Challenging Mathematical Insights into Masonry Domes Over the Centuries

Abstract

We focus on masonry domes which are considered architectural landmarks either in different historical periods and in different cultural contexts. From a mathematical point of view, an approximation of a dome is provided by a rotation solid whose cross-section gives the generating curve. Obviously, a frequent generating curve is the semicircumference, but here we want to highlight the role of parabola and catenary used as generating curves to make the structural load lighter. At the present they are well-studied different curves, but until the 17th century, they were considered the same curve, even though they significantly differ from the point of view of structural properties. Actually, catenary is the curve of a hanging chain, which exhibits a tension strength only. When it is “frozen” and inverted it exhibits a compression strength only, which means that it supports itself. Parabola does not exhibit such structural property, but catenary may differ from a convenient parabola very slightly so that building approximation makes a catenary appear as a parabola and this parabola is so close to a catenary that it approximately retains its structural properties, point by point. Here, we investigate the mathematical connection between catenary and parabola in masonry dome structure, referring in particular to Brunelleshi’s dome in Florence, Saint Peter’s dome in Rome and San Gaudenzio’s dome in Novara.