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Geometric Figuring

Photo (top) by Romulo Fialdini

Throughout the history of Western art, mathematics has enjoyed a schizophrenic reception: revered at times as a model of the “rational” and the “true,” rejected in other eras as a paradigm of bloodless abstraction. In LACMA’s sparklingly pretty show on midcentury minimalism, “Beyond Geometry: Experiments in Form 1940s–1970s,” we find the two sentiments commingling, sometimes in the service of the same artistic ends.

Representing the rationalist camp is the Swiss artist Max Bill, architect of one of the show’s most striking and obviously mathematically inspired works, Tripartite Unity, a flowing steel sculpture of a morphing Möbius strip. During the 1940s and 1950s, Bill advocated “a new form of art” in which “geometry [was] the means of determining the mutual relationship of its component parts.” Bill’s command that art be based in “reason” was taken up by artists like François Morellet, whose wondrous mathematico-mechanical sculpture, Four Self-Distorting Grids, is another of the show’s highlights: Half a dozen large metal lattices mounted on poles open and close like huge engineered flowers.

Bill’s approach was also to have an enormous influence on the development of minimalist art in Central Europe and South America, and one of the great joys of this show is the chance to see modern art that originated in places other than North America and Western Europe. Brazilian artists such as Jesús Rafael Soto, Lygia Clark and Hélio Oiticica, inspired by Bill, created works at once rigorous and playful — a magical mathematicalism.

In Soto’s Kinetic Structure With Geometric Elements, a Plexiglas canvas painted with a grid of lines is observed through a glass sheet painted with a differing grid, creating the effect of what mathematicians call moiré patterns. As you walk past, observing from different positions, the lines appear to beat and pulse against one another, generating the optical illusion of movement within the space of the work.

But the mathematical spirit that hovers indelibly over this exhibition was also emphatically rejected by other artists represented here, particularly the American minimalists. In her catalog essay, curator Lynn Zelevansky notes that Bill’s approach was disdained by Frank Stella, Donald Judd and Robert Morris, artists who reacted against what they saw as a lifeless and quintessentially European form of abstraction. Despite the fact that the work of all three draws heavily on forms the rest of us would think of as “geometric” — squares and boxes, for example — these artists rejected that label. In the terms of French philosopher Merleau-Ponty, geometry represented to them an “ideal” rather than a “presence.” Or, as Zelevansky puts it, something “disconnected from the physical immediacy and temporality [they] sought.”

Stella, Judd and Morris specifically aimed to create works that blurred the boundary between object and viewer, a goal they pursued through sculptures that they set directly on the floor, or made to protrude from the wall, or hang from the ceiling. Believing that the beholder was not just a pair of eyes and a mind, but a perceiving and embodied organism, they insisted that “the viewer interact with the work as a body in space.”

For these American artists, the geometry Bill embraced represented the antithesis of physical engagement and embodied presence. Paradoxically, it was precisely those things that the Renaissance pioneers of perspectival painting sought to achieve by using geometry as the basis of their art. From the 14th through 16th centuries, the artists exploring geometrical techniques for representing three-dimensional space on a two-dimensional surface were also playing with ideas about embodiment. Like Stella et al., the great Renaissance masters also wanted to break down the barrier between the art object and the viewer to give their audiences an experience of being physically present within the space of the work.


This new motivation is evident in what is generally regarded as the first great triumph of perspective style, the Arena Chapel in Padua. Here, Giotto painted an epic cycle of images of the life of Christ, each event depicted as if it is a little diorama or theater set. In contrast to the early medieval imagery, in which objects vary in size according to their spiritual status — with angels being larger than men and Christ being the largest figure of all — in the Arena Chapel cycle, all figures appear at the same scale. Instead of the iconic gold backgrounds of the early Middle Ages, Giotto painted landscapes with solid-looking mountains and botanically observed trees. True, the whole effect is somewhat naive to modern eyes, but there is no denying that this is an attempt to depict three-dimensional space as a geometrically coherent whole.

In his book The Heritage of Giotto’s Geometry, art historian Samuel Edgerton argues that through the application of mathematical techniques, Giotto and his followers wanted to create images so psychologically powerful they would make viewers feel as if they were looking at the actual events depicted. In effect, Edgerton suggests, perspectival imagery might be seen as a kind of medieval virtual reality. (Today’s video games also draw on the same techniques, simulating three-dimensional scenes using geometrical principles.)

Edgerton cites a treatise written by the Franciscan monk Roger Bacon in 1267, just a decade before work began on the Arena Chapel, in which the mathematically inclined friar urged Pope Clement IV to encourage painters to adopt the new style, which he called “geometric figuring.” Through this technique, Bacon believed, the events of Scripture could be literally rendered real for viewers, who would now be able to witness for themselves the life of Christ and the saints. Bacon’s goal was nothing less than a reinvigoration of Christianity; such images, he said, could serve to convert heathens and would inspire believers to launch a new crusade to the Holy Land.

The new crusade never happened, but the effect of “geometric figuring” on European psychology was electrifying. Edgerton suggests, rightly I believe, that perspectival representation paved the way for the re-imagining of space that was so critical a foundation for the scientific revolution. During the three centuries following Giotto, artists like Piero della Francesca, Alberti, Dürer and Raphael seriously explored geometrical techniques, in the process retraining European minds to see the world around them through a Euclidian lens. During the 15th century, artists were among the leading mathematicians of the day, and Piero della Francesca eventually gave up painting to devote himself full time to geometrical research.


One of the great innovations of perspectival imagery was precisely that it recognized the onlooker as a body present in physical space — a turn that would be repeated by the American minimalists half a millennium later. Unlike early medieval images, which could be viewed equally from anywhere, a perspective image encodes within it the location for the viewer, the so-called “point of origin.” Initially, the point of origin was directly in front of the image, so the onlooker perceived the scene “as if through an open window,” to use Alberti’s famous phrase.

But by the height of the quattrocento, painters well understood that perspective was an illusion, and they used this illusive power to manipulate audiences both physically and mentally, producing images in which the point of origin was located in bizarre places. For example, Andrea Mantegna’s Saint James Led to Execution, where it is below the image, and Leonardo’s The Last Supper, in which it is 15 feet in the air. In The Psychology of Perspective and Renaissance Art, Michael Kubovy argues that in such works Mantegna and Leonardo aimed to give the viewer an experience of being transported out of the body. Such an effect was possible only because the viewer’s perception of themselves in space could be manipulated by the work.

Like Jesús Rafael Soto and the American minimalists, perspective painters used the relationship between viewer and object to create works in which the observer became part of the whole — not just a pair of eyes and a mind, but a perceiving and embodied presence.