Editing Wallace–Bolyai–Gerwien theorem (section)

== Degree of decomposability ==
Consider two equidecomposable polygons ''P'' and ''Q''. The minimum number ''n'' of pieces required to compose one polygon ''Q'' from another polygon ''P'' is denoted by σ(''P'',''Q'').

Depending on the polygons, it is possible to estimate upper and lower bounds for σ(''P'',''Q''). For instance, [[Alfred Tarski]] proved that if ''P'' is convex and the [[Diameter of a set|diameters]] of ''P'' and ''Q'' are respectively given by d(''P'') and d(''Q''), then<ref name=":0">{{Cite book|title=Alfred Tarski|last1=McFarland|first1=Andrew|last2=McFarland|first2=Joanna|last3=Smith|first3=James T.|date=2014|publisher=Birkhäuser, New York, NY|isbn=9781493914739|pages=77–91|language=en|doi=10.1007/978-1-4939-1474-6_5}}</ref>
:<math>\sigma(P,Q) \ge \frac{d(P)}{d(Q)}.</math>
If ''P<sub>x</sub>'' is a rectangle of sides ''a''{{thin space}}·{{thin space}}''x'' and ''a''{{thin space}}·{{thin space}}(1/''x'') and ''Q'' is a square of side length ''a'', then ''P<sub>x</sub>'' and ''Q'' are equidecomposable for every ''x'' > 0. An upper bound for σ(''P<sub>x</sub>'',''Q'') is given by<ref name=":0" />
:<math>\sigma(P_x,Q) \le 2 + \left\lceil \sqrt{x^2 - 1} \right\rceil, \quad\text{for } x \ge 1.</math>
Since σ(''P<sub>x</sub>'',''Q'') = σ(''P''<sub>(1/''x'')</sub>,''Q''), we also have that
:<math>\sigma\left(P_\frac{1}{x},Q\right) \le 2 + \left\lceil \frac{\sqrt{1-x^2}}{x} \right\rceil, \quad\text{for } x \le 1.</math>