Table Of ContentVisualizing Mathematics with 3D Printing
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Visualizing Mathematics
with 3D Printing
H E N RY S E G E R M A N
JOHNS HOPKINS UNIVERSITY PRESS BALTIMORE
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© 2016 Johns Hopkins University Press
All rights reserved. Published 2016
Printed in China on acid-free paper
9 8 7 6 5 4 3 2 1
Johns Hopkins University Press
2715 North Charles Street
Baltimore, Maryland 21218-4363
www.press.jhu.edu
Library of Congress Cataloging-in-Publication Data
Names: Segerman, Henry, 1979–
Title: Visualizing mathematics with 3D printing / Henry Segerman.
Description: Baltimore : Johns Hopkins University Press, 2016. | Includes bibli-
ographical references and index.
Identifi ers: LCCN 2015043848| ISBN 9781421420356 (hardcover : alk. paper) |
ISBN 9781421420363 (electronic) | ISBN 142142035X (hardcover : alk. paper)
| ISBN 1421420368 (electronic)
Subjects: LCSH: Geometry—Computer-assisted instruction. | Mathemat-
ics—Computer-assisted instruction. | Geometry—Study and teaching. |
Geometrical constructions. | Three-dimensional imaging. | Three-dimen-
sional printing.
Classifi cation: LCC QA462.2.C65 S44 2016 | DDC 516.028/6—dc23 LC record
available at http://lccn.loc.gov/2015043848
A catalog record for this book is available from the British Library.
Special discounts are available for bulk purchases of this book. For more informa-
tion, please contact Special Sales at 410-516-6936 or [email protected].
Johns Hopkins University Press uses environmentally friendly book mate-
rials, including recycled text paper that is composed of at least 30 percent
post-consumer waste, whenever possible.
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Contents
Preface vii
Acknowledgments xi
1 Symmetry
1
2 Polyhedra
23
3 Four-Dimensional Space
33
4 Tilings and Curvature
69
5 Knots
97
6 Surfaces
111
7 Menagerie
149
Appendix A: Commentary on Figures 159
Appendix B: How I Made These Models 177
Index 181
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Preface
Welcome to my book, dear reader. Before anything
else, let me fi rst encourage you to visit the companion
website to this book, 3dprintmath.com.
This is a popular mathematics book, intended for
everyone, no matter his or her mathematical level.
This book is a little diff erent from other popular math
or science books. In this book, whenever it makes
sense, the diagrams are photographs of real-life 3D
printed models. Almost all of these models are avail-
able virtually on the website—they can be rotated
around on the screen so that you can view them from
any angle. They are also available to download and 3D
print on your own 3D printer or purchase online at
the website.
With these models, you, the reader, can experi-
ence three-dimensional concepts directly, as three-
dimensional objects. They let me describe some very
beautiful mathematics, including some topics that,
although accessible, are diffi cult to explain well using
only two-dimensional images. I’ve tried hard to make
things understandable with only the book, but ideally
you should be reading while holding the 3D printed
diagrams in your hands or using the virtual models
on the website.
Because this book is built around 3D printed
diagrams, the topics we will look at tend toward the
geometric. The fi rst chapter is about the diff erent ways
that three-dimensional objects can be symmetric.
Chapter 2 is about some of the simplest shapes: the
two-dimensional polygons and the polyhedra, their
three-dimensional relatives. Chapter 3 builds off chap-
ter 2, reaching up to the four-dimensional relatives
vii
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of polygons and polyhedra and investigating how we
can see four-dimensional objects by casting shadows
of them down to three dimensions. Chapter 4 is about
tilings and curvature—whether a surface is shaped
like a hill, a fl at plane, or a saddle. Chapter 5 is about
knots and thinking topologically—looking at geomet-
ric objects but not caring about the precise shapes,
as if everything were made of very stretchy rubber.
Chapter 6 continues the topological theme by looking
at surfaces and then later on thinking about geome-
try again by putting tilings on surfaces. Chapter 7 is
a menagerie, of mathematical prints I couldn’t resist
including in the book.
Appendix A lists credits and technical details for
the fi gures and 3D prints. Some of these include
parametric equations that the adventurous reader
may want to use to create her own visualizations. If
you’re interested in how I go about making models,
see appendix B.
There isn’t much in the way of tricky notation or
calculations in this book. It’s more about getting a
visual sense of what’s going on. Having said that, some
things might be a bit diffi cult to follow. If you get
stuck on something, feel absolutely free to skip over it
and come back later.
Why 3D printing? 3D printing is a technology
that gives unprecedented freedom in the creation
of three-dimensional physical objects. A 3D printer
builds an object layer by layer in an automated addi-
tive process, based on a design given to it by a com-
puter. 3D printers are particularly suited to producing
mathematically inspired objects, in part because
designs can be generated by programs written to pre-
cisely represent the mathematics. Because production
is automated, the physical models you obtain closely
approximate the mathematical ideals. Small produc-
tion runs of 3D objects and production on demand
aren’t as possible with other manufacturing technol-
ogies. There’s no way I could have made all of the
diagrams in this book without 3D printing.
One last comment before we get started: 3D print-
viii Preface
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ers are so good at producing mathematical models
that I sometimes run into an interesting problem. A
photograph of a physical 3D printed object is so close
to the mathematical ideal that viewers sometimes
assume that the photograph is actually a computer
render. All of the pictures in this book that look like
photographs of real objects are indeed photographs
of real objects, sometimes with some color added to
the image to highlight a feature. I have deliberately left
occasional imperfections in the photographs to prove
their reality. Or at least, this seems like an excellent
excuse/reason for any fl aws you may fi nd.
ix Preface
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