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The arc length of every supercircle is an infinite series of hypergeometric functions, and the circle case yields a new series for π.

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2026-07-11 16:44 UTC pith:3U7CQQRE

load-bearing objection Clean, self-contained hypergeometric series for supercircle arc length that fills a documented gap; modest but correctly done classical analysis. the 1 major comments →

arxiv 2607.04592 v1 pith:3U7CQQRE submitted 2026-07-06 math.CA math.MG

On the Arc Length of a Supercircle and a Hypergeometric Formulation of π

Yomber Montilla , R. Omar Rodriguez , Brexys Linares This is my paper
classification math.CA math.MG MSC 33C0526A0651M25
keywords arc lengthsupercirclehypergeometric functionconstant πgeneralized binomial coefficientLamé curve
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

Supercircles are plane curves that morph continuously from a cross-like shape through a diamond and a circle to a square as a single shape parameter n varies. Their areas are elementary, but their perimeters generally are not. This paper derives a single infinite series that gives the exact arc length for every positive n and scale a. The series is built from generalized binomial coefficients and Gauss hypergeometric functions, with a slight change of formula according to whether n is less than or greater than 1. The series converges for every n > 0, recovers the known perimeters of the square, the cross, the rhombus, the astroid and the circle, and, when specialized to the circle and divided by the diameter, produces a hypergeometric series for π itself. A reader who needs an exact, parameter-dependent perimeter for these shapes now has a closed-form expression that can be truncated to any desired accuracy.

Core claim

For every a > 0 and n > 0 the arc length of the supercircle r( heta) = a(|cos heta|^n + |sin heta|^n)^{-1/n} is given exactly by the series L = (8a/n) ∑_m (1/u_m) binom(1/2,m) _{2}F_{1}(1+1/n, u_m; u_m+1; -1), where the auxiliary index u_m equals 1 + 2m|1-n|/n and the two regimes n ≤ 1 and n ≥ 1 are distinguished only by that absolute value.

What carries the argument

The binomial expansion of (1 + ξ^λ)^{1/2} inside the polar arc-length integral, followed by term-by-term identification with Euler’s integral representation of the Gauss hypergeometric function _{2}F_{1}.

Load-bearing premise

The interchange of sum and integral that produces the hypergeometric series is justified only after the fact by the absolute-convergence estimate; no separate domination argument is given at the moment of interchange.

What would settle it

Compute the series for any of the four classical cases (parabolic star n = 1/2, astroid n = 2/3, rhombus n = 1, circle n = 2) to high precision and check whether it matches the known closed-form perimeter to machine accuracy; a mismatch would falsify the formula.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • Every supercircle perimeter is now available as a single, uniformly convergent series rather than a numerical quadrature.
  • The circular case supplies an explicit hypergeometric series for π that can be truncated to any prescribed number of correct digits.
  • The same construction recovers the elementary perimeters of the square, the limiting cross and the rhombus as special or limiting values of the series.
  • The truncation order needed for a fixed tolerance is maximal at n = 1 and decreases rapidly as n moves away from 1.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same binomial-plus-hypergeometric route should extend, with only notational changes, to the arc length of superellipses that have unequal semi-axes.
  • Acceleration techniques such as Aitken Δ^{2} or Padé approximants could turn the n = 2 series into a practical computational formula for π, even though the raw series is slow.
  • Because the series is analytic in n for n > 0, derivatives of length with respect to the shape parameter become available by term-wise differentiation and could be used for shape-optimization problems.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

1 major / 5 minor

Summary. The paper derives an exact infinite-series formula for the arc length L of the supercircle r(θ)=a(|cos θ|^n+|sin θ|^n)^{-1/n} (a>0, n>0). After reducing by symmetry to the polar integral over [0,π/4] and a change of variables ξ=tan^n θ, the radical is expanded by the binomial series for (1+ξ^λ)^{1/2}; term-by-term integration then produces L=(8a/n)∑_{m=0}^∞ (1/u_m) binom(1/2,m) ₂F₁(1+1/n,u_m;u_m+1;-1), where the auxiliary sequence u_m takes two different linear forms according to the regimes 0<n≤1 and n≥1 (Theorem 3.1). Absolute convergence of the series is proved by comparison with ζ(5/2) (or ζ(3/2) when n=1). The formula recovers the elementary lengths of the limiting square and cross (both 8a), the rhombus (4√2 a), and, to high numerical precision, the known closed-form lengths of the parabolic star, astroid and circle; the circular case n=2 yields a hypergeometric series for π.

Significance. The result supplies a single, explicit special-function series that covers the entire one-parameter family of supercircles, a quantity previously accessible only by numerical quadrature for generic n. The derivation uses only classical tools (polar arc-length, binomial series, Euler’s integral for ₂F₁) and is fully elementary; the absolute-convergence proof, the careful extraction of the m=0 term in the rectilinear limits, and the independent numerical recovery of six exact special cases constitute strong internal checks. The accompanying Python truncation code and the tabulated high-precision comparisons further enhance reproducibility. While the π series is not competitive for digit computation (as the authors themselves note), it is a clean geometric byproduct. Overall the paper is a solid, self-contained contribution to classical analysis and special functions.

major comments (1)
  1. [Theorem 3.1] Proof of Theorem 3.1 (immediately after Eq. (23)): the binomial series for (1+ξ^λ)^{1/2} is integrated term-by-term against the remaining positive factors on [0,1] before absolute convergence of the resulting series is established in Proposition 3.2. Although the interchange is valid (uniform convergence on every compact subinterval [δ,1] plus integrability of the singularity at ξ=0 already controlled by Proposition 2.4), the manuscript should either reorder the arguments or insert a short, self-contained justification (Weierstrass M-test on the truncated intervals plus dominated convergence, or an appeal to the absolute-convergence estimate already proved later) at the moment the interchange is performed.
minor comments (5)
  1. [Corollary 3.5] In the proof that ∂L/∂n vanishes at n=1 (Corollary 3.5, Eqs. (57)–(58)), the resulting numerical series ∑ binom(1/2,m)(2m ln 2 − ½ ln 2) is simply declared equal to zero. A one-line evaluation via the generating function (1+x)^{1/2} and its derivative at x=1 would make the cancellation fully explicit.
  2. [Section 4 / Figure 2] Figure 2 embeds full Python source code as a figure panel. For archival purposes it would be preferable to place the code in a supplementary file or an appendix and retain only a short algorithmic description in the main text.
  3. [Section 4.1] The truncation criterion (61)–(63) relies on successive relative differences of an alternating series. While adequate for the reported numerical experiments, a brief remark that the absolute remainder is controlled by the first omitted term (or by the O(m^{-5/2}) bound of Proposition 3.2) would strengthen the error analysis.
  4. [Appendix B] Appendix B evaluates the auxiliary series S_1 by reducing it to a definite integral that is then stated to equal 2 ln 2 − √2 ln(1+√2). A short indication of the antiderivative (or a reference to a standard integral table) would complete the argument.
  5. [Introduction and references] Typographical consistency: the manuscript mixes “Lamé” / “Lam ´e” and occasionally omits the accent; a uniform spelling should be adopted throughout.

Circularity Check

0 steps flagged

No significant circularity: the arc-length series is obtained by elementary substitutions and textbook special-function identities, and the π series is simply the n=2 specialization divided by the diameter.

full rationale

The derivation chain is self-contained and non-circular. Starting from the polar definition of the supercircle (Eq. 1), the authors reduce the arc-length integral by axial/diagonal symmetry (Lemma 2.2), introduce the auxiliary z( heta) and the substitution ξ = tan^n heta (Proposition 2.3), rewrite the radical via the binomial series for (1 + ξ^λ)^{1/2} (Eq. 23), and identify the resulting integrals with the Euler integral representation of 2F1 (Eqs. 25–26). Absolute convergence is proved afterwards by comparison with ζ(5/2) and ζ(3/2) (Proposition 3.2). The elementary special cases (square, cross, rhombus) are recovered by taking limits or setting n=1 inside the same series and using only standard hypergeometric identities (Proposition 3.4); the π series (Corollary 3.6) is obtained by setting n=2 and dividing by the diameter 2a. No parameter is fitted to data, no uniqueness theorem is imported from the authors’ prior work, and no known empirical pattern is merely renamed. The numerical tables serve only as independent verification against closed-form lengths already known for the parabolic star, astroid, rhombus and circle. Consequently the central claim does not reduce to its inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 4 axioms · 0 invented entities

The entire derivation rests on textbook real-analysis and special-function identities applied to the classical polar arc-length integral; no free parameters are introduced and no new mathematical objects are postulated. The only background facts required are the binomial series for exponent 1/2, Euler’s integral representation of ₂F₁, elementary Gamma-function asymptotics, and the comparison test for absolute convergence.

axioms (4)
  • standard math Binomial series (1+x)^{1/2} = ∑ binom(1/2,m) x^m converges for |x|≤1
    Invoked in the proof of Theorem 3.1 to expand the radical factor on the unit interval.
  • standard math Euler integral representation of the Gauss hypergeometric function ₂F₁(v,u;w;γ)
    Used to identify the remaining definite integral after the binomial expansion (Eq. 26).
  • standard math Asymptotic Γ(m+v)/Γ(m+w) ∼ m^{v-w} for fixed v,w and large m
    Supplies the O(m^{-3/2}) decay of the generalized binomial coefficient needed for absolute convergence (Appendix A).
  • domain assumption Axial and diagonal symmetries of the supercircle reduce total length to eight times the arc on [0,π/4]
    Stated in Lemma 2.2 and used throughout; follows immediately from the polar equation.

pith-pipeline@v1.1.0-grok45 · 20668 in / 2451 out tokens · 31634 ms · 2026-07-11T16:44:34.569508+00:00 · methodology

0 comments
read the original abstract

We obtain an infinite-series representation for the arc length of a supercircle in terms of the scale parameter $a$ and the shape parameter $n$. The resulting expression is constructed by means of generalized binomial coefficients and Gauss hypergeometric functions, distinguishing two regimes associated with the value of $n$. We also analyze the absolute convergence of the resulting series. We verify the consistency of the formulation from limiting cases and particular configurations of the family of supercircles: when $n\to0^+$ and $n\to\infty$, the length converges to the value $8a$, corresponding to the limiting rectilinear geometries, whereas for $n=1$ we recover the perimeter of the rhombus with diagonals of length $2a$. In addition, as a validation against supercircles with exact arc length, the formulation reproduces with high numerical precision the arc length of the parabolic star, the astroid, and the circle. Finally, by specializing the circular case $n=2$ and normalizing the length by the diameter $2a$, we obtain a series representation, in terms of hypergeometric functions, for the constant $\pi$.

Figures

Figures reproduced from arXiv: 2607.04592 by Brexys Linares, R. Omar Rodriguez, Yomber Montilla.

Figure 1
Figure 1. Figure 1: Family of supercircles with a = 1: n → 0 (cross), n = 1/2 (parabolic arcs), n = 2/3 (astroid), n = 1 (rhombus), n = 2 (circle), and n → ∞ (square). The dashed lines illustrate the angular partition into octants of width π/4 [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Code in Python 3.13: a) iterative numerical calculation of the arc length L(n) by truncating Eq. (21); b) iterative calculation of the hypergeometric representation of π, Eq. (59) [PITH_FULL_IMAGE:figures/full_fig_p013_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Behavior of the arc length L of the supercircle as a function of the parameter n on a logarithmic scale [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Truncation order m∗ as a function of the parameter n on a logarithmic scale, with a = 1, εs = 10−6%. orders of the order of a few hundred, whereas the case n = 1, corresponding to the rhombus, requires a truncation order of tens of thousands of terms [PITH_FULL_IMAGE:figures/full_fig_p014_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: summarizes the convergence behavior. In panel (a), the first 16 partial sums oscillate around π, alternating above and below the reference value with progressively smaller amplitudes. This oscillation is due to the alternating character of the binomial coefficients 1/2 m  for m ≥ 1. In panel (b), the absolute error |Sm − π|, represented on a logarithmic scale, decreases as the number of partial sums consi… view at source ↗

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Hypergeometric Series Representations for the Perimeter of Lam\'e Superellipses

    math.CA 2026-07 accept novelty 6.0

    The perimeter of a Lamé superellipse admits exact hypergeometric series representations for s>1 (conditionally convergent) and 0<s<1 (Abel-summable), with the rhombus at s=1 uniquely minimizing length.

Reference graph

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