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

Apostol, Tom M. and Mnatsakanian, Mamikon A. (2013) New Balancing Principles Applied to Circumsolids of Revolution, and to n-Dimensional Spheres, Cylindroids, and Cylindrical Wedges. American Mathematical Monthly, 120 (4). pp. 298-321. ISSN 0002-9890. http://resolver.caltech.edu/CaltechAUTHORS:20130509-103547032

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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.


Item Type:Article
Related URLs:
URLURL TypeDescription
http://dx.doi.org/10.4169/amer.math.monthly.120.04.298 DOIUNSPECIFIED
http://www.jstor.org/stable/10.4169/amer.math.monthly.120.04.298PublisherUNSPECIFIED
Additional Information:© 2013 Mathematical Association of America.
Classification Code:MSC: Primary 51M25
Record Number:CaltechAUTHORS:20130509-103547032
Persistent URL:http://resolver.caltech.edu/CaltechAUTHORS:20130509-103547032
Official Citation:Tom M. Apostol, Mamikon A. Mnatsakanian The American Mathematical Monthly Vol. 120, No. 4 (April 2013), pp. 298-321
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:38381
Collection:CaltechAUTHORS
Deposited By: Tony Diaz
Deposited On:13 May 2013 23:54
Last Modified:13 May 2013 23:54

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