An Eleven-Dimensional Geometric Reading of Lenz's 6π^5 Approximation to the Proton–Electron Mass Ratio — A Doodle with the Unit 11-Ball and Twenty Equatorial Cross-Polytopes

That the proton-to-electron mass ratio m_p/m_e ≈ 1836.15 can be approximated by 6π^5 = 1836.118 is a well-known numerical coincidence, dating from an extremely short 1951 note by F. Lenz. The present paper shows that 6π^5 need not be read as mere number-matching: it can be drawn, in eleven dimensions, as the ratio of the volume of the unit ball to the total volume of twenty congruent, non-overlapping cross-polytopes (the eleven-dimensional relatives of the octahedron). Concretely, one of the eleven coordinates is labeled t (a label only) and on the remaining ten axes I take the 2 × 10 = 20 points that I call the equatorial configuration. At each of them I place a translated copy of the standard eleven-dimensional cross-polytope of circumradius 1. If the proton is the unit 11-ball, and the electron the twenty copies taken together, the ratio of the two volumes is exactly 6π^5; the agreement with the measured mass ratio remains approximate, with relative error 1.88 × 10^-5.Constructions that realize 6π^5 as an exact ratio of volumes already exist in the literature, built from the volumes of Lie groups and related objects (Section 1); the present one reaches the same value with elementary geometry alone, in a dimension that modern physics singles out for independent reasons (eleven-dimensional supergravity, M-theory). This paper is a piece of recreational mathematics, and has nothing to do with any of my professional duties. On the other hand, a numerical control experiment confirms that a finite toolbox of small primes and π produces, in abundance, agreement as good as that of 6π^5 for every mass ratio tested here. Under the geometric constraint that each expression be a ratio of volumes of elementary bodies, all taken in one and the same dimension, exactly one ratio in the finite search space, with a single term in the denominator, matches its target as closely as 6π^5 matches its own: fp/e ≈ 6π^5 in eleven dimensions. In the end the one merit of this construction is not how well the value agrees, but that 6π^5 alone can be drawn as a single geometrically realizable picture. Drifting each target within its band of measurement uncertainty leaves that verdict unchanged, and the reciprocal of the fine-structure constant, 1/α, stays out of reach even in the vocabulary of length ratios — the form in which α properly appears in physics. Section 7 then lists candidly the fatal problems this construction has as a physical theory, from the disagreement with measurement itself to the absence of Lorentz covariance. Licensing: the paper (this PDF) and the seven figures are licensed under CC BY 4.0. The source code and the data files inside proton_electron_geometry_supplement.zip are licensed under the MIT licence. See the LICENSE file inside that archive.