Conformality in Hawksmoor's ceiling, the Roman Pantheon and Mercator's projection

Abstract

The dome of the Roman Pantheon is coffered with ribs surrounding sunken lacunaria, thus forming a grid. How this is achieved given the curvature of the dome has long been a subject for study and speculation. Although detailed measurements now exist, thus far no single principle has emerged which fixes the overall geometry. Similar coffering occurs in Hawksmoor's design for the ceiling of the Buttery in All Souls College, Oxford. Both these examples are doubly curved, making their tiling with (almost) square coffers problematic. We address this using methods of differential geometry. Observing that the ribs intersect orthogonally, we hypothesize that they are images under a conformal (angle preserving) mapping of a uniform M×N tiling of a planar rectangle with squares. This is unique and is the inverse of Mercator's projection of the ceiling to the rectangle. It minimizes an energy functional which captures the sum of the elastic and gravitational energies in the long wavelength limit. Thus, any construction method which is stable to long wavelength deformations necessarily leads to a conformal coffering. This then gives quantitative predictions with no adjustable parameters for the relative sizes and locations of each coffer, which are in good agreement with photographic data, and consistent with direct measurements of the Pantheon.