Fig. 8. Reconstruction of the plan of Raphael’s ‘School of Athens’. The reconstruction assumes that the floor tiling (red) is based on a square motif, as shown. Locations of the pillar, the table and the globes are indicated. The archway in the far distance is, omitted in the interests of scale.

Despite the accuracy of the painting’s geometry, it is evident that it was not designed with its final location in mind. As a fresco painted into the plaster of the wall, it must have been intended for this particular location in the central of the three Stanze at the Vatican. However, the reconstruction of Fig. 8 reveals that the center of projection falls at a distance of 9 m away from the painting, whereas as the room is only 8 m in length. The natural distance at which to locate the center of projection would be at eye height at the center of the room, 4 m in front of the painting. (The only way to avoid this implication is to assume that the floor tiles were based on a rectangular motif with a 2:1 width/depth ratio). Instead, Raphael locates the center of projection behind the far wall, a point that is inaccessible both in distance and in height. In this, he follows the majority of the Quattrocento frescoists, who rarely sited their perspective constructions so as to be conveniently viewed from the true center of projection. It was only with the advent of the seventeenth century that the great ceiling artists Pozzo, Veronese and Tiepolo began to really pay attention to the viewer’s location for their masterworks. 

On the other hand, consideration of the alternative suggests that Raphael may have adopted this construction intentionally, because it is the only way to design a view of the scene so that the floor and its inhabitants are visible into the distance. Since the bottom of the frame is above the spectators’ eye height, a literal construction would not allow any of the floor to be visible. The scene would have to be painted with the viewers looking up at the edge of a ‘stage’ and able to view only the philosophers near the front from a low viewpoint. Perhaps Raphael was aiming to evoke a robustness of viewing position to a point where the distortion was hardly noticeable, allowing the deep view into the distance and the natural layout of the buildings arcades. On this interpretation, the shallow angle of the lower floor construction would then be a compromise intended to lead the eye into the scene without the floor seeming too steep. By spreading the oblique projection points more widely, the foreground flooring may be seen as better approximating the level that it would have realistically had.

It is of interest to reconstruct the three-dimensional geometry of the architecture of Plato’s Academy from the information given in the painting, something that has never been previously published. An outline attempt at this reconstruction is developed in Fig. 9 (in orthographic perspective). Most of the features of Raphael’s design may readily be captured, but the interwoven hexagonal and diamond-shaped coffers in the barrel vault proved too daunting for this task. Presumably Raphael used a perspective base grid and a beam compass in the plaster to lay out the geometry of this pattern, but it would seem to challenge the geometry of his time even to define the hexagonal geometry, let alone to project it in perspective on a curved vault. The skeptical reader may wish to spend a little time developing this construction with the tools and understanding of 21st century mathematics in order to appreciate Raphael’s achievement. Perhaps it was this that Bramante is depicted as laboring so hard on analyzing at the lower right of the scene!

		Fig. 9. Three-dimensional (orthographic) reconstruction of the architecture intended for the School of Athens.

The School of Athens has historically been of interest not only for its linear perspective but for its treatment of the projection of spheres removed from the central viewing axis. The projection of spheres is a problem that has been historically contentious since the time of Leonardo (Gill, 1975; Pedoe, 1976; Kemp, 1990). Pirenne (1970), for example, felt the need to resort to pinhole photography to prove that spheres project as ellipses away from the line of sight. The geometry for which the elliptical result is valid is the classic perspective point projection through a plane (the geometry implemented by a pinhole camera with a flat film plane). Pirenne shows that these elliptical distortions produce the correction depth impression when viewed through a pinhole located at the center of projection, although he argues that the human eye does not correct for this distortion in free viewing. One reason for the historical confusion is that spheres always project as circles on the human retina, without elliptical distortion. It is only in the plane of the picture that the circle is elongated to an ellipse. Fig. 11 makes clear the relation between the circular cone of projection at the eye and its elliptical intersection with a picture plane at some angle to that cone.

The regularity and classicism of Raphael’s architectural design impels one to look for sources of inspiration for this imposing architectural composition. One of the dominant humanists of the age was the Leon Battista Alberti, who had even written a book on the rational approach to bringing up a family. It was not long since Alberti had completed his final architectural work, the Church of Sant’Andrea in Mantua (Fig. 10). While certainly not identical, it has many elements in common with Raphael’s design, including solid Romanesque style, a central barrel-vault, symmetrical alcoves, a circular dome in the center with vertical windows and full-length steps in the front. It seems quite plausible that Raphael would have seen this striking design and found it suitable for his purposes.

Fig. 10. Alberti’s Church of Sant’Andrea in Mantua, which could have provided part of the inspiration for Raphael’s architectural design..

The School of Athens has historically been of interest not only for its linear perspective but for its treatment of the projection of spheres removed from the central viewing axis. The projection of spheres is a problem that has been historically contentious since the time of Leonardo (Gill, 1975; Pedoe, 1976; Kemp, 1990). Pirenne (1970), for example, felt the need to resort to pinhole photography to prove that spheres project as ellipses away from the line of sight. The geometry for which the elliptical result is valid is the classic perspective point projection through a plane (the geometry implemented by a pinhole camera with a flat film plane). Pirenne shows that these elliptical distortions produce the correction depth impression when viewed through a pinhole located at the center of projection, although he argues that the human eye does not correct for this distortion in free viewing. One reason for the historical confusion is that spheres always project as circles on the human retina, without elliptical distortion. It is only in the plane of the picture that the circle is elongated to an ellipse. Fig. 11 makes clear the relation between the circular cone of projection at the eye and its elliptical intersection with a picture plane at some angle to that cone.