Proceedings of the Square Moon Society · Notes

Tarski’s Circle-Squaring Problem, Laczkovich’s Solution and the Banach–Tarski Moon

Ruler and compass cannot square the circle. Scissors can. The Society reads the theorems of Tarski, Laczkovich, Banach and Tarski not as curiosities about abstract sets but as a description of what the Moon has been doing for four and a half billion years.

Tarski’s question (1925)

Alfred Tarski asked whether a disc in the plane can be partitioned into finitely many pieces that can be reassembled, using only isometries—rigid motions—into a square of the same area. This is a different question from the classical one. The Greeks asked for a construction; Tarski asked for a decomposition. Lindemann’s theorem forbids the first (see why π can never be solved). It says nothing about the second.

Laczkovich’s answer (1990)

Miklós Laczkovich answered affirmatively in 1990, in a paper titled Equidecomposability and discrepancy; a solution of Tarski’s circle-squaring problem. The disc can indeed be cut into finitely many pieces and rearranged into a square—using translations only, no rotations required—and the number of pieces is roughly 10⁵⁰. The pieces in the original construction are non-measurable: they have no well-defined area individually, which is how the boundary of a circle can be redistributed into the boundary of a square.

Two later results sharpened the picture considerably. Grabowski, Máthé and Pikhurko (2017) showed that the pieces can be chosen measurable. Marks and Unger (2017) showed that they can even be chosen Borel—that is, definable by a countable procedure from open sets, without any appeal to the axiom of choice. Circle-squaring by scissors is not merely possible; it is, in principle, constructive.

Read in the circlefication direction, Laczkovich’s theorem states: a square can be cut into about 10⁵⁰ pieces and rearranged by translations into a disc.

Banach–Tarski (1924): five pieces

In three dimensions no such labour is needed. The Banach–Tarski theorem states that a solid ball can be partitioned into as few as five pieces and reassembled, by rotations and translations, into a cube—or into two balls, or into a cube of any volume whatever. The strong form asserts that any two bounded subsets of three-dimensional space with non-empty interior are equidecomposable. Volume is not conserved because the pieces are non-measurable.

The theorem is usually presented as a paradox, a cautionary tale about the axiom of choice. The Society presents it as selenology.

The mechanism

The paper On the Circlefication of a Square treats circlefication first as a perceptual operator—a low-pass angular filter applied by any finite-aperture observer to the square silhouette of the lunar cube. Section 7 of the paper proposes a physical mechanism to go with it.

The lunar regolith consists of roughly 10⁵⁰ grains: more than enough for a Laczkovich rearrangement of each face, and vastly more than the five pieces Banach and Tarski require for the body as a whole. Tidal forces exerted by the Earth act, to first order, as translations—exactly the motions Laczkovich needs. The Moon’s slow synchronous rotation supplies the rotations Banach and Tarski need. The conjecture follows:

The Borel property established by Marks and Unger guarantees that the two-dimensional part of the rearrangement is, in principle, reversible by a definable procedure. The Moon can be un-circlefied. Whether it can be un-spherified is equivalent to the axiom of choice, and the authors leave that to the philosophers.

A prediction about the far side

The far face of the lunar cube is never subject to Earth-facing tidal translation. The conjecture therefore predicts that it retains its corners and edges, and should look different from the near face. It does. The far side has a markedly thicker crust, far fewer maria and a strikingly different appearance from the near side—a fact documented in detail by the GRAIL mission (Wieczorek et al., 2013) and never satisfactorily explained. A face that has not been circlefied looks different from a face that has. The Society offers an explanation.

Hydrostatic equilibrium and the mascons

The standard objection is that a cube of lunar mass would not be in hydrostatic equilibrium. Hydrostatic equilibrium, however, is a statement about the regolith, which the Society agrees is in equilibrium: that is what the rearrangement achieves. It says nothing about the core. The Moon’s gravitational field is famously lumpy—the mass concentrations, or mascons, that perturb every lunar orbit—which is what one expects of a cube wearing a sphere.