Optical Illusions in Rome: A Mathematical Travel Guide

92 Pages·2019·4.782 MB·English

by  Kirsti Andersen

AMS / MAA | SPECTRUM VOL 99
OPTICAL ILLUSIONS IN ROME
A Mathematical Travel Guide
Kirsti Andersen
Translated by
Viktor Blåsjö
Committee on Books
Susan G. Staples, Chair
2019 Editorial Committee
James J. Tattersall, Editor
Michael Barany, Andrew Beveridge, Virginia M Buchanan, Thomas L. Drucker, Evan D. Fisher,
Donna L. Flint, Richard K. Guy, Dominic Klyve, John Lorch, Cayla Danielle McBee
2010 Mathematics Subject Classification. Primary 01A40, 00A66, 00A67; Secondary 51-03.
For additional information and updates on this book,
visit www.ams.org/bookpages/spec-99
Cover art courtesy of Søven Halse.
Library of Congress Cataloging-in-Publication Data
Names: Andersen, Kirsti, author. | Blåsjö, Viktor, 1982 – translator.
Title: Optical illusions in Rome: A mathematical travel guide / Kirsti Andersen; translated by Viktor Blåsjö.
Other titles: Romerske synsbedrag. English
Description: Providence, Rhode Island: MAA Press, an imprint of the American Mathematical Society, [2019] | Series:
Spectrum; volume 99 | Originally published in Danish by the Danish Association of Mathematics Teachers:
Romerske synsbedrag (København: Matematiklaererforeningen, 2016). | Includes bibliographical references.
Identifiers: LCCN 2019022967 | ISBN 9781470452674 (paperback)
Subjects: LCSH: Optical illusions in art—Miscellanea. | Trompe l’oeil painting—Italy—Rome—Miscellanea. |
Architecture—Italy—Rome—Miscellanea. | Optical illusions—Problems, exercises, etc. | Trompe l’oeil painting—
Problems, exercises, etc. | Architecture—Problems, exercises, etc. | AMS: History and biography [See also the
classification number—03 in the other sections]—History of mathematics and mathematicians—15th and 16th
centuries, Renaissance. | General—General and miscellaneous specific topics—Mathematics and visual arts,
visualization. | General—General and miscellaneous specific topics—Mathematics and architecture. | Geometry
{For algebraic geometry, see 14-XX}—Historical (must also be assigned at least one classification number from
Section 01).
Classification: LCC N7430.5 .A5413 2019 | DDC 701—dc23
LC record available at https://lccn.loc.gov/2019022967
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10 9 8 7 6 5 4 3 2 1 24 23 22 21 20 19
iii
CONTENTS
TRANSLATOR’S INTRODUCTION v CHAPTER 3. THE ANAMORPHOSIS IN
TRINITÀ DEI MONTI 29
ACKNOWLEDGMENTS ix 1. The order of the Minims 29
2. Niceron’s construction of an
INTRODUCTION 1
anamorphic grid 31
3. Maignan’s anamorphosis 31
CHAPTER 1. TROMPE L’ŒIL ON WALLS 5
4. Mirror anamorphoses 35
1. Expansions of rooms in antiquity 5
2. Expansions of rooms in the CHAPTER 4. CEILINGS AS IMAGE SURFACES 37
Renaissance 7
1. The order of the Jesuits 37
3. Peruzzi and the Villa Farnesina 7
2. Pozzo, the master of illusionistic art 38
4. The virtual expansion of the Sala
3. The dome of Sant’Ignazio 40
delle prospettive 9
4. The decoration of the central nave
5. The Renaissance impact of Peruzzi’s of Sant’Ignazio 45
Sala delle prospettive 13
6. Theatre decors 14 CHAPTER 5. SOME RESULTS FROM
PERSPECTIVE THEORY 51
CHAPTER 2. THREE-DIMENSIONAL
CHAPTER 6. EXERCISES 61
TROMPE L’ŒIL 17
1. Exercises for Chapter 1 61
1. Borromini and Bernini 18
2. Exercises for Chapter 2 62
2. Borromini’s colonnade in Palazzo
3. Exercises for Chapter 3 65
Spada 18
4. Exercises for Chapter 4 66
3. The eye point of Borromini’s
5. Exercises for Chapter 5 67
colonnade 20
4. The length of the apparent colonnade 21
NOTES FOR THE TRAVELLER 69
5. A later addition to Borromini’s
ENDNOTES 71
colonnade 23
BIBLIOGRAPHY 73
6. Bernini’s royal staircase 23
7. Bernini’s design of St. Peter’s Square 26 SOURCES OF THE ILLUSTRATIONS 78

v
TRANSLATOR’S INTRODUCTION
Using the geometrical principles of perspective to makes us wander all over the place in confusion,
deceive the eye is an old art. Vitruvius noted that often changing our minds about the same thing and
“by this deception a faithful representation of the regretting our actions and choices with respect to
appearance of buildings might be given in painted things large and small ... . The art of measurement,”
scenery, … so that, though all is drawn on a vertical by contrast, “would make the appearances lose their
flat facade, some parts may seem to be withdrawing power” and “give us peace of mind firmly rooted in
into the background, and others to be standing out in the truth.”5 A rousing case for mathematics! But Plato
front.”1 perhaps drew his conclusions a step too far, rejecting
Such optical illusions are no mere party tricks. categorically the role of observational data in science:
In fact, they have arguably played a pivotal role in “there’s no knowledge of … sensible things, whether
several major developments in Western thought. To by gaping upward or squinting downward.”6 Science
Plato they raised profound epistemological conun- must be based on “the naturally intelligent part of
drums. He was concerned that “trompe l’œil painting” the soul,” not observation. For example, “let’s study
has “powers that are little short of magical, ... because astronomy by means of problems, as we do geometry,
they exploit this weakness in our nature,” bypassing and leave the things in the sky alone.”7 With such
“the rational part of the soul.”2 The solution to this attitudes, perhaps it is no wonder that the Greeks
problem, as Plato saw it, was a solid mathematical excelled more in mathematics than in the sciences.
education. Since “sense perception seems to produce When the principles of perspective were redis-
no sound result” with these “trompe l’œil paintings,”3 covered in Renaissance Italy, they were again at the
“it makes all the difference whether someone is a heart of major scientific developments, but this time
geometer or not.”4 “The power of appearance often on the side of empirical science. According to many
1 Vitruvius, De Architectura (1st century BC), VII.11, [Vitruvius (1914), 198]. 2 Plato, Republic, X.602d, [Plato (1997), 1207].
3 Plato, Republic, VII.523b, [Plato (1997), 1140]. 4 Plato, Republic, VII.526d, [Plato (1997), 1143]. 5 Plato, Protagoras, 356d, [Plato (1997), 785].
6 Plato, Republic, VII.529b, [Plato (1997), 1145]. 7 Plato, Republic, VII.530b, [Plato (1997), 1146].
vi
recent scholars, “the invention of perspective by the be won. One anecdote perhaps speaks for the age as
Renaissance artists, … by demonstrating that math- a whole: “It was said of Uccello that the discovery
ematics could be usefully applied to physical space of perspective had so impressed him that he spent
itself, [constituted] a momentous step … toward the nights and days drawing objects in foreshortening,
general representation of physical phenomena in and setting himself ever new problems … . He was so
mathematical terms.”8 Thus “the mathematization engrossed in these studies that he would hardly look
of the sublunary world begins not with Galileo but up when his wife called him to go to bed, and would
with Alberti,”9 who wrote on the geometrical prin- exclaim ‘What a sweet thing perspective is!’”13
ciples of perspective in painting in the 15th century. Art historians have argued that “our pleasure
More generally, “Renaissance developments in prac- in illusion … rests precisely in the mind’s effort in
tical mathematics predated the intellectual shifts in bridging the difference between art and reality.” It
natural philosophy.”10 “Real science is born when … follows that “pleasure is destroyed when the illusion
the experimental method of the craftsmen overcomes is too complete” or “so easy as to be automatic.”14
the prejudice against manual work and is adopted by “When the painter packs a vast expanse into a narrow
rationally trained university-scholars. This is accom- space, when he leads me across the depths of the
plished with Galileo.”11 “What enabled Galileo to infinite on a flat surface, … I love to abandon myself
overcome [this divide] … seems easily explicable in his illusions, but I want the frame to be there, I
upon considering Galileo’s background in the arts want to know that what I see is actually nothing but
and crafts.”12 Perhaps it is no coincidence, then, that a canvas or a simple plane.”15 The examples we shall
Galileo was a countryman of the pioneering painters meet in Chapter 4, for instance, direct the viewer to
and architects—or applied mathematicians, if you a specific vantage point prominently marked on the
will—whom we shall meet in this book. floor. In such a case, at least, it is fair to say that
But the art we shall encounter was not devised for “illusionism … requires of the viewer a willing partic-
science or philosophy. We shall see many works that ipation, amounting to a complicity with the artist
make use of the power of perspective to supplement … . No great illusionistic scheme ever truly deceived,
modest buildings with grandiose virtual expansions or attempted to deceive. The artist’s function was to
of their architecture, when the real thing would have fling open the illusionistic portals into the domain of
been vastly more expensive to build or indeed even the imagination.”16
impossible due to spatial limitations. But, more than If anything, this element of explicitly embracing
a cost benefit analysis to the effect that marble is the artifice of the illusion is arguably becoming more
cheaper to paint than to sculpt, it was perhaps the prominent as we progress chronologically through
sheer joy of the illusion that was the real pleasure to the book. Anamorphoses—that is to say, as Shake-
8 [Conner (2005), 270]. 9 [Wootton (2015), §5.8]. 10 [Bennett (1991), 176]. 11 [Zilsel (2000), 5]. 12 [Cohen (1994), 349]. 13 [Gombrich
(1989), 190]. 14 [Gombrich (2002), 236]. 15 [Quincy (1837), 128], quoted from [Gombrich (2002), 236]. 16 [d’Otrange Mastai (1975), 11].
vii
speare put it, “perspectives which, rightly gaz’d
upon, show nothing but confusion; [yet] ey’d awry,
distinguish form”17—may be seen as the logical
conclusion of this attitude taken to its extreme. Gone,
at this point, is any pretence of architectural thrift, yet
the art retains its appeal—suggesting, perhaps, that it
was the thrill of playing havoc with the senses that
was the primary purpose all along.
We follow in the footsteps of many a generation
when we travel to Rome to come face-to-face with the
finest achievements of past ages. A “Grand Tour” of
Europe’s classical sites and artistic treasures—invar-
iably culminating in Rome—was long considered a
sine qua non of a refined education among Northern
Europeans of means. “A man who has not been in
Italy is always conscious of an inferiority, from his
not having seen what it is expected a man should see
… . All our religion, almost all our law, almost all
our arts, almost all that sets us above savages, has
come from the shores of the Mediterranean.”18 The
aspiring connoisseurs who completed this cultural
pilgrimage quickly realized that “Rome was a city
which demanded a guide … employing one of the
established ciceroni or an ‘antiquarian’ was both a
practical measure and a statement of one’s serious-
ness as a student of antiquities and correct taste.”19
We are fortunate to have Kirsti Andersen, the leading
scholar in the field,20 as cicerone on our journey.
Viktor Blåsjö
Utrecht, February 2018
17 Shakespeare, Richard II (1595), II.2. 18 Dr. Johnson, 1776, quoted in Boswell’s Life of Samuel Johnson. 19 [Sweet (2012), 101].
20 [Andersen (2006)].
viii
References for Translator’s Introduction [Gombrich (2002)] E. H. Gombrich, Art and Illusion:
A Study in the Psychology of Pictorial Representa-
[Andersen (2006)] Kirsti Andersen, The Geometry of
tion, 6th ed., Phaidon.
an Art: The History of the Mathematical Theory of
[Plato (1997)] Plato, Complete Works, ed. John M.
Perspective from Alberti to Monge, Springer, 2006.
Cooper, Hackett, 1997.
[Bennett (1991)] J. A. Bennett, The challenge of
[Quincy (1837)] A. C. Quatremère de Quincy, Essai
practical mathematics, in S. Pumfrey, P. L. Rossi, &
sur la neture, le but et les moyens de l’imitation dans
M. Slawinski (eds.), Science, Culture and Popular
les beaux arts, Paris, 1823.
Belief in Renaissance Europe, Manchester University
Press, 1991, 176–190. [Sweet (2012)] Rosemary Sweet, Cities and the
Grand Tour: The British in Italy, c. 1690–1820,
[Cohen (1994)] H. Floris Cohen, The Scientific
Cambridge University Press, 2012.
Revolution: A Historiographical Inquiry, University
of Chicago Press, 1994. [Vitruvius (1914)] Vitruvius, The Ten Books on Archi-
tecture, translated by Morris Hicky Morgan, Harvard
[Conner (2005)] Clifford D. Conner, A People’s University Press, 1914.
History of Science: Miners, Midwives, and Low
[Wootton (2015)] David Wootton, The Invention of
Mechanicks, Nation Books, 2005.
Science: A New History of the Scientific Revolution,
[d’Otrange Mastai (1975)] Marie-Louise d’Otrange Allen Lane, Penguin Random House, 2015.
Mastai, Illusion in Art: A History of Pictorial Illu-
[Zilsel (2000)] Edgar Zilsel, The Social Origins of
sionism, Abaris Books, New York, 1975.
Modern Science, edited by D. Raven, W. Krohn, &
[Gombrich (1989)] E. H. Gombrich, The Story of Art, R. S. Cohen, Boston Studies in the Philosophy of
15th ed., Phaidon. Science 200, Kluwer, 2000.
ix
ACKNOWLEDGMENTS
This book first appeared in Danish in 2016. Many which Touborg added three. Kurt Finsten applied his
people have encouraged me to produce an English great experience in designing the layout. To each of
edition as well, but I have not had the time. So I was the five mentioned, I extend my heartfelt gratitude.
very happy when Viktor Blåsjö offered to undertake On one subject where I have limited background,
the translation. I am very grateful to him for this. the colonnades of St. Peter’s Square, I am very
My Danish book was published by the Danish grateful to Ivan Tafteberg Jakobsen for his efforts in
Association of Mathematics Teachers. Here I came initiating me into the subject. He has written instruc-
in contact with a group that was very helpful. Dorte tive notes on the subject, which he let me read, and
Krammer, Jens Peter Touborg, and Sven Toft Jensen he also helped me understand the colonnades through
were extremely good in inspiring me to present my correspondence.
ideas in a clearer and more elegant way. Toft Jensen Regarding the colonnades on St. Peter’s Square,
has also drawn a number of fine illustrations to the Anker Tiedemann kindly gave permission for Toft
book. Sources for the illustrations are at the end Jensen to base a drawing on one of his.
of the book. I am grateful to the Danish editors for
supplying the exercises in Chapter 6.
Kirsti Andersen
Søren Halse, in an impressive achievement, took
Amsterdam, February 2018
the majority of the excellent photos of the book to