Proceedings of the Square Moon Society · Notes
Why π Can Never Be Solved: Lindemann, Wantzel and the Residual Corner Angle
Everyone knows that π “goes on forever”. Fewer people ask what that sentence is actually about. The Square Moon Society’s answer: it is about a corner that has not yet been rounded.
Defined, yes. Constructed, no.
The number π is perfectly well defined: it is the ratio of a circle’s circumference to its diameter, and it has been since Archimedes. A common objection to the lunar-cube hypothesis is that π is therefore not “unsolved” in any meaningful sense. The distinction the Society draws is between defining a quantity and constructing it, and that distinction is not ours. It belongs to Ferdinand von Lindemann, who proved in 1882 that π is transcendental, and it is the whole content of the paper On the Circlefication of a Square.
Three theorems, one direction
Three results from the eighteenth and nineteenth centuries settle the classical status of π.
Lambert (1761) proved that π is irrational: its decimal expansion never terminates and never repeats. Every digit ever recorded—3.14159 26535 89793…—is followed by another, and no pattern rescues the sequence.
Wantzel (1837) characterised the numbers that can be constructed with ruler and compass. A length is constructible only if it lies in a tower of quadratic extensions of the rationals, which means it must be algebraic of degree 2m. This single theorem disposed of two of the three great problems of Greek geometry: the trisection of the angle, and the Delian problem of doubling the cube, which needs the cube root of 2.
Lindemann (1882), with a proof simplified by Hilbert in 1893, showed that π is not algebraic at all. It is not the root of any polynomial with rational coefficients, of any degree. Combined with Wantzel’s theorem this settles the third Greek problem: since √π is not algebraic, no finite ruler-and-compass construction produces a square whose area equals that of a given circle.
The conventional reading of these theorems is “the circle cannot be squared.” The Society points out that every one of the proofs begins with a square—the unit square, the rational lattice, the field of algebraic numbers—and shows that the circle cannot be reached from it. Not one of them starts from a circle. Read in the direction in which they are actually proved, they are theorems about the incompleteness of circlefication: you cannot circle the square.
The incompleteness theorem
Theorem 1 (Incompleteness of circlefication). There is no finite geometric construction, and no finite sequence of arithmetic operations, that maps the square of side s onto the disc of radius s/√π. Every physical circlefication is an approximation πn of π, with error δn = |π − πn| > 0.
The identity s = r√π links the side of the visible lunar face to the radius of its circlefied image. To complete the circlefication one would need to construct √π exactly, and Lindemann says that cannot be done. Every physical observation—a photograph, a telescope, an eye—is a finite sequence of algebraic operations, and therefore stalls at some algebraic πn.
From a digit to a corner
Suppose an observer’s circlefication has reached πn. The observer believes the equal-area square has side sn = r√πn, whereas the true side is s = r√π. The relative side error is approximately δn/2π, and a side error on a square leaves, at each of its four corners, a residual wedge of angle
εn ≈ δn / 2π .
This is the residual corner angle at the n-th stage of circlefication. Because δn is never zero, εn is never zero either. Every digit of π not yet recorded is a corner not yet rounded.
| Approximation | Value | δn | εn (rad) |
|---|---|---|---|
| 22/7 (Archimedes) | 3.142857 | 1.3 × 10−3 | 2.0 × 10−4 |
| Kochański (1685) | 3.141533 | 5.9 × 10−5 | 9.4 × 10−6 |
| 355/113 (Zu Chongzhi) | 3.1415929 | 2.7 × 10−7 | 4.2 × 10−8 |
| (2143/22)1/4 (Ramanujan) | 3.14159265258 | 1.0 × 10−9 | 1.60 × 10−10 |
| 355/113 · (1 − 0.0003/3533) (Ramanujan) | 3.14159265358979 | 1.3 × 10−15 | 2.1 × 10−16 |
What this means for the Moon
Theorem 2 (Transcendental Residual Theorem). For every physically realisable observer there exists ε > 0 such that the observed lunar silhouette possesses four corners of angular residual at least ε.
A physically realisable observer performs finitely many measurements of finite precision, and therefore computes with some algebraic πn ≠ π. The residual corner angle follows directly, and the four-fold symmetry of a square’s Fourier spectrum—which contains only harmonics of order 4k—distributes it equally over four corners. Not three, not five: a harmonic of order 3 or 5 would require a term the spectrum does not contain. The Moon, as it appears to any observer whatsoever, has exactly four corners. How large they are, and how they might be measured, is the subject of a separate note.
The digit hunt, re-read
From Archimedes’ 3 10/71 to the trillions of digits computed today, the history of π is usually told as a story of ever-better approximation to a fixed truth. In the Society’s reading it is the record of a circlefication in progress. Each new digit rounds the corner a little further; the corner is never gone. The two great impossibilities of Greek geometry—the circle that cannot be squared and the cube that cannot be doubled—meet in the single formula for the lunar edge, aM = rM(4π/3)1/3, which contains both the transcendental π and the Delian cube root. That formula is the Moon.