### Laczkovich Squares the Circle

The phrase squaring the circle’ generally denotes an impossible task. The original problem was one of three unsolved challenges in Greek geometry, along with trisecting an angle and duplicating a cube. The problem was to construct a square with area equal to that of a given circle, using only straightedge and compass.

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Squaring the Circle in Ancient Greece

The problem of squaring the circle remained open for two thousand years. A circle and square have equal areas iff the ratio between the side of the square and the radius of the circle is equal to $\sqrt{\pi}$.

In 1882, Ferdinand von Lindemann showed that the number ${\pi}$ is transcendental and therefore not constructable. Two line seqments with ratio ${\pi}$, or ratio ${\sqrt{\pi}}$, cannot be constructed using ruler and compass, so the circle cannot be squared by the methods of Euclidean geometry.

Mathematicians then asked if it is possible to cut up a circular disc and reassemble the pieces to form a square. In technical terms, they asked if the circle and square are dissection congruent’. Boundary points may be ignored in this approach. Nevertheless, this `scissors method’ of squaring the circle was shown to be impossible (Dubins-Hirsch-Karush, 1964)

Outline proof of Wallace-Bolyai-Gerwein theorem (image from Marks and Unger, 2017).

Any two polygons of the same area are dissection congruent. The idea of the proof is shown in the Figure above, taken from the presentation of Andrew Marks and Spencer Unger (2017). This result does not generalize to three dimensions. In 1902, Max Dehn showed that a cube and a regular tetrahedron are not dissection congruent.

Tarski’s Problem

In 1925, Polish logician Alfred Tarski proposed his circle-squaring problem: given a disc ${\{x : |x-x_0|\le r \}}$ in the plane ${\mathbb{R}^2}$, dissect it into a finite number of pieces and reassemble them to form a square of area equal to that of the disc. In other words, Tarski asked whether a disc is equidecomposable to a square: can it be partitioned into finitely many parts which can be rearranged to form a square?

The problem was inspired by the extraordinary Banach-Tarski theorem, which states that any two bounded sets in ${\mathbb{R}^3}$ with non-empty interiors are equidecomposable. In particular, any ball is equidecomposable to any cube — of any size!

Miklos Laczkovich in 2011 [photo from Wikimedia Commons].

Tarski’s problem was proven by the Hungarian mathematician Miklos Laczkovich to be solvable. The theorem of Laczkovich states that, if ${A}$ and ${B}$ are bounded sets in ${\mathbb{R}^k}$ with the same volume [strictly, the same positive Lebesgue measure, and having boundaries with upper Minkowski dimension less than ${k}$], then ${A}$ and ${B}$ are equidecomposable.

Laczkovich’s proof was non-constructive, and depended critically on the axiom of choice. Moreover, the pieces he used were not measurable. Some pieces are like jig-saw pieces, others are curves and others are collections of points. Laczkovich estimated that the number of pieces in his decomposition was approximately ${10^{50}}$.

Laczkovich actually proved the construction of the square does not require rotation of the pieces: only translations are needed. A consequence of his proof is that any simple plane polygon can be decomposed into finitely many pieces and reassembled to form a square of equal area, using only translations.

Schematic illustration of the dissection of a disc and rearrangement as a square (from Marks and Unger, 2017).

A Constructive Proof

Just a few years ago, Andrew Marks and Spencer Unger (2017) presented a constructive solution of the circle-squaring problem. They describe the background to the problem, outline the proof of Laczkovich and then present a constructive proof. The pieces they used were measurable (Borel sets, or elements of the ${\sigma}$-algebra generated by the usual topology of open sets in the plane). Their partition required something like ${10^{200}}$ pieces. The dissection is illustrated in the Figure above. For more details, see the presentation of Marks and Unger (2017).

Sources

${\bullet}$ Laczkovich, Miklos, 1990: Equidecomposability and discrepancy: a solution to Tarski’s circle squaring problem, J. reine angew. Math., 404, 77-117.

${\bullet}$ Marks, Andrew and Spencer Unger, 2017: A constructive solution to Tarski’s circle squaring problem. UC Berkeley Logic Colloquium, 25 Aug 2017. URL: https://www.math.ucla.edu/~marks/talks/circle_squaring_talk.pdf.

${\bullet}$ Ruthen, Russell, 1989: Squaring the circle. Sci. Amer., July, 1989, pp. 22-24.