New Balancing Principles Applied to Circumsolids of Revolution, and to n-Dimensional Spheres, Cylindroids, and Cylindrical Wedges

Abstract

Archimedes’ mechanical balancing methods led him to stunning discoveries concerning the volume of a sphere, and of a cylindrical wedge. This paper introduces new balancing principles (different from those of Archimedes) including a balance-revolution principle and double equilibrium, that go much further. They yield a host of surprising relations involving both volumes and surface areas of circumsolids of revolution, as well as higher-dimensional spheres, cylindroids, spherical wedges, and cylindrical wedges. The concept of cylindroid, introduced here, is crucial for extending to higher dimensions Archimedes’ classical relations on the sphere and cylinder. We also provide remarkable new results for centroids of hemispheres in n-space. Throughout the paper, we adhere to Archimedes’ style of reducing properties of complicated objects to those of simpler objects.

Additional information

Notes on contributors

Tom M. Apostol

TOM M. APOSTOL joined the Caltech mathematics faculty in 1950 and became professor emeritus in 1992. He is director of Project MATHEMATICS! (http://www.projectmathematics.com), an award-winning series of videos he initiated in 1987. His long career in mathematics is described in the September 1997 issue of The College Mathematics Journal. He is currently working with colleague Mamikon Mnatsakanian to produce materials demonstrating Mamikon's innovative and exciting approach to mathematics.

Mamikon A. Mnatsakanian

MAMIKON A. MNATSAKANIAN received a Ph.D. in physics in 1969 from Yerevan University, where he became professor of astrophysics. As an undergraduate, he began developing innovative geometric methods for solving many calculus problems by a dynamic and visual approach that makes no use of formulas. He is currently working with Tom Apostol under the auspices of Project MATHEMATICS! to present his methods in a multimedia format.