The circular loxodromic lines of the torus – Brunelleschi's constructive principle?

Abstract

The umbrella vault of Pazzi Chapel in Santa Croce church in Florence has been the subject of many, 3d survey based researches, but it’s shape has never been described in satisfactory manner. The descriptions given so far, verified through three-dimensional models, despite whether drew near to geometry, did not give any explanation of building process, that as we know, often guides design choises and principles. In the current research, attempts are made to formulate new hypotheses on the logic and form-finding processes that could have determined the choosen shape of the sail. The existing technical literature and treatises on masonry do not give any information about the building processes in the sources, which is a core issue for explaining the peculiar shape of vaults. The analyses are done by correlating detailed surveys and geometric analyses of the vault, comprising the curves, surfaces and the possible masonry texture with the hypothesized form.

The form of the inner sail is obtained by a three-dimensional transformation of a toric surface, with respect to the fixed point (dome’s oculus) and constant lenght (torus radius), like the Nicomedes concoid in 2d. The vault is made by the inner sail (the concoid surface) and the outer sail that is not visible in this moment (the hypothesized toric surface).

In this research, and especially in the verification steps, we could recognize as very useful certain measures that, at first, seemed to have no meaning. The inner sail of the umbrella vault of the Pazzi Chapel is built on the basis of a regular dodecagon, with sides of about 4.90 Florentine braccia, while the diameter of the circumscribed circle is about 8.67 braccia, the measures that could seem a bit unusual for Brunelleschian architecture practice. Even the width of the ribs has a not expected measure of 0.38 braccia.

When, on the other hand, we go on to describe the genesis of this sail (the inner sail), which is given by a three-dimensional transformation of a toric surface (the outer sail) we suddenly arrive at a hypothetical torus with the major diameter of exact 10 Florentine braccia and the minore one that measures exactly 2 braccia and a half. This form has been verified by overlaying the thus constructed NURBS 3d model to the point cloud MESH model, and the result was very satisfactory (of course for the inner sail because the outer one is not visible at this time).

A further study, of this "hypothetical" torus, and in order to make a purely constructive hypothesis, made us understand how it would also be possible to build by using only chords, because it has some very special features. The ribs of the dome present the Villarceau's circumferences of the toric sail, which are also the rhumb lines of the torus. As we know, a rhumb line is described by a constant step (in the ratio between length and height) on any surface. Translated in the "standard" size of bricks, these curves may have the slope of 1:2 or 1: 4 (the brick put crosswise), while it is particularly complex to achieve different gradients.

In our case, of sail built onto a dodecagon's side, the slope of the circular rhumb line is given by half the internal angle (360°/12/2=15°), which corresponds to the slope of 1: 3.73 ratio. By making the slope of 1: 4 ratio - from the center of the polygon, we reach a point on its side that is located

0.19 braccia of its vertex. That measure corresponds exactly to the size of the detected rib's width of 0.38 braccia (2x0.19).

This particular circular rhumb line, might have given the enormous constructive advantages in phase of realization. In fact it allows (in theory) to determine, all the points in the space that belong to the toric surface, by using only a cord attached in the center of the related circumference and the correct position of bricks. So, that way, the construction place becomes a research site on these particular curves, that produced particular knowledge that might be used later during the construction of the dome of Santa Maria del Fiore.