In this book, Rebecca Zorach takes us on a lively hunt for the triangle’s embedded significance. From the leisure pursuits of Egyptian priests to Jacopo Tintoretto’s love triangles, Zorach explores how the visual and mathematical properties of triangles allowed them to express new ideas and to inspire surprisingly intense passions. Examining prints and paintings as well as literary, scientific, and philosophical texts, The Passionate Triangle opens up an array of new ideas, presenting unexpected stories of the irrational, passionate, melancholic, and often erotic potential of mathematical thinking before the Scientific Revolution.
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THE PASSIONATE Triangle
By Rebecca Zorach
University of Chicago PressCopyright © 2011 The University of Chicago
All right reserved.
Chapter OneThe Passionate Triangle
As deluged as we may be with digital images, art historians stubbornly maintain that nothing replaces acquaintance with original works of art. For some it may be a matter of an irreducible spiritual experience. It may be the conviction that we can't grasp the labor that goes into the making of an object when we receive it as pixels on a screen. Sometimes this view is simply a utilitarian one: you can't wholly trust reproductions when trying to identify basic visual elements of a large and detailed painting. This is partly, but not only, a matter of scale. Photography is not purely objective, and a trick of light, the texture of painted surface, or imperfect color calibration—all can render a significant detail illegible.
So at the tail end of an all-too-businesslike trip to Europe, with just two days in Munich, I made my way to the Old Masters picture gallery, the Alte Pinakothek, with a bit of sleuthing to do. My goals were modest: I wanted simply to figure out what was going on in the upper right-hand corner of Tintoretto's Vulcan, Venus, and Mars (plate 1), a dramatic Renaissance depiction of the most famous love triangle in Greco-Roman mythology.
As it happens, my pilgrimage was rewarded. What in reproductions had been an unidentifiable dark dome turned out to be the cavernous opening of a fireplace; what had seemed like steps leading up to it were actually the bricks of its base. This matters to the painting's meaning: as Vulcan, the blacksmith god, bursts in, seeking signs of his wife's infidelity, he enters from the upper right—direct from his labors. Seen in this way, the room pulsates with parallel heat: heat from the lovers and heat from the fire of the forge. Tintoretto insists on the point that the artisan's place of work is contiguous with the place of his betrayal. A small point, but one of many components that taken together form the texture of art-historical scholarship.
But there was more to this visit. Wandering about the museum's Italian Renaissance wing, I was surprised by a voice: "Typische Dreieckskomposizion." 1 "Typical triangular composition": the words that startled me could have come from the ghost of Heinrich Wölfflin, the turn-of-the-twentieth-century art historian who strongly associated triangular composition with the High Renaissance. Wölfflin trained in Munich and knew this collection well; it provided him with many of his main examples. The voice came, however, from one of the earnest, Birkenstock-shod teenagers standing next to me in front of Raphael's Canigiani Holy Family (plate 2) and corroborated my sense that the idea of triangular composition still had currency in textbooks and survey classes. I paused: perhaps my new test subjects would expound further. But they moved on. The phrase was the end of description, not the beginning of it. And indeed, this is the problem with the thing we call "triangular composition." Along with "perspective," it has come to constitute a stereotyped view of Renaissance image conventions. An observation that stands in for argument, it seems more obvious than it is.
Viewing the Italian Renaissance paintings in the museum together, as a group, I could also think about the particular view of the Renaissance that they present. This collection that furnished Wölfflin with so many important examples also happens to include many examples of geometric, and particularly triangular, arrangements in its Renaissance paintings. Collected largely in the mid-nineteenth century, the Italian paintings gathered in Munich reflect choices made by German aristocrats in the age of Hegelian idealism and of the Nazarenes—artists who sought to respiritualize painting by taking inspiration from Italian Renaissance art, a process later echoed in art history by Wölfflin and in art by his contemporary Kandinsky.
You may now be asking, "Love triangles and triangular composition: what does one really have to do with the other?" In fact, I continue to ask myself this question: it is one of the central questions of this book. The relationship between abstract mathematical figures and their concrete instantiations was also a pressing question for Renaissance philosophy, and the triangle served philosophers in the Renaissance as the example. But this is more than a difference between abstract and concrete. My investigation at the Alte Pinakothek dealt with a love triangle, a metaphor playing out over a pronounced perspectival geometry, full of meaning and pathos. My other topic, triangular composition, seems now a rather sterile visual device, a trick for art history exams. The fact that the students saw fit to declare the composition triangular but had nothing more to say about it was telling. Geometry in art has been either severed from content or connected to it in the most suspect and overdetermined ways. But geometry in the Renaissance, indeed the triangle in the Renaissance, was connected to stories, myths, myriad practical uses—and yes, emotions, devotions, desires, and passions.
Sixteenth-and seventeenth-century writers like Lomazzo and Félibien discussed the triangle in painting as a technical precept. In the late nineteenth and early twentieth century Heinrich Wölfflin established it as a pedagogical and hermeneutic principle. In 1971 Ernst Gombrich—rather roughly—characterized Wölfflin's precepts thus: "Where you see a Madonna, I see an equilateral triangle, and that is what you ought to see, or attend to." Gombrich caricatures, but the triangle does haunt Wölfflin's discussion of Leonardo and Raphael. Particularly in their iterations of the Christian Holy Family, Wölfflin finds a decided triangular or pyramidal form. He identifies it with the High Renaissance—"classic art"—and with idealism, seeing it as part of a reaction against and refinement of the (admittedly appealing) naturalism of the Quattrocento.
Art historians ever since have considered triangular composition a hallmark of the Renaissance. A typical and rather judicious example is Harald Keller's account: "Beginning with the first stirrings of the Renaissance, articulation of surfaces, especially when achieved by means of human figures, was effected by applying simple formulas derived from everyday geometry. Among those formulas, none was used more extensively than was the triangle." Textbooks tell us that the triangle reflects High Renaissance ideals and interest in geometry and that it provides stability, hierarchy, coherence, or focus.
As a gimmick of the introductory survey lecture, the triangle is simple and identifiable. It is also, significantly, something one can say about visual form, independently of content; without a crutch, even professional art historians can be inarticulate about form. The triangle fits well with the writing exercises we give beginning students—"formal analysis" papers in which we ask them to set aside any ideas they may have about a picture's content and instead to focus on form: visual properties like line, color, and texture. With Renaissance art, it is justly difficult for students to set aside subject matter in describing what they see. To ask them to do so is a deeply, and ideologically, modernist exercise. Modernism, particularly though not only in the tradition of Clement Greenberg's criticism, asks artists (and their viewers) to sacrifice content to form—indeed, to rid art of subject matter entirely. Most Renaissance works are not, however, amenable to an argument that completely disregards content. They were made to communicate subject matter, to make arguments, to impress with their lavish materials, to prompt offerings and meditation.
In this last aspect, Renaissance art possesses a surprising parallel with modernism. Formal analysis may be part of a modernist colonization of earlier periods of European art, but the earlier periods have their sly revenge: this ascetic operation of denying oneself "content" is typical of devotional exercises practiced by medieval and Renaissance believers. One might even say that the focus on form is itself a spiritual practice, as Wölfflin's contemporary Wassily Kandinsky, one of the founders of modern art, could tell us. Kandinsky—like Wölfflin resident in Munich in the 1890s—derived the principle of the triangle from past art in his effort to respiritualize form. (He even used some of the same triangular examples from art history as Wölfflin had, and referred to Wölfflin in his own teaching.) In the pages that follow I will argue that Renaissance philosophers and artists indeed imagined the possibility of progressing, by a process of abstraction from the visual, toward pure mathematical form and finally contemplation of the divine—via the triangle. And yet the triangle was not a simple, pure, celestial form, for it was equally grounded in mundane things. The equilateral triangle based on the ground suggests transit between earth and sky. The human condition as understood in the Renaissance was to be in between: aspiring to spirit yet insistently dragged back to the world.
The compositional triangle begets other triangles. At times art historians have been fond of tracing lines—perspectival and compositional—on images as a tool for interpretation. But once we begin looking for triangles and tracing them on images, they seem so ubiquitous as to become meaningless. How to think about them within the context of art? We might start by jettisoning the phrase "triangular composition" and think instead in terms of shapes and arrangements. (We can allow the phrase back into our discussion where it really makes sense.) While it's often possible to identify shapes and arrangements convincingly, the notion of triangular composition relies on questionable assumptions about the way painters "mapped" images. Did complicated arrangements of invisible lines really underlie images? Existing preparatory drawings don't really support this idea. Many have criticized the idea that paintings possess complicated and secret geometries. 6 The fact that geometries can be traced on a picture does not mean they were part of the work's conception; in fact it might be better to consider these geometries as works of art in and of themselves, inspired by the original works they purport to diagram.
But the plotting of a real or symbolic geometry of the picture plane is not an impossibly strange idea, since astronomers constructed constellations connecting points in the sky, surveyors imagined diagonals when making measurements, and navigators plotted points on maps. Artists did plot the geometries of bodies and buildings, dividing them into shapes. Geometric order was imposed on images to depict conceptual relationships, and invisible lines were made visible in diagrammatic drawings and in paintings that use lines of force to map affinity, dependence, and desire.
Significant geometries, simple or more complex, do manifestly appear in works of art. We can "diagnose" a symbolic triangle formed by the wings of two doves in a Ferrarese manuscript (fig. 1.1)—they form a perfectly equilateral "Alpha" adorned with a musical scroll that forms an "Omega"—or a triangular trinity of rosebushes on a Belgian tapestry (fig. 1.2). As early as the ninth century, a crucifixion mural in Trier was composed with (still discernible) snapping lines that determined a triangular composition with Christ's heart as its apex. Describing the composition, Warren Sanderson notes that "this equilateral triangle was located so precisely at the center of the lunette it must have served more than just practical compositional purposes." The triangular arms and torso of Christ in Leonardo's Last Supper provide a good example of an "unnecessary" triangle: the shape is not required by the subject matter. Is it, then, a satisfying decorative motif that just happens to hint at the Trinity? The Virgin Mary's body produces an equilateral triangle in Bellini's Madonna of the Meadow (plate 3) and in Andrea del Sarto's Madonna of the Sack in the Santissima Annunziata in Florence (fig. 1.3); Garofalo's Saint Jerome in Meditation is similarly triangular (fig. 1.4) to a degree that can hardly be accidental. The outline of a group of figures might follow an unnecessarily triangular shape—as in Raphael's Holy Family or Diana Mantuana's print depicting Latona giving birth to Diana and Apollo on the floating island of Delos (fig. 1.5). Supporting figures in a triad group might lean in to suggest the lines of the triangle strongly with their bodies, as in the Quattrocento paintings we will see in chapter 4. Unnecessary triangles also include the nearly equilateral one formed by the corner of the table in Bronzino's portrait of Ugolino Martelli (fig. 1.6), the bright triangle of light on the musical score in the foreground of Carpaccio's Vision of Saint Augustine (plate 4), or the folded cloth in front of Joseph in a Holy Family by Garofalo (fig 1.7). Perhaps we can see such triangles as "ensigns," following a suggestion by Hubert Damisch: "an iconic element which without performing any syntactical or constructive function, broadcasts, as it were, its operations by miming them" and thus "proclaims and reflects the work of the painting."
A problem arises when in order to reveal a triangle we must select subjective points and plot the shape as if on a graph. How does one choose which points count and which don't? John Canaday argues that Perugino's triptych of the Crucifixion with Saints (fig. 1.8) in the National Gallery in Washington is triangularly composed; Canaday plots points and draws lines between them to make two intersecting triangles. Yet Raphael's Transfiguration (fig. 1.9), he asserts, can't be mapped the same way: "An attempt to diagram the composition would result in an incomprehensible and irrational web of lines stretching arbitrarily from point to point." This happens because he is trying to diagram too much in the composition. Given the angle of the hovering prophets, the Transfiguration can easily be mapped (should one wish to do so) as a blank triangle revealing Christ inside a circle—heavenly shapes mounted atop a roiling earthly cube. The Transfiguration doesn't fit Canaday's definition for two reasons: the triangle is not comprehensive—it applies only to part of the painting—and the painting itself, taken in its entirety, veers away from the High Renaissance purity and simplicity that Canaday associates with triangles.
To establish ways of talking about triangular arrangement, I ask, "Triangular in relation to what?" and "How does one triangle differ from another?" In this book, I work with constellations of related images, learned and popular, that "think." They define, construct, and express philosophical ideas—not only visual ones. Sometimes this thinking can be documented in textual evidence, and sometimes it can't; images must then constitute their own form of evidence.
To assert that the triangle matters in its own right, as opposed to being simply a feature of composition, might seem to promise only limited consequences for how we understand the meaning and significance of images. It lacks the chronological linearity of an artist's or patron's life story, or the material or institutional consistency of an object, the temporal and spatial stability of particular places and periods. As an abstraction, it is neither topic, nor genre, nor exactly technique. The triangle is a kind of emblem of geometry and related philosophical questions, and thus can be used as an interpretive tool for getting at a certain set of issues. It is also a surprisingly "real" topic. It can also help us escape the frames we traditionally use to study Renaissance art—biography, nation, genre, theme. These frames are useful but they are conditioned by modern institutions; the triangle, as a tool, helps us open up questions from a different point of view.
Medieval and Renaissance Triangles
Once one enters the theoretical world of the Renaissance triangle, things get complicated quickly. For philosophers the triangle is a topic, an example, even a primal site of origin; for theologians it is a conscious and overt trinitarian symbol; for perspectivalists it is the basis of vision; for artists it is a convention for organizing a picture plane. For many of those who thought about it, it was a way of mapping relationships. From the early fifteenth century to the mid-sixteenth century, interests in geometry and in ancient philosophers (Plato, Pythagoras, and various followers) converged with new theological ideas, as in the work of the fifteenth-century German philosopher and cardinal Nicholas of Cusa. Medieval diagrams played a role in medieval scientific culture (constructing diagrams was part of training in geometry), though the results were variable in quality. At the same time geometric diagrams played an exegetical role in theology. Michael Evans argues that diagrams were "the way medieval thinkers approached some of the problems most important to them"—tools for conveying theological ideas through the "logic and exactitude of technical illustrations."
Late in the fifteenth century new printing techniques began to reproduce diagrams with a high degree of accuracy. The first printed edition of Euclid was produced in 1482 in Venice by Erhard Ratdolt, who used shaped thin strips of metal (probably what we call printer's rule) to create the hundreds of diagrams the text required. This was a watershed moment for the history of science and of the visual representation of knowledge, and surely had an effect on the symbolic as well as technical uses of geometry over the decades that followed. Interest in mathematics may also have been inspired by the semi-millennium itself; the work of Dürer and Leonardo particularly exemplifies this. Luca Pacioli wrote in his Summa that "Holy Theology cannot reach our intellect well without mathematics." The triangle in particular appeared in early printed diagrams as a tool for understanding the relations between microcosm and macrocosm, human and world. Diagrams schematized knowledge, abstracting linear principles. Alongside illustrations in editions of Euclid and other "strictly" mathematical texts, Renaissance diagrams used triangles to understand and define the "human."
Excerpted from THE PASSIONATE Triangle by Rebecca Zorach Copyright © 2011 by The University of Chicago. Excerpted by permission of University of Chicago Press. All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.
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Table of Contents
List of Illustrations vii
1 The Passionate Triangle 1
2 The Triangles of Perspective 27
3 The Triangle and the Trinity 51
4 Equality, Hierarchy, and the Birth of Triangular Composition 69
5 A Flood of Tears 93
6 The Third Place and the World 129
7 Jealous Geometries 153
Epilogue in the Female Eye 181