The word “geometry” means earth measurement and the subject evolved from measuring plots of land accurately and also from the work of builders and carpenters. So the geometry that we call Euclidean emerged from the needs of artisans. Another form of geometry – projective geometry – was inspired by artists wishing 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.
Western artists rediscovered perspective during the early Renaissance. Piero della Francesco wrote on the use of vanishing points to depict depth 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 Leon Battista 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 talking 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 ceiling and mosaic floor, all rendered in proper spatial relationship. There are some 40 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 assumed to be Pythagoras and Euclid. The picture dates from about 1510.
The discovery of perspective led to a new form of geometry, called projective geometry. Under projection, distances and angles are distorted. Parallel lines become intersecting lines, just as railway tracks viewed from a bridge converge towards the horizon. Certain properties, however, remain unchanged: a point is projected to a point, a line to a line. In this new geometry, we concentrate on the properties that remain unaltered, or invariant, under projection.
Since the Impressionists, artists have sought to depict the essence of subjects, rather than exact likenesses. Cubist painters of the early 20th century sought to paint what they knew to be there rather than what they could see. They developed 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.
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. Not for the first time, mathematics developed in one context is proving invaluable in another.
Peter Lynch is professor of meteorology at UCD. He blogs at thatsmaths.com