The Symmetries of the Baptistery and the Leaning Tower of Pisa

Abstract

The Leaning Tower of Pisa consists of a ground floor with a blind arcade, six tiers of open arches, and a smallish tamburo. The 15-fold and 30-fold symmetry of the Leaning Tower is extremely rare, if not unique. The Baptistery consists of four levels, and counting a garland of arches, there are five elements to be analyzed. The Romanesque ground floor consists of a blind arcade of 20 arches resting on 20 pilasters. The second level displays a striking “symmetry clash”: a 12-fold symmetry. The third level returns to a 20-fold symmetry. The symmetry clashes help reconstruct the history of the Baptistery construction. In 1185 the original plan for the exterior was replaced by that of Guidolotto. His design involved the regular pentagon and pentadecagon, which had appeared for the first time in the construction of the Leaning Tower. Circa 1221, the quarrels ended with the 12-fold symmetry triumphant. A third reversal, circa 1278, revived the icosagonal symmetry.

First published as: David Speiser , “The Symmetries of the Baptistery and the Leaning Tower of Pisa ”, pp. 135–146 in Nexus I: Architecture and Mathematics, ed. Kim Williams, Fucecchio (Florence): Edizioni dell’Erba, 1996. This is a slightly abridged text of an earlier publication: “The Symmetries of the Baptistery and the Leaning Tower of Pisa,” Annali della Scuola Normale Superiore di Pisa, Classe di Lettere e Filosofia Serie III, vol. XXIV, 2–3 (1994): 511–564, with 7 plates.