and Yackel list several goals for their book: enable readers to construct mathematical objects; illustrate mathematical concepts; identify the mathematics embedded in a given needleart; use mathematics to resolve problems that arise in needlework; and explore the history and applications of various needlearts. That's an ambitious set of goals for a 200-page book, but Belcastro and Yackel are to be credited for their zealous attempt to pull it off.

The book's projects include well-explained patterns for the aforementioned Möbius strip, tori, and a Holbeinian graph, all of which solidly illustrate their respective mathematical concepts. A chapter on the use of Diophantine equations for picking up stitches evenly in knitting is both instructive and quite useful. (As a knitter who has only recently entered the simultaneously terrifying and gratifying world of knitting sweaters, I will no doubt be returning to that chapter many times in the future.)

Unfortunately, in attempting to cater to so many different audiences---math novices, math professionals, needlework novices, crafting pros (and within the latter two groups, knitters, cross-stitchers, etc.)---the book sacrifices some accessibility. Each chapter covers such a wealth of material, including basic over-views, in-depth (and demanding) mathematical discussions, teaching ideas, and the relevant patterns, that it's simply impossible for a given reader to get the most out of every section of the book.

Mathematical and needlework concepts are thoroughly intertwined in the text, so that readers must be at least somewhat familiar with both to make total sense of a chapter. A reader with only a casual interest in mathematics may be forced to skip parts of the mathematical explanations, detracting from his understanding of the related needlecraft discussions. Similarly, a mathematically inclined crafter with an interest in crochet or knitting but no experience in cross-stitch may not get much out of the excellent chapter on graph theory, which features cross-stitch patterns. (To be fair, the sections are for the most part clearly defined, so that math novices reading through a needlework-centric passage should not stumble on unfamiliar mathematical concepts or scary equations, and vice-versa.)

For the myriad math-and-needlework enthusiasts out there, though---and they are out there, as evidenced by the enthusiastic crowd drawn to Taimina and Henderson's demonstration of their crocheted models in New York City a few years back---this book will no doubt earn a place of honor on the shelf. Readers with a broad interest in mathematics will appreciate the range of topics discussed---topology, number theory, and combinatorics, among others---as well as the effort on the part of the authors to illustrate those topics and concepts in creative ways, incorporating so many different crafts.

But perhaps the greatest beneficiaries of this text---a group Belcastro and Yackel undoubtedly had in mind in putting the project together---are math educators. It's been demonstrated time and time again that for some people, translating an abstract concept into a concrete physical application moves that last mental tumbler into place---making the difference between a vague idea of graph theory, for example, and an understanding of what a graph actually means. To that end, the book contains several sections labeled "Teaching Ideas," with suggestions for variations on the patterns presented as well as basic ideas for completely separate projects. The teaching section of chapter 9 ("The Graph Theory of Blackwork Embroidery"), for example, offers ideas for students ranging from the very young all the way up to college level.

If you're still not entirely sold on this text---even I admit that knitted hyperbolic infant pants (chapter 10) will probably not be the next craze at Baby Gap---I encourage you to let the authors' passion for their projects speak for itself. It takes more than exclamation points to convey enthusiasm, and these authors have it leaping off the page throughout the book.

*Michelle Sipics is a contributing editor at SIAM News.*