Perspective in Mathematics and Art

Fig. 30 – Photograph of the Basilica at Paestum. The Basilica
is a Greek Doric temple dating from the sixth century B.C.

Fig. 30 shows a photograph of the Doric columns at Paestum, near Naples, Italy. The row of columns is parallel to the plane of projection. Curiously, the columns exhibit the counter-intuitive property that those further away from the centre of projection appear wider, and not thinner. This apparent paradox can be explained by Fig. 31.

Fig. 31 – Schematic diagram of the Doric columns, with the centre
of projection O. The surface of projection is the line CD.

The angle subtended at the centre of projection O by the width of the columns becomes smaller for columns further away from the centre. For example, q₂ is smaller than q₁. However, the lengths due to the intersection of the limiting light rays of the columns by the projection surface CD become wider for columns further away from O. This is due to the increasing obliquity of the light rays. For instance, ab is longer than cd. Ultimately, this means that columns further away from O will project a larger width onto the projection surface CD.

In practice, most artists omit this weird perspective effect from their paintings, instead giving equal width to all their columns. However, a row of square pillars does not give rise to this problem. As shown in Fig. 32, when projected onto the projection surface EF, the rectangular fronts of the pillars all yield projections of the same shape and size. This is because the projection of a plane figure onto a plane parallel to it always yields a figure that is geometrically similar to the original. The proof for Fig. 32 uses similar triangles, and was discussed under the section on Piero della Francesca (Fig. 20).

Fig. 32 – Schematic diagram of a row of square pillars.
The centre of projection is at O, while the projection surface is EF.

The problem arises for columns because their surfaces are curved, and our eyes find it difficult to distinguish what is the “front” of each column. By contrast, even if we look at a square pillar from an oblique angle, it is easy to tell how much of the “front” we are observing.