1. Prelude 2. What is Topology? 3. Twist and Join 4. Core Graphs and Ribbon Graphs 5. Polyhedra and Polylinks 6. From Boundary to Surface 7. Saddles 8. Topology Revisited 9. Adventurous Projects
Biography
Shiying Dong grew up in rural China, where bamboo weaving was an inseparable part of life, which kindled her passion for mathematical craft. Before she started creating original shapes with her hands, she devoted most of her energy to the traditional paper-and-pencil way of studying spacetime. Shiying has a PhD in theoretical physics from the University of Illinois Urbana-Champaign and a Master’s in mathematics from the University of Pennsylvania. She won a Silver Medal and the Best Female Participant Award in the first Asian Physics Olympiad in 2000. In November 2023, Shiying was one of the two grand prize winners of the Einstein Mad Hat Awards, organized by MoMath and the UK Maths Trust. She was also the recipient of the Best Textile, Sculpture or Other Medium Award in the JMM Mathematical Art Exhibition, 2025, featured in the NY Times. Since 2023, she has been exhibiting her work at MoMath, Gathering 4 Gardner, JMM, Bridges, Vogue Knitting Live, and various museums. She has given dozens of talks on topological crochet and held more than a hundred workshops on crochet and mathematical art and craft. She was the Innovator-in-Residence of the Innovation Lab in Greenwich Library, Greenwich, CT, from September 2025 to November 2025.
Eve Torrence has loved art, crafting, and mathematics since she was a child. She has been very lucky to build a career that merged these passions. She has a PhD in mathematics from the University of Virginia and is professor emeritus at Randolph-Macon College, where she was a member of the faculty from 1994 to 2021. Eve designed and taught many mathematical arts courses at R-MC, in which she used knitting, crochet, bead crochet, and origami to introduce, explain, and illustrate mathematical topics. She has exhibited her mathematical artwork at numerous juried shows at the JMM and Bridges conferences. Eve's sculpture Day was awarded the People’s Choice Award at the 2015 Bridges Mathematical Art Exhibition. She is currently a member of the Board of the Bridges Organization and serves as the managing editor of the Bridges Conference Proceedings. Eve is the author of Cut and Assemble Icosahedra: Twelve Models in White and Color and co-author of The Student’s Introduction to Mathematica and the Wolfram Language. In 2013, Eve received a State Council of Higher Education for Virginia Outstanding Faculty Award. She has served as the chair of the Maryland-DC-Virginia Section of the MAA and President of Pi Mu Epsilon National Mathematics Honor Society.
“Models in this book look magical. And as you cannot learn magic just from watching it being performed, so it is with topological crochet. In this book the magic of topological crochet is carefully revealed to the reader step by step. Follow it by making your own models and you will be rewarded with a deep understanding of topology and your own collection of beautiful crochet sculptures.”
—Daina Taimina, mathematician and author of Crocheting Adventures with Hyperbolic Planes: Tactile Mathematics, Art and Craft for all to Explore
“Eve and Shiying have created a beautiful book. They have been generous in sharing mathematical insights in accessible language, detailed instructions for over sixty topological crochet projects, personal insights for overcoming possible obstructions—conceptual or in one’s stitches—and numerous photos showing step-by-step processes and remarkable finished fiber art pieces to admire. If you aspire to understand topology better, to crochet new models or to learn from great teachers in action, you will love this book.”
—Katherine Seaton, mathematician and author of A Stitch in Line: Mathematics and One-stitch Sashiko
“In recent years there has been no more exciting new body of work in mathematical art exhibitions than Shiying Dong’s crocheted topological sculptures. These objects, simultaneously graceful and mathematically sophisticated, captivate viewers regardless of their background. Here she teams with mathematician and fellow fiber artist Eve Torrence to create an absorbing book that is richly illustrated and eminently readable.”
—Robert Fathauer, engineer and artist, author of Tessellations: Mathematics, Art, and Recreation
“This wonderful book introduces crochet crafters to fun and intriguing mathematics, and mathematicians with a crafty bend to making beautiful and fun objects illustrating symmetry and topology, to their own delight and maybe that of their friends and students! Shiying and Eve are crochet magicians—and their boundless energy and sense of fun is now yours to enjoy as well in this great book.”
—Ingrid Daubechies, winner of numerous mathematics awards, including the National Medal of Science, and first female president of the International Mathematical Union
“If it's beautiful, awe inspiring, conveys interesting math, and can be explored both tactilely and visually—that is the holy grail of mathematical art for me. The crocheted objects described in this book are a slam dunk on all counts. Shiying Dong with her almost magical ability to embody topological ideas in her craft, and Eve Torrence with her contagious inquisitiveness and deep experience conveying and teaching mathematical concepts, are the perfect duo to write this gem of a book. Beyond the artwork itself is the gift of learning through making as they explain the math and the craft to enable us to create these beautiful awe-inspiring objects for ourselves.”
—Ellie Baker, computer scientist and co-author of Crafting Conundrums: Puzzles and Patterns for the Bead Crochet Artist
“This book will appeal to anyone interested in mathematics or crochet. Not only does it show how crochet is an excellent way to explore mathematics, but also how it can be used to create beautiful and fascinating models.”
—Hanne Kekkonen, mathematician and fiber artist known for her precisely crocheted mathematical sculptures
“A wonderfully practical introduction to the art and craft and math of Shiying's virtuoso sculptures, laying out exactly the math you need, with detailed project instructions to follow as you learn. There are ingredients from knot and graph theory, geometry and topology, and so many forms to explore, from elegant loops to wild fanciful polyhedra. I've always needed a model in my hands to understand geometry, and crocheting is such a friendly way to visit with these beautiful shapes.”
—Bathsheba Grossman, pioneering mathematical sculptor known for her 3D printed work in steel
“This delightful book combines mathematics and crochet to create something truly special. Readers can craft beautiful crochet models that bring ideas from topology, graph theory, and knot theory to life. The projects are carefully explained and accessible to adventurous beginners, while the more advanced designs will delight experienced crafters. It’s a wonderful invitation to explore mathematics through making.”
—Laura Taalman, mathematician and co-author of Taking Sudoku Seriously: The Mathematics Behind the World’s Most Popular Pencil Puzzle
“The discovery that crochet is a way to model mathematical concepts has produced some beautiful, tactile, informative artworks. Shiying Dong has taken this yarn-based approach to mathematical complexity to the next level. Luckily for us, she has shown us her inner workings. Everything is explained and beautifully illustrated; from first steps in making crochet chains to detailed instructions, enabling us to follow along on the journey towards creating and understanding these beautiful mathematical shapes.”
—Alison Martin, artist renowned for her large-scale geometric sculptures in woven bamboo and textiles
“Topology has long been a curiosity of mine, even as a non-mathematician. Now, with nothing more than familiar tools—yarn, hook, and stitch markers—Shiying guides us through topological structures that make sense in our hands, even when the math seems out of reach. This isn't a quick skim-and-stitch pattern book, but a new way of making fiber art. Your hands will understand more than you expect!”
—Edie Eckman, fiber artist, knitting and crochet teacher, and author of several best-selling knitting and crochet books, including The Crochet Answer Book: Solutions to Every Problem You'll Ever Face; Answers to Every Question You'll Ever Ask
“This book is a treasure trove of crocheted models, shown in hundreds of photographs, which are accompanied by detailed instructions on how to create these designs. Pliable crochet pieces provide an excellent medium for modeling structures dominated by topological connectivity rather than precise geometric measurements. The 64 projects illustrate many topological concepts. The reader does not need previous knowledge of either topology or crochet, since the book contains gentle introductions to both these domains, and it is a very enjoyable guide to creating your own intriguing sculptures.”
—Carlo H. Séquin, computer scientist and a Fellow of the ACM, specializing in Computer-Aided Design and geometrical sculptures. One of the founders of the annual Bridges Mathematics and Art Conference and a Bridges board member for almost 30 years
“This book weaves mathematics and crochet together in beautiful ways while explaining it all clearly. The scope and clarity of this book are quite impressive.”
—David Press, 3D string-art sculptor
