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An Euler-type volume identity

Published online by Cambridge University Press:  17 April 2009

Kevin Callahan  and
Kathy Hann
Kevin Callahan
Affiliation:
Department of Mathematics and Computer ScienceCalifornia State UniversityHayward CA 94542, United States of America e-mail: callahan@csuhayward.edu
Kathy Hann
Affiliation:
Department of Mathematics and Computer ScienceCalifornia State UniversityHayward CA 94542, United States of America e-mail: khann@csuhayward.edu
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Abstract

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In this paper we present an elementary proof of a congruence by subtraction relation. In order to prove congruence by subtraction, we produce a dissection relating equal sub-polytopes. An immediate consequence of this relation is an Euler-type volume identity in ℝ3 which appeared in the Unsolved Problems section of the December 1996 MAA Monthly.

This Euler-type volume identity relates the volumes of subsets of a polytope called wedges that correspond to its faces, edges, and vertices. A wedge consists of the inward normal chords of the polytope emanating from a face, vertex, or edge. This identity is stated in the theorem below.

Euler Volume Theorem. For any three dimensional convex polytope P

This identity follows immediately from

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 59 , Issue 3 , June 1999 , pp. 495 - 508
Copyright
Copyright © Australian Mathematical Society 1999

References

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