History of the “Squircle”

Architect Peter Panholzer coined the term “squircle” in the summer of 1966 while working for Gerald Robinson. Robinson had seen a Scientific American article on the superellipse shape popularized by Piet Hein and suggested Panholzer use the shape in a project.

Piet Hein used the term superellipse for a compromise between an ellipse and a rectangle, and the term “supercircle” for the special case of axes of equal length. While Piet Hein popularized the superellipse shape, the discovery of the shape goes back to Gabriel Lamé in 1818.

Squircle with p = 3.162034

You can find more on the superellipse and squircle by following these links, but essentially the idea is to take the equation for an ellipse or circle and replace the exponent 2 with a larger exponent. The larger the exponent is, the closer the superellipse is to being a rectangle, and the closer the supercircle/squircle is to being a square.

Panholzer contacted me in response to my article on squircles. He gives several pieces of evidence to support his claim to have been the first to use the term. One is a letter from his employer at the time, Gerald Robinson. He also cites these links. [However, see Andrew Dalke’s comment below.]

Optimal exponent

As mentioned above, squircles and more generally superellipses, involve an exponent p. The case p = 2 gives a circle. As p goes to infinity, the squircle converges to a square. As p goes to 0, you get a star-shape as shown here. As noted in that same post, Apple uses p = 4 in some designs. The Sergels Torg fountain in Stockholm is a superellipse with p = 2.5. Gerald Robinson designed a parking garage using a superellipse with p = e = 2.71828.

Panholzer experimented with various exponents [1] and decided that the optimal value of p would be the one for which the squircle has an area half way between the circle and corresponding square. This would create visual interest, leaving the viewer undecided whether the shape is closer to a circle or square.

The area of the portion of the unit circle contained in the first quadrant is π/4, and so we want to find the exponent p such that the area of the squircle in the first quadrant is (1 + π/4)/2. This means we need to solve

\int_0^1 (1 - x^p)^{1/p}\, dx = \frac{\Gamma\left(\frac{p+1}{p}\right)^2}{\Gamma\left(\frac{p+2}{p} \right )} = \frac{1}{2} + \frac{\pi}{8}

We can solve this numerically [2] to find p = 3.1620. It would be a nice coincidence if the solution were π, but it’s not quite.

Sometime around 1966 Panholzer had a conference table made in the shape of a squircle with this exponent.

Computing

I asked Panholzer how he created his squircles, and whether he had access to a computer in 1966. He did use a computer to find the optimal value of p; his brother in law, Hans Thurow, had access to a computer at McPhar Geophysics in Toronto. But he drew the plots by hand.

There was no plotter around at that time, but I used transparent vellum over graph paper and my architectural drawing skills with “French curves” to draw 15 squircles from p=2.6 (obviously “circlish”) to p=4.0 (obviously “squarish”).

Related posts

[1] The 15 plots mentioned in the quote at the end came first. A survey found that people preferred the curve corresponding to p around 3.1. Later the solution to the equation for the area to be half way between that of a circle and a square produced a similar value.

[2] Here are a couple lines of Mathematica code to find p.

    f[p_] := Gamma[1 + 1/p]^2/Gamma[1 + 2/p]
    FindRoot[f[p] - (1 + Pi/4)/2, {p, 4}]

The 4 in the final argument to FindRoot is just a suggested starting point for the search.

11 thoughts on “History of the “Squircle”

  1. Rick: Good catch. My first thought was that it was a little more than pi, but it’s much closer to sqrt(10).

    The exact value is between 3.162037 and 3.162040, and so it’s a little less than sqrt(10) = 3.1622…

  2. A Google Scholar search for “squircle” found it being used in 1953 at https://www.osti.gov/servlets/purl/12468785 for a different shape. The Wilkins effect is the “flux increase at the ends of slugs separated by a non-absorbing spacer” says https://www.osti.gov/servlets/purl/4313290 , in the context of nuclear fuel rods. It appears that if you have four rods in a 2×2 formation, then the interior part was called a squircle. There’s a Google Books snippet view of “Heat Transfer 1970: Papers Presented at the Fourth International Heat Transfer Conference. Paris-Versailles, 1970” at https://books.google.se/books?id=7KS2AAAAIAAJ&q=squircle&dq=squircle&hl=en&sa=X&ved=0ahUKEwiQk533wrXhAhVnyaYKHabLCGEQ6AEIUTAJ which shows this image.

    An archive.org search shows another and still different use of squircle in Struble’s 1962 book “Nonlinear differential equations”, page 39.

    Quoting it: Define the double norm of a vector x-bar as the scalar quantity ||x-bar|| = max(|x1|, |x2|, … |x_n|). Show that in two-space, the locus ||x-bar|| = 1 is a square which circumscribes the circle |x-bar| = 1. Thus the double norm equivalent of a Euclidean circle is also a square. The author is indebted to one of his students for suggesting the picturesque name squircle, a “square circle”. On the other hand, the orientation of the “single-norm” square |x-bar| = 1 suggests a diamond, and so one might coin for it the name dircle, a “diamond circle.”

  3. I hadn’t heard the term “squircle” before, but I like it! As you know, each value of p induces a metric defined by using the p-norm. You can therefore define pi(p) to be the ratio of the circumference of the unit squircle to its diameter, where both lengths are measuredin terms of the L^p metric. When p=2, pi(p) = 3.14159…. As p->infinity, pi(p) -> 4. For the Panholzer squircle with p=3.1620, pi(p) = 3.284195. For a graph of pi(p) as well as a longer discussion, see “The value of pi depends on how you measure distance” at https://blogs.sas.com/content/iml/2019/03/13/pi-in-lp-metric.html

  4. Peter Panholzer

    I was unaware of the (unexplained) 1953 use of “squircle” mentioned by Andrew Dalke (above).
    Nevertheless, John Cook’s story in this blog relates to the first use of “squircle” in the context of Lamé’s and Piet Hein’s work – and to subsequent innumerable related uses of the word in scientific publications ever since, particularly in the new millennium.

    While the 1953 and 1962 citations were never adopted later, my dubbing Piet Hein’s supercircle as “squircle”, was.

    I just learned from Heather Cooper that it was her design company heathercooper.com that ordered the squircle boardroom table from architect Gerald Robinson and that it was built from birch by Steven Harris in Toronto in 1966.

  5. These are a pretty popular widget in “shadertoy” circles when it comes to producing signed distance fields.

    e.g. You can use them to tweak the distance field for a torus into one for a nice tire shape, by just picking a norm like L8 instead of L2 for one of the length calculations for the implicit surface of the torus.

    e.g. you can replace

    float torus( vec3 p, vec2 t ) return length( vec2(length(p.xz)-t.x,p.y) )-t.y; }

    with

    float tire( vec3 p, vec2 t ) { return length8( vec2(length(p.xz)-t.x,p.y) )-t.y; }

    This idea is also commonly used to build the bulk of the “uber primitives” people like to use in this space for playing around:

    https://www.shadertoy.com/view/MsVGWG

  6. Peter Panholzer

    I always considered any supercircle following the equation
    |x|^p + |y|^p = 1
    as a squircle.
    p = 3.162 is John D. Cook’s solution for the unique squircle for which is undecidable if it is visually closer to a circle or closer to a square, e.g. whose area is halfway between a circle and a square.
    I would call this unique solution a supersquircle.
    Another interesting is to find the unique solution for
    x|^p + |y|^p = 1
    where the area is maximized in relation to the perimeter
    (or the perimeter minimized in relation to the area)
    perimeter / area for a unit square is 4 (p > > > ∞)
    perimeter / area for a unit cercle is 2 (p = 2)
    for which p is perimater/area a minimum?
    This is reputedly a very tricky computation.
    in 1998 mathforum participants arrived at p = 4.39366
    (Ronald Bruck and Brian Skinner independently)
    The graph for p vs perimeter/area looks like this:
    http://peter.panholzer.org/SQUIRCLE_perimeter.gif
    perimeter/area approaches asymptotically 2 as p travels from 4.39365 to infinity (the square).
    John, a plot of this squircle with p = 4.39366 would be welcome.

  7. Peter Panholzer

    MikeD
    Thank you for the squircle plot p = 4.39365.
    It has indeed the mesmerizing quality of largest area versus perimeter.
    Perfectly suited for an 8 people dining table.

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