Viewpoints: Mathematical Perspective and Fractal Geometry in Art

259 Pages·2011·7.73 MB·English

by  Marc Frantz

VIEWPOINTS
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VIEWPOINTS
Mathematical Perspective and
Fractal Geometry in Art
Marc Frantz
Annalisa Crannell
Princeton University Press
Princeton and Oxford
Copyright c 2011 by Princeton University Press
(cid:13)
Published by Princeton University Press, 41 William Street,
Princeton, New Jersey 08540
In the United Kingdom: Princeton University Press, 6 Oxford Street,
Woodstock, Oxfordshire, OX20 1TW
press.princeton.edu
Cover photo: Winter Road along the Trees, by Wil Van Dorp
All Rights Reserved
Library of Congress Cataloging-in-Publication Data
Frantz, Marc, 1951–
Viewpoints: mathematicalperspectiveandfractalgeometryinart/ MarcFrantz,
Annalisa Crannell.
p. cm.
Includes bibliographical references and index.
ISBN 978-0-691-12592-3(hardback: alk. paper)
1.Perspective—Textbooks. 2.Fractals—Textbooks. 3.Art—Mathematics—Textbooks.
I. Crannell, Annalisa. II. Title.
QA515 .F73 2011
742.0151-dc22 2010053315
′
British Library Cataloging-in-PublicationData is Available
This book has been composed in LATEX
The publisher would like to acknowledge the authors of this volume for providing
the camera-ready copy from which this book was printed.
Printed on acid-free paper.
∞
Printed in the United States of America
1 3 5 7 9 10 8 6 4 2
Contents
Preface vii
Acknowledgments ix
1 Introduction to Perspective and Space Coordinates 1
Artist Vignette: Sherry Stone 9
2 Perspective by the Numbers 13
Artist Vignette: Peter Galante 25
3 Vanishing Points and Viewpoints 29
Artist Vignette: Jim Rose 39
4 Rectangles in One-Point Perspective 43
What’s My Line?: A Perspective Game 55
5 Two-PointPerspective 59
Artist Vignette: Robert Bosch 77
6 Three-PointPerspective and Beyond 85
Artist Vignette: Dick Termes 113
7 Anamorphic Art 117
Viewpoints at the Movies: The Hitchcock Zoom 135
Plates follow page 138
8 Introduction to Fractal Geometry 139
vi Contents
Artist Vignette: Teri Wagner 157
9 Fractal Dimension 161
Artist Vignette: Kerry Mitchell 193
Answers to Selected Exercises 197
Appendix: Information for Instructors 215
Annotated References 223
Index 229
Preface
Viewpointsisanundergraduatetextinmathematicsandartsuitable
formath-for-liberal-artscourses,mathematicscoursesforfineartma-
jors,andintroductoryartclasses. Instructorsinsuchcoursesatmore
than25institutionshavealreadyusedanearlieronlineversionofthe
text, called Lessons in Mathematics and Art. The material in these
texts evolved from courses in mathematics and art which we devel-
oped in collaboration and taught at our respective institutions. In
addition,this materialhasbeentested at,andinfluencedby, aseries
of weeklong Viewpoints faculty development workshops.
Pedagogical Approach
The approach of Viewpoints is highly activity based, much like
the approach of teaching in art school. As many of our workshop
graduateswillattest, the true value of the materialcanonly be fully
appreciated by engaging in these activities, and not by merely read-
ingthebook. Wehaveendeavoredtoincludeproblemsandactivities
of genuine interest and value to art students—problems that go sig-
nificantly beyond what students normally learn in art school. In our
experiencetheauthenticityoftheproblemsmakesthemgenuinelyin-
teresting, not only to art majorsbut to students from a broadrange
of disciplines. We have included a detailed appendix for instructors
which includes advice on the window-taping activity of Chapter 1, a
sampletimetableforafirst-yearseminarcoursebasedonViewpoints,
and a list of additional writing assignments.
Wehaveendeavoredtomakesurethattheproblemsarerealmath
problems. Happily, there is a wealth of problems at the boundary of
mathematicsandarthavinganumberofexcellentpedagogicalprop-
erties: (1) the problems are natural and easily understood; (2) the
problems have multiple solutions of varying difficulty and applica-
bility; (3) the problems admit multiple proofs, both geometric and
algebraic; (4) once arrived at, the solutions are easy to remember
and rewarding to use; and (5) the search for solutions captures the
essence ofmathematicalresearchanddiscovery. (We demonstrate in
depth how this fivefold approach comes to bear on a single problem
viii Preface
in the solutionto Exercise 8 of Chapter 4.) We have soughtto make
the problems in this book embody each of these characteristics,at a
level that is accessible to every undergraduate student.
Artist Vignettes
A special feature of the text is a series of personal essays which
we call Artist Vignettes. During the course of our project we have
been fortunate to meet a number of professional artists who have
generously contributed to the book. Each Vignette contains a short
biography,anartist’sstatement,andsomeimagesoftheartist’swork.
Color images of the artists’ work will also appear in the color plate
section. We feel that the input of practicing artists informed about
mathematics will be an exciting and attractive addition to the text.
Origin of the Text
TheinitialcoursedevelopmentandthefirstViewpointsworkshops
were supported by the Indiana University Mathematics Throughout
the Curriculum project, the Indiana University Strategic Directions
Initiative, Franklin & Marshall College, and the National Science
Foundation (NSF-DUE 9555408). The 2001 workshop was also sup-
ported by the ProfessionalEnhancementProgramof the Mathemat-
ical Association of America. We revised and expanded the text as
partofacollaborationbetweenourtwoinstitutions,supportedbyan
NSF Educational Materials Development grant (NSF-DUE 0439891
and 0439713).
Marc Frantz
Indiana University
Annalisa Crannell
Franklin & Marshall College
Acknowledgments
The first glimmerings of this book appeared fifteen years ago, when
the two authors happened to have the good luck to be in the same
place (IUPUI) at the same time (fall of 1995), just as Indiana Uni-
versityproposeda Mathematics Throughoutthe Curriculumproject
(MTC) to the National Science Foundation. The fearless leader of
that grant was Dan Maki. Since then, he’s been supportive of our
small part of his larger project at every stage: as we started teach-
ingmath-and-artcoursesatourtwohomeinstitutions, aswestarted
writingupourmaterials,asweputtogetherourViewpointssummer
workshopsfor college instructors, and as we hostedour follow-upre-
unions. We couldn’t imagine a better fearless leader than Dan, and
to him goes our heartfelt gratitude.
Asmemoryserves,itwasBartNg,Dan’scoleader,whofirstmen-
tioned the idea of our collaboration. We’re grateful to Bart for his
support and for the spark that led to such a long and rewarding
adventure.
And of course, we’re very grateful to the National Science Foun-
dationforthegranttoMTC(NSF-DUE9555408),andforthegrants
to our own collaborativeproject (NSF-DUE 0439891and NSF-DUE
0439713)to write and disseminate this book.
Tina Straley, Brian Winkel, Pippa Drew, and Dorothy Wallace
invitedustoDartmouthinthesummerof1998forawonderfulwork-
shop on math and the humanities. Pippa and Dorothy have done a
lot of nice workwith symmetry groupsand art,and we learneda lot
from them—including the joys of running a workshop! It’s no coin-
cidence that the first Viewpoints workshop came two summers after
that.
Oneofourfirstandmostdetermined“Viewpointers”(aswecalled
our workshopparticipants)was Sister BarbaraReynolds of Cardinal
Stritch University. It was Sr. Barbara who brought Peter Galante
and Teri Wagner to our attention; she came to our workshop two
yearsin a rowandsentevenmoreofher colleaguesinlater years. In
her position as Editor of the MAA Notes series, she played a pivotal
role in encouraging our manuscript. The world should have more