ThatsMaths

That’s Maths in the Irish Times this week ( TM024: search for “thatsmaths” ) deals with perspective in art and its mathematical expression as projective geometry.

The study of geometry evolved from measuring plots of land accurately and from the work of builders and carpenters. So Euclidean geometry emerged from the needs of artisans. Another form of geometry, called projective geometry, was inspired by artists who wished to represent things not as they are but as they look.

We all know that a circular coin appears oval in shape when viewed from an angle. The Greeks were aware of such distortions. Nothing remains of their drawings and paintings, less durable than their sculpture. We know, however, from literary references, that they understood the laws of perspective and used them in designing realistic scenery for their
dramatic plays.

The discovery of perspective led to a new form of geometry, called projective geometry. Under the process of projection, distances and angles are distorted. Parallel lines become intersecting lines, just as railway tracks, viewed from a bridge, converge. Certain properties, however, remain unchanged: a point is projected to a point, a line to a line, and a tangent to a tangent. In this new geometry, we concentrate on the properties that remain unaltered, or invariant, under projection.

The Renaissance
Western artists rediscovered perspective during the early Renaissance, using vanishing points to depict depth. Piero della Francesco wrote on the use of perspective, and Filippo Brunelleschi, who designed the magnificent dome of the cathedral in Florence, gave artists the mathematical means of realistically representing three dimensions in painting. The laws of perspective were systematised by Alberti, who dedicated his work “On Painting” to Brunelleschi.

Raphael’s masterpiece, The School of Athens, is of particular interest. This fresco, in the Apostolic Palace in the Vatican, shows all the major classical Greek philosophers and mathematicians, engaging in dialogue or immersed in contemplation. The picture gives a brilliant sense of space through the use of perspective. The setting is an imposing hall with lofty arches, ornate ceilings and mosaic floors, all rendered in proper spatial relationship.

There are some forty people in the picture. Plato and Aristotle stand centre-stage, right at the vanishing point of the architectural backdrop. The main figures at left and right foreground are believed to be Pythagoras and Euclid. The picture, painted around 1510, is a splendid exemplar of the High Renaissance.

Left: Plato and Aristotle. Centre: Pythagoras. Right: Euclid (or Archimedes?).

Abandonment of Perspective
From the time of the impressionists, artists have sought to depict the essence of their subjects, rather than to produce exact likenesses. The cubist painters of the early twentieth century were driven by an urge to paint what they knew to be there rather than what they could directly see. Thus, they developed a wide variety of techniques to represent three-dimensional forms from multiple viewpoints, unconstrained by the laws of perspective. For the past century, the role of perspective in fine art has been greatly diminished.

New Applications

Given the visual character of projective geometry, it is no surprise that it emerged from the interests of artists. Today, it is proving vital in another context, computer visualisation. Graphic artists, developing computer games, use projective geometry to achieve realistic three-dimensional images on a screen. And, in robotics, automatons use it to reconstruct their environment from flat camera images. Once again, mathematics developed in one context is proving invaluable in another.

Sources:
Gaiger, Jason, 2008: Aesthetics and Painting. Continuum. 179pp.

Brunelleschi and the re-discovery of linear perspective: maItaly blog