Some reflections on the non-constructible polygon in Santa Maria Novella, Florence

Abstract

The geometric process followed by Leon Battista Alberti when designing the façade of the Santa Maria Novella church in Florence is well-known. This façade contains 48 ornamental elements which were created through the construction of regular polygons. Specifically: 7 elements have a pentagonal base, 3 elements have an hexagonal base, 36 elements have an octagonal base, and 2 elements have an icosikaihexagonal base (26 sides). In our view, it is interesting that Alberti, having designed all ornaments on the façade on the basis of regular polygons which can be constructed using a straightedge and a compass only, decided to top the lateral scrolls with a circular design arising from a 26-sided regular polygon, since this regular polygon cannot be constructed using only a compass and a straightedge. Therefore, in this paper we use a mathematical approach to theoretically compare several approximate methods for constructing an icosikaihexagon using a compass and a straightedge, in order to ascertain which of these methods best suits the point pattern of this special Renaissance ornament.