Crocheting Adventures with Hyperbolic Planes

158 Pages·2009·24.78 MB·English

by  Daina Taimina

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© IFF / Steve Rowell
CCrroocchheettiinngg AAddvveennttuurreess
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Daina Taimin, a

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Library of Congress Cataloging-in-Publication Data
Taimiņa, Daina.
Crocheting adventures with hyperbolic planes / Daina Taimiņa.
p. cm.
Includes bibliographical references and index.
ISBN 978-1-56881-452-0 (alk. paper)
1. Geometry, Hyperbolic. 2. Crocheting--History. I. Title.
QA685.T35 2009
516.9--dc22
2008038302
Printed in India
13 12 11 10 10 9 8 7 6 5 4 3 2
This story started here,
October 1, 1995
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Foreword by William Thurston ix
Acknowledgments xi
Introduction 1
Chapter 1. What Is the Hyperbolic Plane? Can We Crochet It? 9
Chapter 2. What Can You Learn from Your Model? 25
Chapter 3. Four Strands in the History of Geometry 35
Chapter 4. Tidbits from the History of Crochet 63
Chapter 5. What Is Non-Euclidean Geometry? 69
Chapter 6. How to Crochet a Pseudosphere and a Symmetric Hyperbolic Plane 79
Chapter 7. Metamorphoses of the Hyperbolic Plane 87
Chapter 8. Other Surfaces with Negative Curvature: Catenoid and Helicoid 109
Chapter 9. Who Is Interested in Hyperbolic Geometry Now and How Can It Be Used? 119
Appendix Paper Models 133
Endnotes 139
Index 147
Contents vii
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Many people have an impression, based on years of school- cats make—perhaps meowing, purring, squeaking, yowl-
ing, that mathematics is an austere and formal subject con- ing. But can you draw a good picture of a cat, can you
cerned with complicated and ultimately confusing rules give a good description or animation of how a cat moves,
for the manipulation of numbers, symbols, and equations, can you describe or recreate the sounds a cat makes? It’s
rather like the preparation of a complicated income tax a lot harder to communicate a cat than to see a cat, and it
return, where there are myriad unexplained steps, rules, requires serious cultivation of special talents.
exceptions, and gotchas. We learn about cats by watching cats; with most math-
Good mathematics is quite opposite to this. Mathe- ematical ideas, our culture has provided no comparable
matics is an art of human understanding. Billions of years shortcuts. Without a good way for whole-brain communi-
of evolution have given us many extraordinary capabilities cation, understanding is denatured.
that we ordinarily take for granted—but we deny those ca- Non-Euclidean or hyperbolic geometry was a topic of
pabilities at our peril. In the abstract, the mere act of walk- great mystery and confusion for many centuries, as Daina
ing through a room without bumping into other people recounts in this book. Insights people may have developed
or things is a far greater accomplishment than the most were hard to document, so they crumbled away.
sophisticated formal computation ever done by mathema- Why do Daina’s crochet models have such a great
ticians. Computers are far better than humans at formal resonance with so many people? It’s because they break
computations, but humans far surpass current computers through the austere, formal stereotype of mathematics and
at informal and intuitive reasoning. present a path to a whole-brain understanding of a beauti-
Our brains are complicated devices, with many spe- ful cluster of simple and signifi cant but rarely understood
cialized modules working behind the scenes to give us an ideas. The crochet models also break through the stereo-
integrated understanding of the world. Mathematical con- type that mathematics is only relevant to traditionally male
cepts are abstract, so it ends up that there are many differ- interests.
ent ways they can sit in our brains. A given mathematical These models have a fascination far beyond their
concept might be primarily a symbolic equation, a picture, visual appearance. As illustrated in the book, there is actu-
a rhythmic pattern, a short movie—or best of all, an in- ally negative curvature and hyperbolic geometry all around
tegrated combination of several different representations. us, but people generally see it without seeing it. You will
The non-symbolic mental models for mathematical con- develop an entirely new understanding by actually follow-
cepts are extremely important, but unfortunately, many of ing the simple instructions and crocheting! The models
them are hard to share. are deceptively interesting. Perhaps you will come up with
Mathematics sings when we feel it in our whole brain. your own variations and ideas.
People are generally inhibited about even trying to share In any case, I hope this book gives you pause for
their personal mental models. People like music, but they thought and changes your way of thinking about math-
are afraid to sing. You only learn to sing by singing. ematics.
How do you think of a cat? You probably have men-
tal images, you probably can see in your mind how cats Bill Thurston
move, you can probably hear in your mind sounds that Ithaca, NY
Foreword ix