Generalizations of the Squircle-Lemniscate Relation and Keplerian Dynamics

Muthu Veerappan Ramalingam; Zbigniew Fiedorowicz

REVIEW 2 major objections 4 minor 1 cited by

For every positive integer n, the perimeter of the sinusoidal spiral r^n = cos(nθ) is a fixed multiple of the area of the Lamé curve x^{2n}+y^{2n}=1, and the paper extends this to sectors, superellipses, and a central force law.

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2026-08-03 08:18 UTC pith:QGGYAQ5N

load-bearing objection Core identity and sector duality are correct and clean; the central-force law has a real but patchable gap for n>5, and the global Kepler correspondence is asserted more than proved. the 2 major comments →

arxiv 2601.17358 v5 pith:QGGYAQ5N submitted 2026-01-24 math.HO math.DS

Generalizations of the Squircle-Lemniscate Relation and Keplerian Dynamics

Zbigniew Fiedorowicz , Muthu Veerappan Ramalingam This is my paper

classification math.HO math.DS MSC 14H5026B1570F05

keywords Lamé curvessinusoidal spiralssquirclelemniscatesuperellipsespoliclescentral forceKeplerian motion

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.

The paper proves a generalized squircle–lemniscate relation: for any positive integer n, one integral equals a constant times another, meaning the perimeter of the n-leaf sinusoidal spiral is determined by the area of the corresponding Lamé curve. The equality is sector-by-sector, so each radial sector of the Lamé curve has area proportional to a matching arc of the spiral. That duality has a physical reading: a particle moving along the Lamé curve while sweeping area at a constant rate (Keplerian motion) corresponds to a particle moving uniformly along the spiral. The paper also derives the central force that produces such Keplerian motion on Lamé curves and introduces 'policles,' a new family of rounded 2n-gons with the same type of arc/area duality. A sympathetic reader would care because this organizes a family of classical curves under one identity and connects pure curve geometry to orbital mechanics.

Core claim

The central discovery is Theorem 1: for positive integer n, ∫₀¹ dr/√(1−r^{2n}) = 2^{1/n} ∫₀¹ (1−x^{2n})^{1/(2n)} dx. The left integral is the arc-length integral of the sinusoidal spiral r^n = cos(nθ); the right is the first-quadrant area of the Lamé curve x^{2n}+y^{2n}=1. Corollary 6 upgrades this to a sector-wise identity: the length of an arc of the spiral between two radii is 2^{1+1/n} times the area of the corresponding radial sector of the Lamé curve. The paper reads this as a kinematic equivalence: Keplerian motion on the Lamé curve at constant areal velocity maps to uniform motion on the spiral. It further derives a central force law for the Lamé curve and, in a separate construction

What carries the argument

The load-bearing tool is the integral identity of Theorem 1 and its parametrized proof. Writing the Lamé curve as x = cos^{1/n}(nt), y = sin^{1/n}(nt), the paper uses Green's theorem to convert the quadrature area into an arc-length integral of the sinusoidal spiral; the substitution r = sin^{1/n}(nu) collapses the computation. Theorem 5 contributes a larger substitution, r^n = 2v^n/(1+v^{2n}), which turns any radial arc-length integral into a sector-area integral and yields Corollary 6's sector-by-sector duality. The force law is derived through Binet's equation, the standard polar-coordinate orbit equation, with the simplification u+u'' = (2n−1)w^{2n−2}/u^{4n−1}, producing F(r) = −C r^{4n−

Load-bearing premise

For the central-force claim to hold for all n > 1, the angle term w = sinθ cosθ must be recoverable from the radius r alone along the Lamé curve; the paper verifies this only for n = 2, 3, 4, 5, so for larger n the force law may be multi-valued and not a true central force.

What would settle it

For n = 6, write the Lamé curve condition u^{12} = cos^{12}θ + sin^{12}θ and the relation w² = (1−cos4θ)/8; check whether the resulting equation has a unique real solution w² for every u along the orbit. If two different w² values give the same u, then the formula F(r) = −C r^{21} w^{10} is not a single-valued function of r, and the claim of a central force law for all n fails.

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

If this is right

For every positive integer n, the original squircle–lemniscate relation carries over to Lamé curves and sinusoidal spirals: total Lamé area = 2^{1−1/n} ϖ_{2n}, where ϖ_{2n} is the full-leaf arc length of the spiral.
The sector-wise identity gives an exact kinematic dictionary: a particle tracing the Lamé curve at constant areal velocity traces the sinusoidal spiral at constant speed over a full cycle.
Keplerian motion on Lamé curves is generated by an explicit central force; for n = 2, 3, 4, 5 the force is written as a function of r alone, e.g., F(r) = Cr(1−r⁴) for n = 2 and F(r) = −C(1−r⁶)²/r³ for n = 3.
The area formula extends to all superellipses (|x/a|^α + |y/b|^α = 1) for α > 0, giving A = 2^{1−2/α} ϖ_α ab.
The new policle curves r⁴ = n sin²(nθ)/(1−cos^{2n}(nθ)) satisfy the direct arc/area duality l = 2a√n, so the squircle–lemniscate relation has a second, simpler generalization.

Where Pith is reading between the lines

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

For n > 5, the proposed force law may fail to be a single-valued central force; a natural next step is to determine the largest n for which the polynomial relation between u^{2n} and w² is invertible, or to identify the branch structure that still makes the motion integrable.
Because the identity holds for arbitrary positive real α, the same duality may extend to irrational exponents, where the curves are defined only piecewise; this suggests a continuum of non-integer spiral–Lamé pairs with no simple leaf geometry.
The policle construction shows the squircle–lemniscate relation is not unique: many families of curves can be paired to sinusoidal spirals by the same substitution, so further polar equations r^m = f(θ) with a similar duality may exist.
The kinematic correspondence could be tested numerically: simulate uniform motion on the spiral, pull it back to the Lamé curve via the sector map, and verify constant areal velocity, especially for large n where the central-force formula remains open.

Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

Core identity and sector duality are correct and clean; the central-force law has a real but patchable gap for n>5, and the global Kepler correspondence is asserted more than proved.

read the letter

Punchline: the integral identity (Theorem 1) is right, the sector-level correspondence (Corollary 6) is the real payload, and the policle construction is genuinely new. The Kepler/force part is the weakest third: Theorem 8 works but only proves single-valuedness up to n=5, and the global uniform-motion correspondence is sketched rather than proved.

What is new: the n-parameter integral identity is, as you say, the gamma duplication formula in disguise — the authors even acknowledge this in Remark 2. But the geometric packaging is not empty. Corollary 6 gives a clean sector-to-arc duality, and the policles are a nice new family with a one-line proof. Credit where due: the algebra in Theorem 5 and the Green's theorem calculation in Theorem 1 are correct; I checked the substitution and it lands. The paper is honest about limitations and cites Levin and Siegel properly. Self-citation to their own n=2 paper is appropriate, not a red flag.

Soft spots, in proportion: 1. The central-force law (Theorem 8, Eq. 25) is expressed as a function of r and w = sinθ cosθ. For it to be a central force, w must be a single-valued function of r along the orbit. The paper only does this inversion for n=2,3,4,5. For n>5 it claims without proof that the polynomial relation is 'maybe' multi-valued; in fact P_n(z)=z^n+(1−z)^n is strictly decreasing on z∈[0,1/2] (w²=z(1−z)), so a unique inverse exists and the gap is patchable. But as written, the theorem is not fully established for all n>1. A rigorous reader cannot verify it from the paper alone. This is a real gap, not a fatal one. 2. The global Kepler/uniform correspondence (Section 5) is asserted via symmetry and a cycle diagram for n=3. It is plausible, but not proved for general n. The sector-level statement is solid; the global extension needs a short argument mapping the full orbit, not a pattern. 3. Cross-references are genuinely garbled. In Theorem 5's proof the substitution refers to equations numbered 10–13 from a different proof; Section 7 cites "Theorem 2" which does not exist. These should be fixed before publication but do not touch the main results.

Who it is for: people who like classical plane curves and pedagogical bridges between area, arc length, and mechanics. It is a math.HO paper, not a breakthrough, but it is correct in its core and the policle idea is worth knowing.

Recommendation: send it to peer review. It deserves a serious referee — the main identity and sector correspondence are solid, and the referee can request the missing inversion argument and a cleaner proof of the global correspondence. My own verdict would be 'revise and resubmit' with the above as minor-to-moderate revisions.

Referee Report

2 major / 4 minor

Summary. This paper generalizes the known squircle–lemniscate identity to arbitrary n. Theorem 1 establishes ∫_0^1 (1-r^{2n})^{-1/2} dr = 2^{1/n} ∫_0^1 (1-x^{2n})^{1/(2n)} dx, so the perimeter of the sinusoidal spiral r^n = cos(nθ) is a fixed multiple of the area of the Lamé curve x^{2n}+y^{2n}=1. Corollary 4 extends this to general superellipses and real exponents. Theorem 5 and Corollary 6 prove a sector-by-sector correspondence: the arc length of the spiral over a stated sector equals 2^{1+1/n} times the area of a corresponding radial sector of the Lamé curve. The paper then interprets this as a correspondence between Keplerian and uniform motion, derives a central-force formula (Theorem 8), and introduces 'policles' with their own sector/arc-length duality (Theorem 11).

Significance. The central integral identity and sector correspondence are correct, elegant, and proven by elementary substitutions and Green's theorem; they genuinely extend the n=2 result. The paper is self-contained, gives explicit beta-function checks, and the policle construction is a nice addition. The main advertised physical and force-law results are not fully established as written: the central force is not reduced to a function of r alone for n>5, and the global Kepler/uniform correspondence is asserted rather than constructed. These gaps are patchable, so the paper is worthy of publication after revision.

major comments (2)

[Theorem 8 / Eq. (25), Remark 10] The theorem states a central force law F(r) = -C r^{4n-3} w^{2n-2}, w=sinθcosθ, for all n>1. As derived, this is a function of both r and θ; for a central force it must depend on r alone. Remark 10 shows elimination explicitly only for n=2,3,4,5. For n>5, the polynomial relation between u^{2n} and w^2 has degree ≥3 and no invertibility argument is supplied. This leaves the force-law claim unproved for general n. The gap is fixable: writing z=sin^2θ, P_n(z)=z^n+(1-z)^n has P'_n(z)=n(z^{n-1}-(1-z)^{n-1})<0 on z∈[0,1/2), so a unique inverse exists. The authors should add this argument or restrict the theorem.
[Section 5, first paragraph] The passage 'By symmetry this correspondence extends to ... the entire Lamé curve and the entire sinusoidal spiral' is not a proof. Corollary 6 establishes a sector-area/arc-length equality in the first quadrant; extending this to a global correspondence between Keplerian and uniform motion requires a precise definition of the pairing and a verification of boundary identifications. The point-cycle pattern is illustrative but not a rigorous argument. Please provide a proof or state this as a conjectural interpretation.

minor comments (4)

[Theorem 5 proof] The sentence 'Substituting (11), (12) and (13) into the integral on the left hand side of (10)' uses incorrect equation numbers; it should refer to (18), (19), (20) and (17).
[Section 7, first sentence] 'Theorem 2 generalizes [3, Theorem 10]' should be 'Theorem 5 generalizes [3, Theorem 10]'.
[Theorem 1 proof] The displayed Green's theorem computation has missing parentheses and an ambiguous factor: the line '= 1/2 1/n ∫ ...' is hard to follow and the factor 2^{1/n} appears inconsistently. Please rewrite for readability.
[Theorem 8 proof] The symbol C is used for both the intermediate quantity s^{2n-4}+c^{2n-4} and the final constant C=(2n-1)mh^2. Rename the intermediate quantity to avoid confusion.

Circularity Check

0 steps flagged

No significant circularity: the integral and sector/arc identities are derived directly by calculus; the self-citations are motivational, not load-bearing.

full rationale

The paper's central results are self-contained derivations, not fits or renamed inputs. Theorem 1 is proved by a direct Green's theorem computation and an explicit change of variables, with Remark 2 giving an independent beta/gamma verification. Theorem 5 is a change-of-variables identity, and Corollary 6 follows from it by converting the Lamé curve to polar form and comparing the resulting integrals; Remark 7 then derives Theorem 1 as a special case. No parameter is fitted and no quantity is 'predicted' from a fitted input. The force-law derivation (Theorem 8) applies Binet's equation to the prescribed Lamé orbit; it does not assume the target formula. The acknowledged limitation in Remark 10—that explicit elimination of w = sinθ cosθ in favor of r alone is carried out only for n = 2,3,4,5—is a gap in fully exhibiting F as a function of r for n > 5, but it is a correctness gap, not circularity, because the derivation does not depend on that inversion being assumed. References to the authors' prior work [3] and to Siegel [13] provide motivation and historical context; the present proofs do not invoke [3] or [13] as premises for the identities. The policle result (Theorem 11) is likewise a direct integral substitution. Thus the manuscript is not circular; at most it contains minor non-load-bearing self-citation.

Axiom & Free-Parameter Ledger

0 free parameters · 5 axioms · 1 invented entities

The central results are parameter-free derivations; no constants are fitted to data. The main assumptions are standard mathematical tools plus domain-specific geometric and mechanical modeling choices, including the unproved global symmetry extension and the single-valuedness of the force law for general n.

axioms (5)

standard math Standard calculus: polar arc-length formula, Green's theorem, and trigonometric substitutions are valid for the curves in question.
Used throughout Sections 2-4; no special assumptions beyond ordinary real analysis.
domain assumption Binet's equation correctly describes orbits under a central force directed toward the origin.
Invoked in Theorem 8 to derive the force law; standard mechanics, but a physical modeling assumption.
domain assumption The first-quadrant sector correspondence extends by symmetry to a global bijection between the entire Lamé curve and the entire sinusoidal spiral.
Section 5 states the global Kepler/uniform correspondence without a formal proof.
domain assumption Policles, defined by r^4 = n sin^2(nθ)/(1−cos^{2n}(nθ)), are simple closed curves whose sectors have well-defined areas.
The polar equation has a 0/0 limit at θ=0; the paper assumes the curve is regular and encloses sectors.
domain assumption n is a positive integer for global geometric statements, while the integral identities hold for positive real n.
Stated in the introduction; global leaf/petal counting depends on integrality.

invented entities (1)

Policles no independent evidence
purpose: A new family of curves that generalize the squircle and satisfy a simple sector-area/arc-length duality with sinusoidal spirals.
Policles are explicitly defined in Section 7 and their main property is proved there, but there is no external falsifiable handle beyond the paper's own theorem.

pith-pipeline@v1.3.0-alltime-deepseek · 9851 in / 19504 out tokens · 190180 ms · 2026-08-03T08:18:46.776060+00:00 · methodology

0 comments

read the original abstract

This paper establishes a generalized relationship between the arc length of sinusoidal spirals $r^n=\cos(n\theta)$ and the area of generalized Lam\'e curves defined by $x^{2n}+y^{2n}=1$. Building on our previous work connecting the lemniscate to the squircle, we prove an integral identity relating these two curves for any positive integer $n$, which we further generalize to arbitrary positive real exponents and general superellipses. We further extend this correspondence to a geometric relationship between radial sectors of the Lam\'e curve and arc lengths of the spiral, providing a physical interpretation where keplerian motion on the Lam\'e curve corresponds to uniform motion on the spiral. Additionally, we derive an explicit central force law for keplerian motion along the Lam\'e curve. Finally, we introduce policles--a new class of curves generalizing the squircle--and demonstrate a direct geometric mapping between their sectors and the arc lengths of sinusoidal spirals.

Figures

Figures reproduced from arXiv: 2601.17358 by Muthu Veerappan Ramalingam, Zbigniew Fiedorowicz.

**Figure 1.** Figure 1: The Sinusoidal Spiral r 5 = cos(5θ) When n = 1 the sinusoidal spiral is just the circle with cartesian equation x − 1 2 2 + y 2 = 1 4 and when n = 2, the sinusoidal spiral is just the lemniscate with cartesian equation (x 2 +y 2 ) 2 = x 2 −y 2 . The rotation θ 7→ θ − π 2n converts the polar equation of the sinusoidal spiral to the alternative form r n = sin(nθ). In the context of this paper, sinusoidal sp… view at source ↗

**Figure 2.** Figure 2: Relation Between x 6 + y 6 = 1 and r 3 = cos(3θ) Corollary 6. Let 0 ≤ α ≤ π 4 , T = tan(α), and β = 1 n arccos 2T n 1+T 2n . Then (21) l = 21+ 1 n a, where l is the length of the arc of the sinusoidal spiral r n = cos(nθ) within the polar sector β ≤ θ ≤ π 2n and a is the area of the radial sector of the Lam´e curve x 2n + y 2n = 1 within 0 ≤ θ ≤ α. Proof. By equation (7) (22) l = Z R 0 dr √ 1 − r 2n , … view at source ↗

**Figure 3.** Figure 3: Comparison of keplerian motion on the Lam´e curve x 6 + y 6 = 1 and uniform motion on the sinusoidal spiral r 3 = cos(3θ). A full cycle of paired motions on the Lam´e curve and the sinusoidal spiral is described by the following pattern: → (P1, Q1) → (P2, Q2) → (P3, Q3) → (P4, Q4) → (P5, Q5) → (P6, Q6) → (P7, Q1) → (P8, Q2) → (P1, Q3) → (P2, Q4) → (P3, Q5) → (P4, Q6) → (P5, Q1) → (P6, Q2) → (P7, Q3) → (P8… view at source ↗

**Figure 4.** Figure 4: Policle and Sinusoidal Spiral. Theorem 11. Let B be a point in the polar sector 0 ≤ θ ≤ π 2n of the sinusoidal spiral r n = cos(nθ) and let B′ be its radial projection onto the policle r 4 = n sin2 (nθ) 1−cos2n(nθ) . Let C be the point on this polar sector of the sinusoidal spiral such that OC = OBn . Then (34) l = 2a √ n, where l is the arc length of the sinusoidal spiral from C to P = (1, 0) and a denot… view at source ↗

discussion (0)

Forward citations

Cited by 1 Pith paper

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

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

Works this paper leans on

19 extracted references · 2 linked inside Pith · cited by 1 Pith paper

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