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A Congruence Problem for Polyhedra

by Alexander Borisov (University of Pittsburgh), Mark Dickinson (Enthought Inc.), and Stuart Hastings (University of Pittsburgh)

Award: Lester R. Ford

Year of Award: 2011

Publication Information: American Mathematical Monthly, vol. 117, March, 2010, pp. 232-249.

Summary

The authors begin this excursion in combinatorial geometry by posing a simple, natural question: how many simple measurements (of lengths and angles) suffice to determine the congruence class of a convex polyhedron? It is reasonably clear, for example, that the six edge-lengths of a tetrahedron suffice, but are six measurements necessary? It is also reasonably clear that though the twelve edge-lengths of a cube will not suffice (think rhombuses), is there another set of twelve measurements that would work?

The authors prove that the number of edges of the polyhedron is always a sufficient number of measurements to determine a local congruence. This number is not always necessary; for example, there exists a set of nine measurements that suffice for a cube. This main theorem is not new, though the authors' proof of it is. That proof uses only elementary techniques, mostly multivariable calculus and linear algebra. An especially nice feature of the proof is that it provides an effective algorithm for actually constructing a sufficient set of measurements for a given polyhedron (an implementation of the algorithm is available online).

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About the Authors (From the MathFest 2011 Prizes and Awards Booklet)

Alexander Borisov received an M.S. in mathematics from Moscow State University and a Ph.D. from Penn State University. He is currently an assistant professor at the University of Pittsburgh. He has authored and co-authored papers on a wide range of topics in algebraic geometry, number theory, and related areas of pure mathematics.

Mark Dickinson received his Ph.D. from Harvard University in 2000. He has held teaching and research positions at the University of Michigan, the University of Pittsburgh, and the National University of Ireland, Galway. He is a keen programmer, and is one of the core developers of the popular 'Python' programming language. He currently works as a scientific software developer for Enthought, Inc.

Stuart Hastings has worked almost entirely in the area of ordinary differential equations (which these days one has to mislabel as “systems” to sound respectable). A student at MIT of Norman Levinson (a previous winner of this award), he had positions at Case Western Reserve and SUNY, Buffalo before coming to the University of Pittsburgh over twenty years ago. Now retired, he continues to teach part time and is completing a book on classical methods in ordinary differential equations with J. B. McLeod.

Subject classification(s): Discrete Mathematics | Combinatorics | Geometry and Topology
Publication Date: 
Monday, August 22, 2011

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