SB3D (Tour/Viviani)

This is the first image in a series of three presented at the celebration of the hundredth birthday of Prof. Dirk Struik. On September 23, 1994, at Brown University, Prof. Struik gave his own centenary lecture "Mathematicians I Have Known", and the gallery includes a photograph of him during that event. Each of the three images that were presented to Prof. Struik was developed during the summer of 1994 by teams of students working at Brown.

"The Temple of Viviani" is an enhancement of a standard figure from descriptive geometry and graphical solid modeling, depicting the intersection of a sphere with a circular cylinder of half the radius passing through the center of the sphere and tangent at one point of the equator. This example is found in any multivariable calculus book since it is one of the first interesting cases for which it is possible to evaluate both the volume and the surface area of the intersection. It is also one of the most frequently misdrawn illustrations, since many volumes draw only the top half and either make the bottom point of the curve smooth or cuspidal, whereas the computer diagram clearly indicates a figure eight curve with a transversal crossing. From the point of view of Lagrange multipliers, this position represents the non-transversal intersection of a cylinder with the level set of a distance function to a point (or, dually, the intersection of a sphere with the distance function to a line). This image accentuates the intersection curve by presenting it as a small tube. In the exhibition, there was a mention that the same figure was featured in one of the continuously projected videotapes shown in the alcove of the gallery. The history of the image and its use in calculus was presented in the exhibit booklet, and as a link on the electronic version. The image rendering was done by Julia Steinberger and Neel Madan, while the videotape sequence was designed and executed by Ying Wang.

Surfaces Beyond the Third Dimension Last modified: 08 Oct 2000 08:19:40 Comments to: Thomas F. Banchoff (thomas_banchoff@brown.edu)

The Mathematics of the Temple of Viviani Temple of Viviani