arXiv · 2607.24843
A formula is established that counts the number of different balls that can be made by decorating the pentagons and hexagons of a classic football with Truchet-like patterns.
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Thomas Fernique. 2026-09-17. Counting Truchet Tile Balls. https://arxiv.org/abs/2607.24843
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This paper develops a formal theory of musical scales and their harmonic coverings and introduces orbit covers: coverings obtained by translating a fixed subset across a scale via a group action. Orbit covers generalize familiar constructions, such as the covering of the diatonic scale by tertian triads, and are motivated by the search for a generalized harmonic framework extending common-practice tonality. We model modes as group structures associated with pitch-class sets and scales as torsors, introducing scale covers and, in particular, orbit covers. To each orbit cover we associate a nerve complex encoding its intersection structure and associated topological invariants. We classify triadic orbit covers of heptatonic scales up to affine symmetry and nerve isomorphism. These results support a broader theory of harmonic organization with analytical and compositional applications.
We give a self-contained, numerically verified derivation of three equivalent unconditional representations of the Euler--Mascheroni constant $γ$ as a rapidly convergent series in the Taylor coefficients $a_{2n}$ of the Riemann $ξ$-function about its centre of symmetry $s=1/2$ (equivalently, in the Jensen coefficients $c_n$ or in the Csordas--Norfolk--Varga Turan moments $\hat{b}_n$). We also record the corresponding convergent formula for the even-indexed Bernoulli numbers $B_{2r}$, obtained by evaluating the (everywhere convergent) Taylor series of $ξ$ at $s=2r$. We then develop, from the evenness of $Ξ(x)=ξ(1/2+ix)$, an infinite hierarchy of exact relations at $s=1/2$: the even-order relations reduce the central logarithmic derivatives of $ζ$ to closed forms in $π$ and the odd zeta values (base case $2ζ'(1/2)/ζ(1/2)=γ+π/2+\log(8π)$), and the odd-order relations evaluate the power sums over the nontrivial zeros and link back to the coefficients $a_{2n}$. All of these identities hold whether or not the Riemann Hypothesis (RH) is true; consequently they carry no evidential weight for or against RH, and we make the correct, unconditional relationship of these coefficient sequences to the Jensen--Polya program precise. This note is a corrected version of an earlier preprint [arXiv:2112.11228v1]: we retain only the results that are correct, we remove a divergent series of Bernoulli numbers that was previously used as though it were convergent, we correct several attributions, and we withdraw the earlier claim that these computations support RH. A summary of the corrections is given in Section 7.