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Complete Dissections: Converting Regions and Their Boundaries

Apostol, Tom M. and Mnatsakanian, Mamikon A. (2011) Complete Dissections: Converting Regions and Their Boundaries. American Mathematical Monthly, 118 (9). pp. 789-798. ISSN 0002-9890.

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Classical dissections convert any planar polygonal region onto any other polygonal region having the same area. If two convex polygonal regions are isoparametric, that is, have equal areas and equal perimeters, our main result states that there is always a dissection, called a complete dissection, that converts not only the regions but also their boundaries onto one another. The proof is constructive and provides a general method for complete dissection using frames of constant width. This leads to a new object of study: isoparametric polygonal frames, for which we show that a complete dissection of one convex polygonal frame onto any other always exists. We also show that every complete dissection can be done without flipping any of the pieces.

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Additional Information:© 2011 Mathematical Association of America. The authors wish to thank the referees for valuable suggestions that improved this paper. They also wish to thank Dan Velleman for suggesting a way to considerably shorten our proof of Theorem 3, and for suggesting the simple example of isoparametric frames in Figure 13.
Issue or Number:9
Record Number:CaltechAUTHORS:20111111-153809155
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Official Citation:Complete Dissections: Converting Regions and Their Boundaries Tom M. Apostol and Mamikon A. Mnatsakanian The American Mathematical Monthly Vol. 118, No. 9 (November 2011), pp. 789-798
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:27759
Deposited By: Tony Diaz
Deposited On:18 Jan 2012 22:59
Last Modified:03 Oct 2019 03:26

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