Hawksmoor's Ceiling, Mercator's Projection and the Roman Pantheon

TL;DR

This work proposes that Hawksmoor's coffered ceilings arise from conformal mappings of regular square tilings, i.e., the isothermal coordinates that map planar grids to curved surfaces, with the inverse of Mercator's projection governing the pattern. For surfaces of revolution, explicit vertex coordinates are derived, showing that coffer lines align along meridians and parallels and that coffer sizes scale with the radius function $\rho(z)$; the hemispherical Pantheon and the Buttery half-capsule serve as key test cases. The study provides exact formulas, such as $\phi=\frac{n\pi}{N}$, $z_m$ given by an integral involving $\rho(z)$, and, in concrete examples, $z_m=R\tanh(\pi m/N)$ and $\rho_m=R\operatorname{sech}(\pi m/N)$ for the hemisphere, with camera-distortion-corrected comparisons showing excellent agreement with photographs. It also offers a practical construction protocol and discusses broader geometric implications, suggesting a universal, geometry-driven explanation for coffer patterns that extends to other architectural interiors.

Abstract

The ceiling of the Buttery in All Souls College, Oxford, designed by the English Baroque architect Nicholas Hawksmoor, has a vaulted form on an oval base. It is coffered with an array of approximately square sunken lacunaria, whose sizes and positions vary so as to accommodate the constraints of the curved surface and its boundaries. A similar design appears in the dome of the Roman Pantheon. Using methods of differential geometry, we hypothesise that these cofferings should be the images under conformal mappings of regular square tilings of a rectangle or finite cylinder. This guarantees that the coffer ribs meet exactly at right angles and the coffers are close to being square. These mappings are simply the inverse of Mercator's projection of the curved surface onto a plane. For a ceiling which is a general surface of revolution, we derive formulae for the dimensions and location of each coffer. Our results, taking into account camera distortion, are in excellent agreement with photographs of the Hawksmoor ceiling and the Pantheon dome, as well as with recent direct measurements of the latter. We also describe a protocol by which Hawksmoor's ceiling might have been constructed without advanced mathematics.

Figures (19)

• Figure 1: Comparison of the predictions of the conformal hypothesis reported in this paper, adjusted for camera distortion, with modern photographs of Hawksmoor's Ceiling in All Souls College, Oxford (upper photo), and the Pantheon Dome in Rome (lower). The dots mark the predicted relative positions of the midpoints of the intersections of the coffer ridges. Only the ratio of the camera height to that of the ceiling has been adjusted to obtain the best overall fit in each case. See Secs. (\ref{['secbutt']}, \ref{['secpan2']}) for details.
• Figure 2: Interior of the Buttery looking South. The oval plan of the walls and floor is apparent, as is the domed nature of the ceiling. The window and doorway intrude on this structure. Hawksmoor's coffered design for the ceiling uses approximately square coffers which vary in size and orientation, but whose edges always meet at right angles. The size of the grid necessarily diminishes at the polar ends, where it is interrupted by arched niches. The one visible here holds a small bust of Hawksmoor himself. (Photo All Souls College)
• Figure 3: Simplified horizontal section of the Buttery at the level of the cornice which forms the base of the ceiling. It consists of two half-discs connected by a rectangular central section. The ceiling surface is generated by rotating (say) the left half of the perimeter of the base through 180$^\circ$ about the axis of symmetry. The proportions are true: the Hawksmoor coffering of the ceiling constrains the ratio $R'/R$ to be a rational multiple of $\pi$. In fact it is $2\pi/11\approx 0.57$.
• Figure 4: Inferior view of the ceiling. The image is oriented so that the top of the picture is approximately North. It is distorted both by linear perspective and the camera, so the orthogonality of the crossings of the coffer lines is not apparent except near the apex. Note that the lines show as faint cracks in the paintwork in the lower right of the picture. (Photo All Souls College)
• Figure 5: Hawksmoor's barrel vaulted coffered ceiling in Christ Church at Spitalfields. Since the surface has no intrinsic curvature it may be tiled uniformly with regular hexagons or squares.