Art and Linear Perspective

Paul Staiti
Art Department
 Giuliana Davidoff
Mathematics Department

Index| Paintings | Slide List

Art and Linear Perspective was orginally taught as a two-week unit in Art 100, Image and Environment, a team-taught introduction to the history of art. It occurred in November of 1997 at a point in the course after units had been taught on the ancient monument, on the medieval church, and on Islamic art. It was positioned toward the beginning of a fourth unit on European art of the sixteenth and seventeenth centuries. That is, at the point at which pictorial illusion, based on geometric theorems, was being introduced into Western art.

A series of lectures was given. First, illusionistic visual effects were explained, in particular the idea that lines that are parallel in reality appear to converge in illusionistic pictures (see attached slide list). Works by Raphael, the Master of the Barberini Panel, Masaccio, Paolo Uccello, and Piero della Francesca were used to demonstrate the point, which had originally been proposed by the fifteenth-century architect Brunelleschi and theorized by the architect Alberti in Della Pittura. Plates from Albrecht Durer's treatise on painting (1525) illustrated how artists used actual mechanical devices to replicate three dimensions onto two.

Second, actual case studies were examined at length. For example, Perugino's Christ Delivering the Keys to Peter (1480) uses a rigid perspectival grid, but hung fifteen feet off the ground, it is not in a position to be seen illusionistically because every picture in one-point perspective has one and only one observation point. Guercino's Aurora (1621) was then studied. A ceiling painting in Rome, it collapses as an illusion if the viewer strays even a foot from the intended viewing position on the floor. One lesson here is that the dramaturgy of perspective pictures, which at their best involve the viewer in the "scene" that is seemingly taking place on the other side of the pictorial threshold, is fragile and can be spoiled whenever a viewer veers away from the intended observation point.

The idea that there is only one observation point was put to the test by the students themselves, who got to sit behind a framed sheet of gridded plexiglass. In front of the plexi was a grouping of boxes. Students had to outline the boxes from a fixed observation point. When they were done they had to look at those outlined forms from some observation point other than the original observation point. The outlined boxes of course looked like just so many lines. But as soon as they returned to the original observation point the lines coalesced back into the illusion of three-dimensional boxes. Other experiments were run with the plexi box, which is a modern version of the kinds of devices that Leonardo da Vinci and Durer used in the sixteenth century. For example, students studied how perspective lines are altered when the same object is drawn at two very different viewing distances.

The third event in the unit was devoted entirely to the actual geometric proof of the theorem that lines that would be parallel in reality appear to converge to a vanishing point in a picture was conducted in class. This follows the proof of the mathematician Philip Zeeman. This occupied the entire class, which had no prior study of geometry or of proofs, except some from high school.

Fourth, was a class on the power of perspective, in particular perspective that has been toyed with by artists. For example, Henri Matisse's Harmony in Red (1909) extinguishes all the perspectival lines of a table, so that the table appears to be part of the wall pattern behind it. In Rene Magritte's Promenades of Euclid (1955) a picture within the picture shows a scene that is exactly contiguous with the "actual" scene shown through a window behind it. A tower rising to a point and a street in perspective are next to each other. Each seems, however, like the other, as it is visually possible to read each as a receding object, or as a tower. Giorgio di Chirico's Melancholy and Myster of a Street (1914) achieves its sense of deranged reality in part because two arcaded building have two different vanishing point that not only terminate on different sides of the picture, but also on different planes (i.e. there are two observation points, but only one viewer). William Hogath's False Perspective (1754) actually makes a joke of it all, as clapboard on one building recede to a different vanishing point from another building, or, a man is seen standing on brick pavement that has parallel line that expand as they move into space, creating a topsy-turvy world.

In this last unit we also cautioned about using linear perspective as a criterion for the "correctness" of a picture. It would be historically and culturally distortive to claim, for example, that Ambrogio Lorenzetti's Allegory of Good Government (1339) or that the Mogul painting of Majnun Evesdropping on Layla (1560) are in any way inferior to the work of Guercino or Raphael. Students are reminded that illusion was not always the desired effect, and that perspectively-intensive pictures have their own "shortcomings," such as the fixed observation point and the constant threat that the illusion will crumble, defeating the intended purpose.

In reflection, the unit on perspective successfully opened to students the historic convergence of art and mathematics; it introduced them to geometric proofs; and it gave them a way of understanding that pictures are made, that if one has the mental ability to "occupy" them, it is the result of a number of technologies, including linear perspective. Some students lost interest in the geometric proofs. They were willing simply to take our word that the theorem was provable. We insisted nonetheless, but tried to find ways to make it less of a visiting math class and more integrated with the rest of the unit. When the unit was repeated in October-November of 1998, there was the addition of a visiting scholar, Professor Marc Frantz from the Department of Mathematics of Indiana University. Professor Frantz, a former art student, has developed over the years some pedagogical methods that draw students quickly into the mathematics of perspective, to "prove" it in conjunction with the collection of the Mount Holyoke College Art Museum, where classes were held. The website for Professor Frantz's course at Indiana University is: www.math.iupui.edu/m290/

 

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Copyright © 1999 Mount Holyoke College. This page created by Math Across the Curriculum and maintained by Jennifer Adams. Last modified on August 8, 1999.