Proceedings of the Square Moon Society · Notes
How Big Is the Lunar Cube? Dimensions, Corners and How to Measure Them
The IAU gives the Moon a mean radius of 1737.4 km. If the object behind that number is a cube rather than a sphere, how large is it, where are its corners, and why has nobody bumped into one?
Two ways to size a cube from a sphere
Two independent estimates of the edge aM are available, and the paper On the Circlefication of a Square uses both.
The first conserves volume. Taking the IAU mean lunar radius rM = 1737.4 km as the radius of the spherified ball, the cube of equal volume has
aM = rM · (4π/3)1/3 = 1.6120 rM = 2800.7 km.
This single formula contains both classical impossibilities of Greek geometry: the transcendental π of the circle-squarers and the cube root of the Delians, who were asked by the oracle to double a cubical altar and could not.
The second conserves mean radius. A cube’s surface, expressed in spherical coordinates about its centre, has a mean radial distance of about 0.6107 a. Setting that equal to rM gives aM = 1737.4 / 0.6107 = 2844.9 km. The two estimates agree to 1.6 %; the residual is of the order of the Vitruvian offset, and the paper adopts the volume-conserving figure.
| Quantity | Value |
|---|---|
| Edge aM | 2800.7 km |
| Half-edge (centre to face) | 1400.3 km |
| Half-diagonal (centre to vertex) aM√3/2 | 2425.5 km |
| Vertex protrusion beyond apparent limb | 688.1 km |
| Face-centre recess behind apparent limb | 337.1 km |
| Faces / edges / vertices | 6 / 12 / 8 |
| Angular size of vertex protrusion from Earth (before circlefication) | 1.79 × 10−3 rad = 0.10° |
| Corner protrusion at the limb after circlefication (επ · d) | 6.15 cm |
Where the corners are
The cube has half-diagonal 2425.5 km, so its eight vertices protrude 688.1 km beyond the apparent limb, while its six face centres lie 337.1 km behind it. From the Earth, at a mean distance of 384 400 km, a 688 km protrusion subtends 0.10°—one fifth of the lunar diameter—before circlefication. That so large a feature is invisible is a measure of how thorough the circlefication has been.
The recessed face has an observational correlate. The near side of the Moon is dominated by the low, dark, flat maria; the far side is not. A recessed face turned permanently towards the Earth is exactly what the cube predicts.
Why the astronauts saw no edges
Six Apollo 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.
Six centimetres
After circlefication to the Ramanujan stage, the residual corner angle is the Lunar Circlefication Constant επ = 1.60 × 10−10 rad (see Ramanujan’s squaring of the circle). At 384 400 km this corresponds to a linear corner protrusion at the lunar limb of
ℓ = d · επ = 384 400 km × 1.60 × 10−10 = 6.15 cm.
The corners of the Moon, as seen from the Earth, protrude by six centimetres. This is below the resolution of any telescope, which is why the Moon looks round. It is above the precision of lunar laser ranging, which now operates at a few millimetres (Murphy, 2013), which is how the Society proposes to detect them.
Libration: the cube turning its edge
The Moon rocks against the line of sight by up to 6.9° in longitude and 6.7° in latitude (optical libration). A cube tilted by an angle β about an edge presents a rectangle of sides a and a(cos β + sin β); tilted about a face diagonal, an irregular hexagon. For β = 6.9° the silhouette area grows by a factor of 1.113, i.e. the apparent diameter grows by 5.5 %.
It is a well-known fact that the apparent diameter of the Moon varies by some 14 % over a month, conventionally attributed entirely to the eccentricity of the lunar orbit. The paper proposes that 5.5 of those 14 percentage points are the cube turning its edge towards us.
Two testable predictions
Altimetry. A spherical-harmonic analysis of the lunar shape from laser altimetry—the LOLA instrument on the Lunar Reconnaissance Orbiter, for instance—should show power at degrees ℓ = 4 and 6 but none at ℓ = 3, 5 or 7. That is the signature of the cube’s octahedral symmetry group Oh, whose invariant harmonics have only degrees 0, 4, 6, 8, 10, 12, ….
Limb profile. A Fourier analysis of the limb should exhibit power at harmonic 4 exceeding the power at harmonics 3 and 5 by a factor of order 1017. Any measured ratio exceeding unity will, of course, be taken as confirmation.
One respect in which the conventional view is correct
Euler’s relation for the cube gives V − E + F = 8 − 12 + 6 = 2. The Moon is therefore topologically a sphere. The paper notes that this is the only respect in which the conventional view is right.