Is it possible to dissect a square into finitely many pieces, which can then be rearranged to form a disc of equal area?

Note that, unlike in three dimensions where the Banach-Tarski paradox exists, dissections must respect the two-dimensional Banach measure and therefore preserve areas. Hence, it is *impossible* to rearrange the pieces to form a disc of larger or smaller area than the original square.

Obviously, if the pieces are sufficiently well-behaved (e.g. have simple piecewise-smooth boundaries), then there is no solution; the proof is extremely elementary. In 1989, Laczkovich found an axiom-of-choice-based dissection into approximately 10^50 pieces, answering Tarski’s question in the affirmative.

Anyway, my friend and colleague Tim Hutton, inspired by discussion with his 6-year-old daughter, decided to ask how close you can manage with *n* non-pathological pieces, for varying values of *n*. More precisely, he asked the following question:

What is the supremum area *a(n)* of *n* interiors of topological discs with piecewise smooth boundary, such that they can be packed in both a circle of unit area and a square of unit area?

It is trivial to observe that as *n* tends to infinity, *a(n)* monotonically approaches 1. Also trivial is establishing the value of *a*(1):

He’s also launched a collaborative project to find dissections giving lower bounds for *a(n)*. For instance, with six pieces it is possible to achieve 0.9860:

Undoubtedly, more can be found on Tim Hutton’s Google+ page.

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