Tarski's Adaptation of Wojtowicz's Argument on Optimal Dissection of a Unit Square

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This Demonstration shows a reconstruction of a theorem of Tarski. An optimal dissection uses the smallest number of pieces. The theorem states that the number of pieces in an optimal dissection of a unit square into a rectangle of dimensions and has an upper bound , where denotes the ceiling of , that is, the smallest integer greater than or equal to .


In the case , this gives a six-piece dissection.


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



Before World War II, the famous logician Alfred Tarski worked as a high school teacher and as assistant to Jan Łukasiewicz. His contributions to elementary mathematics are described in [1].

Polygons that have the same area are called equivalent. In the article "On the Degree of Equivalence of Polygons," Tarski defines the degree of equivalence of two equivalent polygons and as the smallest natural number for which there exists an -piece dissection of to . The function is denoted as .

If is a square of side and is a rectangle of sides and , Tarski introduces and states a theorem about the function .

Tarski did not prove the theorem, but he gave some hints with reference to Wojtowich's book for high schools. The theorem of this Demonstration has been proved using Mathematica graphics [1, pp. 128, 129].


[1] A. McFarland, J. McFarland and J. T. Smith, eds., Alfred Tarski: Early Work in Poland: Geometry and Teaching, New York: Birkhäuser, 2014.

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