The idle ramblings of a Jack of some trades, Master of none

A few weeks ago, Shefaly speculated about the popular view of a dichotomy between art and science. She and others pointed out that there are notable confluences (Maurits Escher being a wonderful case in point). I remembered later that the development of perspective was, to my mind, possibly one of the most apropos examples of the interplay between art and science, with early work by artists resulting in great advancements in geometry, and simultaneous mathematical ideas informing much of the development of Western art.

Perspective, in retrospect, appears to be a fairly intuitive concept. Even kids recognise that an alley of trees drawn on paper will recede to a point on the horizon if it is to have the semblance of depth and realism. While the great artists of the Classical Period in Greece or Rome, or the Ancient Egyptians or Indians or Chinese or Persians did depict distant objects as smaller, they did not adopt the law of regular diminution of objects as they become distant. The idea of a vanishing point where infinite lines met, the fixed framework in which we represent a view, was not adopted by antiquity. It took almost a thousand years for this knowledge to be applied.1

In some of Eastern art, e.g. Persian miniatures, the size of an object is directly proportional to its importance. There is no perception of depth at all. This is not so much the case in Greek art, where foreshortening was applied to good effect, even if not rigorously. It was Brunelleschi (1377 - 1446), he of the Duomo in Florence, who developed the first mathematical principles of perspective. One of the first paintings made according to these rules was by Masaccio (1401-1428) who created a stir with his wall-painting of the Holy Trinity in the Santa Maria Novella church in Florence. As seen to the left here, there is a very real sense of depth, a hole in the church wall. The hitherto popular grandiose style of his predecessors such as Giotto is alluded to here, but the figures are much heavier, solid forms, more realistic. Even the nails are in perspective! Masaccio kept the vanishing point at eye level to the viewer, which he felt enabled the best illusion of reality.

Interestingly, Masaccio's early paintings do not show linear perspective, whereas the last three of his short life all do. It is not known for sure where he learnt the trick, although there is suggestive evidence that Brunelleschi taught him and Donatello the theory. There are no written records, so Brunelleschi must have done so orally.

The earliest known depiction of linear perspective is this: Brunelleschi painted the Baptistery of the Duomo on a panel, on the back of which he made a hole. He made the viewer peep through the hole and placed a mirror on the other side so that the viewer could see the reflection of the painting. He would then remove the mirror to reveal the actual Baptistery, and the viewer would be stunned by the sheer realism of Brunelleschi's image. Brunelleschi's biographer, Antonio Manetti, describes the peephole: He had made a hole in the panel ... which was as small as a lentil on the painting side ... and on the back it opened pyramidally, like a woman's straw hat, to the size of a ducat or a little more.2

So how did Brunelleschi develop the technique? He never documented it, as I mentioned above, but his friend Alberti (1404-1472) did so in a learned treatise named Della Pittura, which was dedicated to him. The method 3 was as follows:

On paper, choose a centric point (the vanishing point where lines meet at infinity), C. (See figure, taken from reference 3.) This (following Masaccio) should be at eye-level to the viewer. Divide the bottom edge AB of the paper into equidistant portions, and draw lines from these points a0 = A, a1, ..., an = B to C. These lines are called orthogonals, as they are meant to represent the dimension perpendicular to the plane of the painting. Now, draw a line through C parallel to AB. Where it meets the right edge of the paper, N, extend it to a point R, such that NR is the distance between the painter and the paper, called the viewing distance. Draw lines from R to the points ai. Where they meet the right edge of the paper, draw lines across the paper parallel to its bottom edge. We now have a grid such that the spacing diminishes as they approach C. The angles are correct representations of lines of sight towards infinity. The painter can now paint the grid into accurately foreshortened tiles.

Of course, what Brunelleschi and his cohorts achieved was only single-point perspective. Still, it was a conceptual leap forward and it remained for a while the preserve of Italian artists. Contemporaries of Donatello and Masaccio in the north such as Jan Van Eyck were still constructing their illusions of depth in the old ad-hoc manner.

The ideas of the Italian school penetrated northern Europe with the publication in 1525 of Albrecht Dürer's Underweysung der Messung mid dem Zyrkel und Rychtscheyd, in which he discussed the geometric aspects of perspective and proposed drawing machines to teach its use. Alberti, though, asked the seminal question that led to the development of projective geometry
If two drawing screens are interposed between the viewer and the object, and the object is projected onto both resulting in two different pictures of the same scene, what properties do the two pictures have in common? 4
A systematic development of the methods to answer this and similar questions was done by Girard Desargues (1591-1661) in his treatise Broullion project d`une atteinte aux événemens des renconteres du cône avec un plan (1693).

Desargues's Theorem describes the properties of two triangles that are in perspective from a point, that is if the lines joining corresponding vertices meet at a point. They are in perspective from a point if and only if they are in perspective from a line, that is if the extended lines of corresponding sides meet at three points that are collinear. The wonderful aspect of projective geometry is that it is full of such dualities, and in fact demonstrates the equivalence of a points or lines based axiomatic system. An elementary proof of the theorem can be found here.

It is fairly intuitive that a circle in perspective would appear to be an ellipse. But how to prove this? Desargues urged the young Blaise Pascal to do so.

Pascal's Hexagon Theorem: Let a hexagon be inscribed in a (nonsingular point-) conic. Then the three points of intersection of pairs of opposite sides are collinear. Conversely, if the opposite sides of a hexagon, (of which no three vertices lie on a straight line) intersect on a straight line, the six vertices lie on a non-singular point-conic.

Check out the elementary proof here.

Unfortunately, Desargues' theory didn't receive much attention in his time. Perhaps this had something to do with the highly technical development. Or perhaps the contemporaneous theory of analytic geometry created by Descartes had become more popular. But these are equivalent systems of equal power. And you can see how when you study the analytic proof of Pascal's Theorem.


1. Ernst Gombrich, The Story of Art. p. 114.
2. Paul Calter, Geometry in Art and Architecture, Dartmouth College.
3. Kevin H. S. Guan, Perspective in Mathematics and Art
4. Andrejs Treiberg, The Geometry of Perspective Drawing on the Computer.


Shefaly said...

Feanor: First, thanks for the hat tip! Secondly, it *does* show that art and science really are not divided up by as clear a red line as many purists will have us believe :-)

Fëanor said...

Shefaly: hope you had a good Easter? Re: your comment, I'm not sure that I have come across any such red line. After all, one finds (e.g.) statisticians claiming that there is an art to determining the most appropriate model for a particular scenario (there is no necessarily black-box approach to determining said model). Contrariwise, installation artists, for instance, need to know quite a bit of science to concoct their various creations, no?

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