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The Double Subdivision-Network Method

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Two polygons and are equivalent () if they have the same area. In the article "On the Degree of Equivalence of Polygons", Tarski defines the degree of equivalence of two equivalent polygons and as the least integer for which there exists an -piece dissection of to .

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He then gives a few elementary properties of this concept. For example, if and , then .

This Demonstration shows an example where the limiting equality holds. A unit square and a certain triangle are both equivalent by dissection to a parallelogram . Since and , there is four-piece dissection of to —that is, .

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Contributed by: Izidor Hafner (May 2017)
Open content licensed under CC BY-NC-SA

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The method used in the proof of the theorem that two polygons equivalent to a third are equivalent to each other is called the double subdivision-network.

Reference

[1] A. McFarland, J. McFarland and J. T. Smith, eds., Alfred Tarski: Early Work in Poland—Geometry and Teaching, Basel: Birkhäuser, 2014 pp. 137–138.

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