Feature Column from the AMS

Mathematics and Art

2. Mathematical tools for artists

Art is born of the attempt by humans to express themselves about the experience of life. Art can take the form of writing, painting, and sculpture, as well as a variety of other forms of expression. There is art in the combining of function and aesthetics in such everyday objects as plates, cutlery, and lamps. Mathematicians have been able to assist artists by creating "tools" of various kinds for them. Such tools sometimes consist of theorems which show the limitations of what artists can do. One can not attempt to represent more than 5 regular convex polyhedra in Euclidean 3-dimensional space because mathematics shows there are only 5 such regular solids. Dodecahedra can be used to put one month of a calendar on each face but the number of faces of a regular convex polyhedron in Euclidean 3-space is 4, 6, 8, 12, or 20. There can be no others.

A much more important issue is the realism with which artists can draw on a flat piece of paper what they perceive when they look out at their 3-dimensional world. If one looks at attempts at scene representation in Egyptian and Mesopotamian art, one sees that phenomena that are associated with the human vision system are not always respected. We are all familiar with the fact that objects that are far away from us appear smaller than they actually are and that lines which are parallel appear to converge in the distance. These features, which are a standard part of the way that 3-dimensional objects are now usually represented on a planar surface, were not fully understood before the Renaissance. It is common to refer to artists as using "perspective" (or "linear perspective") to increase the realism of their representations. The issues and ideas involved in understanding perspective are quite subtle and evolved over a long time.

The interaction between scholars and practitioners with regard to ideas about perspective parallels the interactions between theory and application that goes on in all the arenas where mathematical ideas are put to work. An artist may want to solve a problem better than he or she did in the past and will not always be concerned with the niceties of proving that the technique used always works or has the properties that the artist wants. An analogy for a more modern situation is that if the current system used to route email packets takes on average 7.2 units of time and one discovers a way of doing the routing in 6.5 units of time on average, one may not worry that one can prove that the very best system would do the job in 6.487 units of time.

Questions about perspective are very much in the spirit of mathematical modeling questions, since in the usual approach one is concerned with the issue of the perception of, say, a scene in 3-dimensional space on a flat canvas under the assumption that the scene is being viewed by a "single point eye." Yet, we all know that humans are endowed with binocular vision! We are attacking such binocular vision questions today, because we have the mathematical tools to take such questions on, while the artist/mathematicians of the past had to content themselves with simpler approaches.

A variety of people whose names are known to mathematicians (but perhaps not to the general public) have contributed to a theory of perspective. Though every calculus student knows the name of Brook Taylor (1685-1731)

for his work on power series, how many mathematicians know that Taylor wrote on the theory of linear perspective? On the other hand every art historian will recognize the name of Piero della Francesca (c. 1412-1492), yet how many of these art historians (or mathematicians) will be familiar with his contribution to mathematics? Similarly, Girard Desargues (1591-1661) is a well known name to geometers for his work on projective geometry (in plane projective geometry there are no parallel lines), but few people involved with art are familiar with his work. The diagram below (a portion of a "Desargues Configuration"), familiar to students of projective geometry, can be thought of as a plane drawing of an "eye" point viewing triangles which lie in different planes.

Another contributor to the theory of perspective was Johann Lambert, also known for having produced results which would follow from assuming that Euclid's Fifth Postulate does not hold. Though perspective is a well-mined area, this does not stop the flow of continued thoughts on the subject. For those accustomed to work on one-point or two-point perspective, there is the monograph of D. Termes who treats one- through six-point perspective!

Related to the tool of linear perspective is the branch of geometry known as descriptive geometry. While descriptive geometry was widely taught in the 19th century, especially in schools of engineering, the subject is not widely known today. The reason in part is that computer software makes it possible for people not familiar with descriptive geometry to perform tasks which make explicit knowledge of it increasingly obsolete. Descriptive geometry provides a set of procedures for representing 3-dimensional objects in two dimensions. The 2-dimensional representation might be on either a piece of paper or a computer screen. These techniques are of great importance to engineers, architects, and designers. The design of, say, a large aircraft may involve tens of thousands of drawings. The roots of the subject lie with people such as Albrecht Dürer (1471-1528) and Gaspard Monge (1746-1818). If an artist, creative designer, sculptor, or architect can not get across his/her conception of how to manufacture or otherwise assemble a "creative" design, then the work involved might go unrealized. Descriptive geometry supports both constructive manufacture and creative design by giving procedures for showing how to represent proposed 3-dimensional creations on a flat surface.

To give some of the flavor of the issues involved, the diagram below shows in blue how parallel lines "project a triangle onto a line," while the red lines show how the same triangle is "projected" from the "eye" onto the same line:

Parallel projection and projection of a triangle onto a line

A', B', and C' show where the vertices are moved by "parallel" projection and A'', B'', and C'' show where the vertices are moved by "conical" projection.

Here is a rather cute result which grew out of this interaction between mathematics and the needs of artists to represent three dimensions on a flat plane. The result is known as Pohlke's Theorem. Karl Wilhelm Pohlke (1810-1876) was a German painter and teacher of descriptive geometry at an art school. He formulated this result in 1853, though the first proof seems to be due to K.H.A. Schwarz (1843-1921) in 1864. The theorem is quoted in various levels of generality. Here is one version:

Theorem (Pohlke): Given three segments (no two collinear) of specified length (not necessarily the same) which meet at a point in a plane, there are three equal length line segments which meet at right angles at a point in 3-dimensional space such that a parallel projection of these segments maps them onto the three chosen line segments in the plane.

Intuitively, this means that if one wants to draw a cubical box in the plane, one can draw any triad of lines for a corner of the cube because there is some position of a cube in 3-space which maps to the given triad. Thus, in the diagram below the triad on the left can be completed to form a "cube," and there is some set of three orthogonal segments in 3-space which can be mapped using parallel projection onto the triad on the left.

Sometimes Pohlke's Theorem is referred to as the Fundamental Theorem of Axonometry. Axonometry, like descriptive geometry, deals with the problem of drawing 3-dimensional objects in the plane. Here is a recent item coauthored by Roger Penrose in this spirit.