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

This paper provides an elementary proof using basic integral calculus and a geometric interpretation for a generalized relation connecting the areas of sectors of the squircle ( $x^4+y^4=1$ ) to the arc lengths of segments of the lemniscate, extending the known connection between their total area and length without relying on elliptic integrals or the gamma function.

Original authors: Zbigniew Fiedorowicz, Muthu Veerappan Ramalingam

Published 2026-06-03

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Zbigniew Fiedorowicz, Muthu Veerappan Ramalingam

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Big Picture: Two Strange Shapes and a Secret Connection

Imagine you have two very unusual shapes drawn on a piece of paper.

The Squircle: This is a "square circle." It looks like a circle that has been gently squashed into a square shape. Mathematically, it's defined by the equation $x^4 + y^4 = 1$ .
The Lemniscate: This looks like a figure-eight or a twisted infinity symbol ( $\infty$ ). It's defined by the equation $(x^2 + y^2)^2 = x^2 - y^2$ .

For a long time, mathematicians knew there was a mysterious link between these two shapes. Specifically, they knew that the total area inside the squircle and the total length of the lemniscate were related by a specific number: $\sqrt{2}$ (roughly 1.414).

However, the old proofs for this were like trying to solve a puzzle using a sledgehammer. They required advanced, complex math (elliptic integrals and gamma functions) that only experts could understand.

The authors of this paper, Zbigniew Fiedorowicz and Muthu Veerappan Ramalingam, have found a "simple" way to prove this. They used basic calculus (the kind taught in high school or early college) to show that this relationship isn't just about the total size of the shapes, but about how they grow together, piece by piece.

The Story of the Two Runners

To understand the proof, imagine two runners on two different tracks.

Runner A (The Squircle Runner): This runner is on the "Squircle" track. They are running in a special way called "Keplerian motion." Imagine a planet orbiting a star; as it gets closer to the star, it speeds up, and as it gets farther away, it slows down, but it sweeps out equal areas in equal amounts of time. Our runner does the same thing: they sweep out a "pie slice" of area at a constant rate.
Runner B (The Lemniscate Runner): This runner is on the "Lemniscate" (figure-eight) track. They are running at a constant speed (uniform motion), just like a car on a highway.

The Magic Discovery:
The paper proves that if you start both runners at the same time, there is a perfect synchronization between them.

If Runner A sweeps out a specific amount of area on the squircle, Runner B will have traveled a specific distance along the lemniscate. The relationship is:

The distance Runner B travels is exactly $\sqrt{2}$ times the area Runner A sweeps out.

It's as if the "area" on one shape is secretly the same thing as "distance" on the other shape, just scaled by a factor of $\sqrt{2}$ .

The "Projection" Trick

How do they prove this? They use a geometric trick involving a "shadow" or a "projection."

Pick a point on the Squircle.
Draw a line from the center of the shape through that point until it hits the Lemniscate.
The paper shows that if you look at the distance from the center to the point on the Lemniscate, and square it, it relates perfectly to the position of the point on the Squircle.

The authors show that if you take the area of the "pie slice" on the squircle and multiply it by $\sqrt{2}$ , it equals the length of the curve on the lemniscate, provided you measure the lemniscate curve starting from a specific point determined by the squircle's position.

Why is this a "Big Deal"?

Usually, calculating the area of a squircle or the length of a lemniscate involves integrals that are impossible to solve with simple formulas. You usually need a computer or very advanced math to get a number.

The beauty of this paper is that it doesn't just give you a number; it gives you a geometric story. It says, "You don't need to know the whole shape to understand the relationship. Just watch how these two runners move, and you'll see that their movements are locked together by this $\sqrt{2}$ rule."

The "Double Cover" Analogy

The paper also mentions a fascinating detail about how these shapes connect.

Imagine the Lemniscate is a loop-de-loop track. The Squircle is a simpler track. The paper suggests that if you map the Squircle onto the Lemniscate, the Squircle has to go around twice to make the Lemniscate go around once.

Think of it like a gear system:

The Squircle is a small gear with 2 teeth.
The Lemniscate is a large gear with 1 tooth.
As the small gear turns once, the large gear turns half a time. But because of the way the math works, the "runner" on the Lemniscate actually completes a full lap while the "sweeper" on the Squircle does two laps.

Summary of the Results

The Main Theorem: There is a direct, point-by-point link between the area of a sector on the squircle and the arc length on the lemniscate.
The Formula: If $A$ is the area on the squircle and $L$ is the corresponding length on the lemniscate, then $L = A \times \sqrt{2}$ .
The Method: They proved this using only basic calculus (differentiation and integration), avoiding the heavy machinery of advanced analysis.
The "Physics" View: They interpret this as a connection between two types of motion: "Keplerian motion" (sweeping area) on the squircle and "Uniform motion" (constant speed) on the lemniscate.

In short, the paper takes a complex, mysterious mathematical fact and explains it using a simple, visual story of two runners on different tracks who are perfectly synchronized by a factor of $\sqrt{2}$ .

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

Problem Statement
The paper addresses a known, yet somewhat mysterious, relationship between the geometry of two specific planar curves: the lemniscate, defined by $(x^2 + y^2)^2 = x^2 - y^2$ , and the squircle (a specific case of a Lamé curve), defined by $x^4 + y^4 = 1$ . Historically, it was established by Legendre and Dirichlet that the total arc length of the lemniscate is $\sqrt{2}$ times the total area enclosed by the squircle. While standard proofs rely on the theory of elliptic integrals and the gamma function, this paper seeks to generalize this global relationship to a local one: relating the arc length of a specific sector of the lemniscate to the area of a corresponding radial sector of the squircle. Furthermore, the authors aim to provide an elementary proof of this relation using only basic integral calculus, avoiding the machinery of elliptic functions.

Methodology
The authors employ a geometric and analytic approach centered on polar coordinates and the fundamental theorem of calculus.

Geometric Setup: The study focuses on the first quadrant. Let $B$ be a point on the lemniscate and $B'$ its radial projection onto the squircle. A corresponding point $C$ is defined on the lemniscate such that its radial distance $OC$ is the square of the radial distance $OB $(i.e.,$ OC = OB^2$).
Integral Formulation: The authors express the arc length $l$ $l$ of the lemniscate from the origin to $C$ $C$ and the area $a$ $a$ of the squircular sector defined by $B'$ $B^{'}$ as definite integrals involving the polar angle $\theta$ $θ$ .
- The lemniscate arc length integral involves $\frac{1}{\sqrt{\cos(2\theta)}}$ .
- The squircle area integral involves $\frac{1}{\sqrt{1 + \cos^2(2\theta)}}$ .
Differentiation Strategy: Instead of evaluating the integrals directly (which yield non-elementary antiderivatives), the authors define a function $f(\alpha) = l - \frac{2a}{\sqrt{2}}$ , where $\alpha$ is the angular parameter of point $B$ . They differentiate this expression with respect to $\alpha$ . By applying the chain rule and the implicit relationship between the angles of points $B$ and $C$ (derived from $OC = OB^2$ ), they demonstrate that the derivative is identically zero.
Boundary Condition: Since the derivative is zero, the function is constant. By evaluating the limit as the points approach the origin (or the starting point $(1,0)$ ), the constant is shown to be zero, proving $l = \frac{2a}{\sqrt{2}}$ .
Analytic Interpretation: The geometric results are reinterpreted in terms of "squigonometric" and "lemniscate" analogs of trigonometric functions. The authors relate the standard lemniscate cosine ($cl$) and the $p$ -norm trigonometric functions ( $\cos_4, \sin_4$ ) to derive an identity linking the two systems.
Alternative Proof via Siegel: The paper also presents an alternative proof based on a calculation by Siegel involving a substitution between integrals of the form $\int \frac{dr}{\sqrt{1-r^4}}$ and $\int \frac{dt}{\sqrt{1+t^4}}$ . This connects the lemniscate arc length to the area of the squircle via a different geometric configuration involving a point $D$ on the lemniscate.

Key Contributions and Results

Theorem 1 (Main Result): The authors prove that for a point $B$ $B$ on the lemniscate and its radial projection $B'$ $B^{'}$ on the squircle, if $C$ $C$ is a point on the lemniscate such that $OC = OB^2$ $O C = O B^{2}$ , then the arc length of the lemniscate from the origin to $C$ $C$ is exactly $\sqrt{2}$ $2$ times the area of the squircular sector defined by $B'$ $B^{'}$ .
- Mathematically: $l = \frac{2a}{\sqrt{2}}$ .
Corollary 2 (Global Relation): By integrating over the entire curves, the authors recover the known result that the total area of the squircle is $\varpi\sqrt{2}$ , where $\varpi$ is the lemniscate constant (the arc length of one lobe of the lemniscate).
Elementary Proof: The paper provides a proof that relies solely on basic calculus (differentiation under the integral sign and the fundamental theorem of calculus), bypassing the need for advanced elliptic integral theory.
Analytic Identity (Theorem 9): The geometric relation is translated into an analytic identity relating the lemniscate cosine function $cl$ and the $p=4$ trigonometric functions:
$cl(\sqrt{2}t) = \frac{\cos_4^2(t) - \sin_4^2(t)}{\cos_4^2(t) + \sin_4^2(t)}$
Physical Interpretation: The authors interpret the result physically: a particle moving with "Keplerian motion" (constant areal velocity) around the squircle corresponds to a point moving with uniform speed ( $\sqrt{2}$ ) along the lemniscate. This establishes a double branched cover mapping between the two curves.
Siegel's Connection: The paper demonstrates that Siegel's calculation regarding the doubling of the lemniscate arc provides an alternative, independent proof of the main theorem, linking the area of the squircle to the arc length of the lemniscate through a specific algebraic substitution.

Significance
The paper claims significance primarily in its pedagogical and unifying value. It demystifies a relationship between two distinct geometric objects (the squircle and the lemniscate) that is usually treated through the heavy machinery of elliptic functions. By providing an elementary proof, the authors make the result accessible to those with only a background in basic calculus.

Furthermore, the work highlights a deeper structural connection between different "analog" trigonometric systems. It suggests that the relationship between arc length and area, which is trivial for the circle (where arc length is twice the area of the sector in radians), generalizes in a specific, non-trivial way to the lemniscate and squircle pair. The paper notes that this specific geometric relation was previously discovered numerically by the second author and is implicitly present in Siegel's work on Fagnano's theorem, but this paper offers the first explicit elementary derivation and geometric interpretation of the local sector-to-sector correspondence.