Is the Moon a Cube? Frequently Asked Questions

Yes. The Square Moon Society holds that the Moon is a regular hexahedron—a cube—with six faces, twelve edges and eight vertices. Because the Moon is tidally locked, the same face is always turned towards the Earth, so what an observer sees is the cube's square silhouette. The apparent roundness of that silhouette is the result of a process the paper calls circlefication: a low-pass angular filtering of the square outline by the finite resolving power of the observer, by atmospheric seeing, and, most fundamentally, by the transcendence of π.

Because every observer performs a circlefication. Any eye, camera or telescope has a finite aperture and finite precision, and therefore computes with some algebraic approximation of π rather than π itself. The Transcendental Residual Theorem shows that the corners are never fully removed, only reduced to a residual angle of about 1.6 × 10⁻¹⁰ radians—six centimetres at the lunar limb, far below the resolution of any telescope. The Moon looks round because the corners are small, not because they are absent.

Conserving the volume of a sphere of the IAU mean radius 1737.4 km, the cube has an edge of 2800.7 km. Its eight vertices lie 2425.5 km from the centre and therefore protrude 688 km beyond the apparent limb; its six face centres lie 337 km behind it. From the Earth a 688 km protrusion would subtend 0.10°, one fifth of the lunar diameter, before circlefication.

Every camera has a finite aperture and therefore performs a circlefication with σ > 0. Proposition 1 of the paper—no finite circlefication produces a disc, but every finite circlefication removes the corners below its own resolution—applies to cameras exactly as it applies to eyes.

Six landings, all on the near face, none within 700 km of a vertex. A person standing on a face of a cube 2800 km across sees a horizon 2.4 km away and a surface flat to within one part in a million. They would report exactly what they reported.

Hydrostatic equilibrium is a statement about the regolith, which the Society agrees is in equilibrium: the paper proposes that 4.5 billion years of tidal action have Banach–Tarski-rearranged the roughly 10⁵⁰ regolith grains into a ball-shaped appearance. It says nothing about the solid hexahedral core beneath. The Moon's famously lumpy gravitational field—the mascons—is what one expects of a cube wearing a sphere.

Defined, yes. Constructed, no. Lindemann proved in 1882 that π is transcendental, so √π cannot be produced by any finite ruler-and-compass construction or any finite sequence of arithmetic operations. Every physical observation is such a finite sequence. The distinction between defining π and constructing it is Lindemann's, and it is the whole content of the paper.

The far face of the cube is never subject to Earth-facing tidal translation, so the paper conjectures it retains its corners and edges and has not been circlefied. The far side's markedly thicker crust, far fewer maria and strikingly different appearance—documented by the GRAIL mission—have never been satisfactorily explained. A face that has not been circlefied looks different from a face that has.

Eight, of which four are turned towards the Earth. The visible silhouette has exactly four corners because the Fourier spectrum of a square contains only harmonics of order 4k; a harmonic of order 3 or 5 would be needed for a third or fifth corner, and it is absent. The whole body has eight vertices, twelve edges and six faces, as the octahedral symmetry group requires. Euler's relation 8 − 12 + 6 = 2 confirms that the Moon is topologically a sphere, which is the only respect in which the conventional view is correct.

Because the process runs from the square to the circle. The classical impossibility theorems—Lambert, Wantzel, Lindemann—all start from a square (the unit square, the rational lattice, the algebraic numbers) and show that the circle cannot be reached from it. Read in the direction in which they are proved, they say you cannot circle the square, not that you cannot square the circle. The three-dimensional version, from cube to ball, is called spherification.

Yes, in two ways. A spherical-harmonic analysis of lunar altimetry (e.g. from the LOLA instrument) should show power at degrees 4 and 6 but none at 3, 5 or 7—the signature of cubic symmetry. And lunar laser ranging, which now works to a few millimetres, should be able to detect the six-centimetre corner protrusion predicted by the Lunar Circlefication Constant.

The apparent diameter of the Moon varies by about 14 % over a month, conventionally attributed entirely to orbital eccentricity. A cube tilted by the 6.9° libration angle presents a silhouette 5.5 % wider in diameter than face-on. The paper proposes that 5.5 of those 14 percentage points are the cube turning its edge towards us.

Yes.

In the paper On the Circlefication of a Square: A Geometric, Transcendental and Selenographic Demonstration that the Moon is a Cube, published in the Proceedings of the Square Moon Society, Vol. IV, No. 4. It is available on this site as a web page and as a PDF.