SB3D (Tour/Viviani)

# The Mathematics of Temple of Viviani

The Temple of Viviani consists of the intersection of a solid sphere and a solid cylinder. To learn about the shape of the Temple, we can observe the intersection of the surfaces of a sphere and a cylinder: the intersection is a very interesting figure-eight curve.

The Temple of Viviani is mentioned in Professor Struik's book on differential geometry. The name was intriguing enough for us to want to know more about the origins of the problem. Looking up the name Viviani in the Brown library database, as well as in the indexes of other differential geometry books, we were able to gain further knowledge about the history of this problem. The Viviani (or also Viviani della Robbia) family was an old and well-established family of Florence. There are books written about their history, autobiographies of different members, and mention of them for several centuries.

The Viviani of the temple is a certain Vincenzo Viviani, who lived in Florence from 16... to 1703. He wrote several mathematical books, and had contacts with Galileo. The John Hay library at Brown has a book by Galileo that belonged to Viviani, and they corresponded. In 1692, Viviani wrote the book for which he is most remembered, the one in which he mentions the temple: Aenigma geometricum de miro opificio testudinis quadrabilis hemisphaericae, auctore D. Pio Lisci Posillo Geometra.

What we now know as the Temple of Viviani was first presented as 'the Florentine problem'. Viviani mentions a Greek temple, made of a half sphere intersecting with a cylinder. The idea of this shape was both appealing and intriguing to many mathematicians of his time, and it was referred to through the ages. In the last century and a half, many German geometers and mathematicians devoted entire books, and sections of books, to the analysis of this shape and especially to the analysis of the parametrization of the intersection line. (As references, there are Gregorius Maettig Kloss' Einige Anwendungen des Florentiner Problems (1856), or Leopold Klein's Streifzuege in das Gebiet der Mathematik und Geometrie, Heft 1, Zur Kreislehre, Ueber das Sogenannte Vivianische Fenster (1915).)

One aspect of the problem remained constant through the ages: the exact shape of the intersection always eluded the mathematicians who were studying it. In spite of Viviani's source of inspiration being a concrete three dimensional shape (a building), his successors tried to use their imagination and interpretation of the mathematical formulas of the intersection alone to represent the shape. In this century, two-dimensional representations of three dimensional objects, in views that were such that all aspects of the object are rendered (as opposed to the strictly vertical or strictly horizontal views that can be found even in the Latin books of the end of the 17th century written about Viviani's problem) started to be used in textbooks as a means of enhancing the students' perception of the problems. However, none of these pictures (even the one in Professor Struik's book) were ever exact: the essential part of the problem, the intersection between the sphere and the cylinder, was always rendered incorrectly.

Now, we have the modern day equivalent to Vincenzo Viviani's Greek temple, a very concrete three dimensional object: computer graphics. So maybe for the first time in over 300 years, you will be able to see the exact shape of the temple of Viviani.

 Surfaces Beyond the Third Dimension Last modified: 08 Oct 2000 08:19:41 Comments to: Thomas F. Banchoff (`thomas_banchoff@brown.edu`)