An Elementary Proof of a Remarkable Relation Between the Squircle and Lemniscate

Muthu Veerappan Ramalingam; Zbigniew Fiedorowicz

An Elementary Proof of a Remarkable Relation Between the Squircle and Lemniscate

T0 review · 0 major / 6 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read For every point on the lemniscate, a corresponding squircle sector determines a lemniscate arc of length $2\sqrt2$ times its area.

desk verdict A correct, honest, and nicely geometric sector-level generalization of the squircle-lemniscate relation; the elementary proof checks out, with only minor presentation gaps. read the letter →

arxiv 2411.19864 v4 pith:A4OEHJQ7 submitted 2024-11-29 math.HO math.NT

Zbigniew Fiedorowicz , Muthu Veerappan Ramalingam This is my paper · ORCID

classification math.HOmath.NT MSC 14H5033E0526A06

keywords squirclelemniscateellipticintegralspolarcoordinatessquigonometryconstantelementaryproofFermatcurve

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper proves a geometric relation behind a classical numerical coincidence: for any first-quadrant point $B$ on the lemniscate $(x^2+y^2)^2=x^2-y^2$, radial projection sends $B$ to a point $B'$ on the squircle $x^4+y^4=1$, and the lemniscate point $C$ whose distance from the origin is the square of $B$'s distance from the origin marks out an arc whose length equals $2\sqrt2$ times the area of the corresponding squircular sector. This one-parameter identity, stated as Theorem 1, reduces by symmetry to the classical fact that the whole squircle has area $\varpi\sqrt2$, where $\varpi$ is the lemniscate constant. The proof avoids elliptic integrals, hypergeometric functions, and the gamma function; it uses polar equations, the fundamental theorem of calculus, and a trigonometric identity. The old coincidence is no coincidence: the two curves are related point by point through radial projection and squaring of the radius.

What carries the argument

The engine is a pair of polar integrals made to cancel. The lemniscate $r^2=\cos(2\theta)$ gives the arc-length integrand $1/\sqrt{\cos(2\theta)}$; the squircle, derived from the parametrization $x=\sqrt{\cos s}$, $y=\sqrt{\sin s}$, has polar equation $r^2=\sqrt2/\sqrt{1+\cos^2(2\theta)}$, so a sector area carries the integrand $1/\sqrt{1+\cos^2(2\theta)}$. The radius condition $OC=OB^2$ becomes $\cos(2\beta)=\cos^2(2\alpha)$, and differentiating $l-2a\sqrt2$ with respect to $\alpha$ through the fundamental theorem of calculus yields zero once $\frac{d\beta}{d\alpha}=\frac{2\cos(2\alpha)}{\sqrt{1+\cos^2(2\alpha)}}$ is substituted in.

What would settle it

Take $B$ at polar angle $\alpha=\pi/8$ on the lemniscate and let $C$ be the first-quadrant lemniscate point with $OC=OB^2$, which lies at polar angle $\beta=\pi/6$. Compute $l$, the arc length of the lemniscate from $C$ to the positive $x$-axis point $P$, and $a$, the area of the squircle sector cut off by the ray through the radial projection $B'$, by direct numerical quadrature from the Cartesian equations. The claim is false if $l-2\sqrt2\,a$ is not zero within the quadrature error.

Watch

Extended reading notes

Core claim

The paper's Theorem 1 is the central claim. With $B$, $B'$, and $C$ as above, the identity $l-2a\sqrt2=0$ holds for every first-quadrant $B$, where $l$ is the arc length of the lemniscate from $C$ to the point where the curve meets the positive $x$-axis, and $a$ is the area of the squircle sector bounded by that axis and the ray through $B'$. The proof computes $l=\int_0^\beta \frac{d\theta}{\sqrt{\cos(2\theta)}}$ and $a=\frac{1}{\sqrt2}\int_0^\alpha \frac{d\theta}{\sqrt{1+\cos^2(2\theta)}}$ with $\cos(2\beta)=\cos^2(2\alpha)$, differentiates the difference with respect to $\alpha$, and finds the derivative is identically zero. Since the difference vanishes at $\alpha=0$, it vanishes for every admissible $B$. The paper also derives the same geometric relation through a second integral substitution, shows the two versions describe equal lemniscate arc lengths, and rewrites the relation as an identity between the lemniscate cosine and the squircle sine and cosine.

Load-bearing premise

The proof stands on the derivation of the squircle's polar equation $r^2=\sqrt2/\sqrt{1+\cos^2(2\theta)}$ from the parametrization $x=\sqrt{\cos s}$, $y=\sqrt{\sin s}$ and on the uniqueness of the first-quadrant point $C$ with $OC=OB^2$; if either gives way, the cancellation that makes $l-2a\sqrt2$ constant collapses.

Editorial extensions

If this is right

Corollary 2: the area of the whole squircle is $\varpi\sqrt2$, where $\varpi$ is the lemniscate constant.
The classical identity $\int_0^1 \frac{dx}{\sqrt{1-x^4}}=\sqrt2\int_0^1 \sqrt[4]{1-x^4}\,dx$ follows without gamma functions or the theory of elliptic integrals.
Corollary 4: for all real $t$, $\operatorname{cl}(\sqrt2\,t)=\frac{\cos_4^2(t)-\sin_4^2(t)}{\cos_4^2(t)+\sin_4^2(t)}$.
Theorem 6 supplies a second geometric matching of the same squircle sector to a different lemniscate segment, and Proposition 7 shows the two segment lengths are equal.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

A natural next step, not taken in the paper, is to test whether the same radial-projection construction matches sector areas to arc lengths for the family $x^{2n}+y^{2n}=1$ paired with the generalized lemniscates $r^n=\cos(n\theta)$.
The derivative-zero mechanism suggests a local differential identity between the two curves, not only a global integral one; checking such an identity at corresponding points might extend to other algebraically related polar curves.
Corollary 4's analytic-continuation step could probably be replaced by a purely real, octant-by-octant argument, making the squigonometric identity available without complex analysis.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Both the reader and the stress-test note are right: the paper is sound, and the central proof checks out. The genuinely new thing is Theorem 1, a sector-level generalization of the known area-arc-length relation, with an elementary proof via FTC and a derivative that vanishes. The authors are candid in Remark 5 that their elliptic-function identity (12) reduces to known lemniscate relations, so the novelty is geometric and expository, not a new island of mathematics. That is fine; the paper does exactly what it claims.

The proof of Theorem 1 is the heart, and it is clean. The polar equation of the squircle (7) follows from the parametrization, the implicit differentiation of (8) is valid on (0, pi/4) with endpoints by continuity, and the constant vanishes at alpha=0. The alternate proof in Section 4 via Siegel's substitution is a nice complement, and Proposition 7 correctly shows the two geometric configurations give the same arc length. I checked the claimed independence: the proof of Theorem 6 uses only the polar equation of the squircle and Siegel's substitution, so deriving Theorem 1 from it is not circular.

Soft spots are minor. The point P in Theorem 1 is never defined; the proof and Figure 1 imply it is (1,0), but the statement should say so. The analytic continuation in Corollary 4 is terse: for an elementary calculus paper, relying on complex analytic continuation is a bit of a stretch, though the cited result makes it correct. Also Remark 8 leaves a proof as an exercise; that's fine, but the substitution v = sqrt((1-r)/(1+r)) deserves at least a hint if the reader wants to verify it. None of these affect the correctness.

The citation pattern is honest. The paper cites Legendre, Dirichlet, Whittaker & Watson, Siegel, and the squigonometry sources. Wikipedia citations are used for standard constants; acceptable in a math.HO paper. The authors flag equivalence to known results rather than overclaiming.

Who should read this: anyone teaching or studying squigonometry, classical curve geometry, or the lemniscate. It would be a good expository paper for a journal like the American Mathematical Monthly. I would send it to a serious referee if I were editing the area. My own verdict: accept with minor revisions (define P, tighten the analytic continuation remark).

Referee Report

0 major / 6 minor

Summary. The paper proves a one-parameter generalization of the classical relation between the area of the squircle x^4+y^4=1 and the arc length of the Bernoulli lemniscate (x^2+y^2)^2=x^2-y^2. Specifically, Theorem 1 states that for a point B on the first-quadrant lemniscate, with B' its radial projection onto the squircle and C the lemniscate point satisfying OC=OB^2, the arc length of the lemniscate from C to P equals 2√2 times the area of the squircular sector OPB'. The proof derives the polar equations of both curves, expresses the relevant arc length and area as integrals, and shows by differentiation that the difference l-2a√2 has zero derivative, hence is constant, with the constant vanishing at α=0. The paper also gives an alternate proof via Siegel's substitution, connects the relation to squigonometry and lemniscate elliptic functions, and discusses the relationship to known identities.

Significance. The main result is an attractive, purely calculus-level proof of a classical and somewhat mysterious relation between two special curves. It avoids elliptic integrals and the gamma function entirely for the core theorem, and it provides a clear geometric meaning for the known identity. The proof is fully displayed and can be checked line by line; no parameters are fitted and no target identity is assumed, so there is no circularity in the main argument. The paper also honestly acknowledges, in Remark 5, that the squigonometric formulation is equivalent to existing relations, and it properly credits Legendre, Dirichlet, and Siegel. While not a breakthrough in elliptic-function theory, the paper is valuable as an exposition and could be published in a suitable mathematical journal. Its strengths are the self-contained derivation, the explicit one-parameter family of relations, and the careful separation of the direct proof from the alternate Siegel-based proof.

minor comments (6)

[Theorem 1] In the statement of Theorem 1, the point P is never defined. The proof and Figure 1 imply that P is the point (1,0) on the positive x-axis, but this should be stated explicitly in the theorem, alongside the definitions of B and B'.
[Equations (9)-(10), proof of Theorem 1] The differentiation step is written as though it holds at the endpoints, but the computation divides by cos(2α) and by sin(2β), which is only legitimate for α in the open interval (0, π/4). Since the endpoint values follow by continuity, the proof should either restrict the differentiation to the open interval and then take limits, or explicitly mention that the endpoint case is obtained by continuity.
[Proof of Theorem 6] The proof of Theorem 6 uses α as the polar angle of B, but α is not defined in the statement of Theorem 6 or at the start of its proof. This should be introduced explicitly, e.g., by writing 'let α be the polar angle of B' before the computation involving T=tan(α).
[Corollary 4] The analytic continuation argument is too terse. The lemniscate cosine cl is a meromorphic function, not an entire function, and the identity is asserted for all real t. The authors should add a brief discussion of how the poles of both sides are handled, or provide a more precise reference for the analytic continuation of the squigonometric functions as meromorphic functions.
[Remark 5] The hyperbolic lemniscate sine slh is used without definition or a precise reference. Since the paper is aimed at readers who may not be specialists in elliptic functions, a definition or a more specific citation would improve readability.
[Title and front matter] The title contains broken word breaks ('ELEMENTAR Y', 'LEMNISCA TE') and the abstract/formula at equation (1) appears garbled in the submitted text; the final version should be carefully proofread.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; the proof is self-contained and the cited identities are independent.

full rationale

The paper's central derivation is self-contained and does not reduce to its own inputs. Theorem 1 is proved by deriving the squircle polar equation (2)-(7) from the parametrization x = sqrt(cos s), y = sqrt(sin s), writing the lemniscate arc length l and squircle sector area a as explicit integrals, differentiating l - 2a*sqrt(2) with respect to alpha using relation (8), and showing the derivative is identically zero by the fundamental theorem of calculus and implicit differentiation. The constant is then evaluated at alpha = 0, where both integrals vanish. No parameter is fitted to the target quantity, and the target identity is not assumed at any point. The alternative proof in Theorem 6 uses Siegel's substitution (17)-(18) with an independent verification that R' = OD, and Proposition 7 invokes standard lemniscate function identities (Pythagorean identity and injectivity of sl) cited to external references [11], not to the authors' own prior work. The only self-citation, the YouTube video [7] by the second author, is illustrative and not load-bearing. The paper even explicitly notes that it is not engaging in circular reasoning when deriving Theorem 1 from Theorem 6, and inspection confirms that the cited identities are external and the derivation is independent. The undefined point P in the statement of Theorem 1 is a presentation gap, not a circularity, since the proof and Figure 1 make clear that P is the point (1,0) at theta = 0. Overall, no circular step is present; the result is an elementary, self-contained derivation with external, non-self-cited support for the auxiliary lemniscate function facts.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The core Theorem 1 proof is self-contained and uses no fitted numbers or new entities. Later sections import standard elliptic-function facts from the literature, which are stated and cited. No free parameters or invented entities appear.

assumptions (5)

standard math Polar coordinate formulas for sector area and arc length are valid for the C^1 curves used here.
Used in the proof of Theorem 1 for the lemniscate and squircle sectors; standard derivation.
standard math Fundamental theorem of calculus, chain rule, and implicit differentiation apply to the integrals in the proof.
Used to differentiate l - 2a√2 with respect to alpha; beta(alpha) is smooth because cos(2β) = cos^2(2α) has a unique solution in [0, π/4].
domain assumption The parametrization (11) of the first quadrant lemniscate and the polar equation (7) of the squircle are correct.
Both are derived from definitions in the paper; they are load-bearing for the integral computations.
domain assumption Lemniscate sine and cosine identities from prior literature (Pythagorean, duplication, slh relation) are correct.
Used in Proposition 7, Remark 5, and Corollary 4; standard results cited to [11] and [3].
domain assumption Analytic continuation of the identity in Corollary 4 from a finite interval to all real t is legitimate.
Relies on analyticity of cl and of cos4^2, sin4^2 as cited in [3]; if these functions are only meromorphic, the statement needs qualification at poles.

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Cite this review

Pith. "Pith review of An Elementary Proof of a Remarkable Relation Between the Squircle and Lemniscate." pith.science (2026). https://pith.science/paper/A4OEHJQ7

@misc{pith2026241119864,
  author       = {Pith},
  title        = {Pith review of: An Elementary Proof of a Remarkable Relation Between the Squircle and Lemniscate},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/A4OEHJQ7}},
  note         = {Machine review of arXiv:2411.19864}
}

abstract

It is well known that there is a somewhat mysterious relation between the area of the quartic Fermat curve $x^4+y^4=1$, aka squircle, and the arc length of the lemniscate $(x^2+y^2)^2=x^2-y^2$. The standardproof of this fact uses relations between elliptic integrals and the gamma function. In this article we generalize this result to relate areas of sectors of the squircle to arc lengths of segments of the lemniscate. We provide a geometric interpretation of this relation and an elementary proof of the relation, which only uses basic integral calculus. We also discuss an alternate version of this kind of relation, which is implicit in a calculation of Siegel.

Figures

Figures reproduced from arXiv: 2411.19864 by the authors.

**Figure 1.** Squircle and Lemniscate The relation depicted here is L = A √ 2, where A is the shaded area in the picture and L is the arc length of the bolded portion of the lemniscate. The points B and C in the picture are related by the equation OC = OB2 , where O is the origin. Due to the symmetries of the squircle and the lemniscate it suffices to check the relation for the portions of the areas and lengths lying in the first… view at source ↗

**Figure 2.** Squigonometry and the Lemniscate [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗

**Figure 3.** Another Relation The relation depicted here is L = A √ 2, where A is the shaded area in the picture and L is the arc length of the bolded portion of the lemniscate. This is exactly the same relation as depicted in [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 2 Pith papers

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

Hypergeometric Series Representations for the Perimeter of Lam\'e Superellipses
math.CA 2026-07 accept novelty 6.0 of 10

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.
Generalizations of the Squircle-Lemniscate Relation and Keplerian Dynamics
math.HO 2026-01 conditional novelty 5.0 of 10

For every n, the arc length of the sinusoidal spiral r^n = cos(nθ) is 2^{1/n} times the area of the Lamé curve x^{2n}+y^{2n}=1, with the same correspondence holding sector by sector.

Reference graph

Works this paper leans on

15 extracted references · 15 canonical work pages · cited by 2 Pith papers

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Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [1]

Ueber eine neue Methode zur Bestimmung vielfacher Integrale

Dirichlet, P. G. L., “Ueber eine neue Methode zur Bestimmung vielfacher Integrale”, 1839. In: Kronecker L, ed. G. Lejeune Dirichlet’s Werke, Cambridge Library Collection - Mathematics. Cambridge University Press; 2012:381-390./external-link-alt

work page 2012

[2] [2]

Legendre, A-M, Exercices du calcul int´ egral,Paris, Huzard-Coursier, 1811./external-link-alt

work page

[3] [3]

A Geometric Interpretation of an Infinite Product for the Lemniscate Constant

Levin, Aaron (2006). “A Geometric Interpretation of an Infinite Product for the Lemniscate Constant”. The American Mathematical Monthly. 113 (6): 510–520. /external-link-alt

work page 2006

[4] [4]

What is the area of a Squircle?

Parker, Matt (2021). “What is the area of a Squircle?”. Stand-up Maths. YouTube. /external-link-alt

work page 2021

[5] [5]

Poodiack, Robert D.; Wood, William E. (2022). Squigonometry: The Study of Imperfect Circles. Springer. ISBN 978-3031137822

work page 2022

[6] [6]

Squigonometry, hyperellipses, and supereggs

Poodiack, Robert D. (2016). “Squigonometry, hyperellipses, and supereggs”. Mathematics Magazine. 89(2):92–102. /external-link-alt

work page 2016

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Bernoulli Lemniscate and the Squircle || A remark- able Geometric fun fact!!?

Ramalingam, Muthu Veerappan (2023). “Bernoulli Lemniscate and the Squircle || A remark- able Geometric fun fact!!?”. Act of Learning. YouTube. /external-link-alt

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Siegel, Carl Ludwig (1969). Topics in Complex Function Theory – Elliptic Functions and Uniformization Theory, Volume 1, Wiley. ISBN 9780471608448

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