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Visualizing Mathematics with 3D Printing Visualizing Mathematics with 3D Printing job: 3D数学 内文 pi 160427 jun This page intentionally left blank Visualizing Mathematics with 3D Printing HENRY SEGERMAN JOHNS HOPKINS UNIVERSITY PRESS BALTIMORE job: 3D数学 内文 piii 160427 jun © 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. job: 3D数学 内文 piv 160427 jun 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 n job: 3D数学 内文 pv 160427 jun This page intentionally left blank 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 n job: 3D数学 内文 pvii 160427 jun 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 job: 3D数学 内文 pviii 160427 jun 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 un job: 3D数学 内文 pix 160427 jun This page intentionally left blank Acknowledgments This book would not have happened without many, many other people. First of all, my parents, Eph and Jil Segerman, were instrumental in my existence. Ap- parently, they also had a large part to play in getting me into 3D stuff in the fi rst place, because both my brother, Will, and I have (in very diff erent ways) built our careers around 3D. Will’s current main source of income is as a virtual milliner. Make of that what you will. Huge thanks to my various collaborators. My brother, Will, worked with me on the monkey sculp- tures, and Vi Hart got us thinking about four-dimen- sional symmetries. Keenan Crane fl owed a coff ee mug, Geoff rey Irving fi gured out where to put hinged triangles, Craig S. Kaplan tiled a bunny, Marco Mahler worked with me on mobiles, and Roice Nelson tiled two- and three-dimensional hyperbolic space. Par- ticular thanks to Saul Schleimer, my collaborator in both topology research and mathematical illustration, who is very easy to distract from the former to the latter. Saul and I worked on too many projects to list here, apart from the one with yet another collabora- tor: the parametrization of the fi gure-eight knot, with François Guéritaud. Thanks to the other mathematicians, designers, and artists whose work I featured: Vladimir Bulatov, Bathsheba Grossman, George Hart, Oliver Labs, Carlo Séquin, Laura Taalman, Oskar van Deventer; Jessica Rosenkrantz and Jesse Louis-Rosenberg of Nervous System; the team that worked on the ropelength knots: Jason Cantarella, Eric Rawdon, Michael Piatek, and Ted Ashton; and the team that worked on the fl at xi job: 3D数学 内文 pxi 160427 jun torus: Vincent Borrelli, Saïd Jabrane, Francis Lazarus, and Boris Thibert. Thanks to Bus Jaco (the head of) and the rest of the Department of Mathematics at Oklahoma State University, for their support while I was writing the book. Bus helped me track down the OSU physics and chemistry instrument shop that built the photo rig for me and also found Joyce Lucca and Sam Welch, who loaned me the turntable. The purchases of many of the models were supported by a Dean’s Incentive Grant from the College of Arts and Sciences at OSU. Thanks to Robert McNeel & Associates for making Rhinoceros, the main program I used to design the models, and to the 3D printing service Shapeways for printing them. Jarey Shay designed the companion website to the book, and NeilFred Picciotto acquired the domain names. Stephan Tillmann and an anonymous reviewer both made early suggestions that changed the core focus of the book. I had some useful conversations about nega- tively curved spaces with Chaim Goodman-Strauss. Thanks to Vincent J. Burke, Andre M. Barnett, and everyone else at Johns Hopkins University Press, who turned my manuscript into a book. Moira Bucciarelli, Evelyn Lamb, Craig Kaplan, Rick Rubinstein, Saul Schleimer, Jil Segerman, Carlo Séquin, Rosa Zwier, and the anonymous reviewers read through versions of the book and found lots of ways to make it better and clearer. All errors are, of course, my own. xii Acknowledgments job: 3D数学 内文 pxii 160427 jun Visualizing Mathematics with 3D Printing n job: 3D数学 内文 pxiii 160427 jun This page intentionally left blank 1 Symmetry Symmetrical objects and patterns surround us, in art, architecture, and design.
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