On a BBP-type formula for $π^2$ in the golden ratio base

Benoit Cloitre

Home/Archive/arXiv

arXiv · 2508.03743

On a BBP-type formula for $π^2$ in the golden ratio base

Benoit Cloitre

View original source ↗Read source document ↗Export metadata ↓

Abstract

This paper presents a detailed, self-contained proof of a BBP-type formula for $π^2$ expressed in the golden ratio base, $ϕ$. The formula was discovered empirically by the author in 2004. The proof presented herein is built upon a fundamental geometric identity connecting $ϕ$ to the fifth roots of unity, offering an intuitive and direct path to the result. The power of the underlying methodology is then demonstrated by extending it to establish a new, computationally efficient Machin-like formula for $ζ(3)$, expressed through rapidly converging, hierarchical series involving the golden ratio.

Explore related subjects

math.GM

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Benoit Cloitre. 2025-08-01. On a BBP-type formula for $π^2$ in the golden ratio base. https://arxiv.org/abs/2508.03743

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Geometric Duality Between Constraints and Gauge Fields: Mirror Realization and Reduction Geometry on Principal Bundles

A connection and a nonzero parallel adjoint field determine an invariant hyperplane constraint on a principal bundle. Its sign mirror preserves the hyperplane and reverses its coorientation; global gauge realization is controlled by a twisted stabilizer reduction. For regular fields we identify the normalizing gauge extension as a pushout of the torus-normalizer extension, giving exact lift orders and simultaneous-splitting criteria. In singular rank-two block families, reductions on a fixed trivial bundle form an affine second-Chern lattice whose Weyl stabilizers and finite-order lift spectra detect topology invisible to paired curvature. The reduction framework also determines the structure group and second cohomology of the matched-flag diagonalization space of Friedman and Park, and gives a first- and second-Chern criterion for normal matrices with fixed separated spectrum on four-complexes; every integral solution of their three-eigenline equation on $S^2\times S^2$ is realized. For moving reductions, the projected circle curvature differs from the ambient paired curvature by a covariant-derivative term. Full fatness on a closed four-manifold forces a nontrivial sign-mirror obstruction for every circle reduction; hyperbolic self-dual-form bundles also provide circle reductions in the $y$-fat setting of Florit and Ziller. Contact transgression, bundle automorphism twists, and the natural first-jet Spencer operator complete the geometric picture.

math.GM↗

arXiv2026

A proof of the irrationality of $π$ motivated by nested radicals with roots of $2$

In this work, we prove the irrationality of $π$ motivated by nested radicals with roots of $2$ of the kind $c_k = \sqrt{2 + c_{k - 1}}$ and $c_0 = 0$. Sample computations showing how the rational approximation tends to $π$ as the integer $k$ increases are presented.

math.GM↗

arXiv2026

Ramanujan-Type Series of Signature 2: Analytical Evaluation via Degree-2 Transformations and Associated Harmonic Expansions

We provide an explicit analytical evaluation of the known rational Ramanujan-type series for the theory of signature 2. Focusing on the singular moduli $k_r$ for $r \in \{2, 3, 4, 7\}$, we demonstrate that the underlying elliptic identities can be established through modular transformations of degree 2. In particular, we showcase a family of rational harmonic Ramanujan-type series for $1/π$ involving higher-degree polynomials

math.GM↗