How To Really Use The Superellipse For Elegant Designs

The term “Superellipse” is not a common occurrence in our day-to-day lives. “Straight lines” and “curves”, on the other hand, are much more common. If you think about it, we write on paper that has straight lines as edges using pens that have curved surfaces as writing tips. We play football with balls that have circular cross-sections on rectangular pitches. We live in buildings that have straight contours and look through rectangular windows at the circular moon. I could keep going, but you get the picture.

We perceive the world with a distinction between straight lines and curves. What is interesting about our experience is the fact that we perceive the marriage of straight lines and curves as “beauty”. And beauty is a word that goes hand-in-hand with the word “design”.

Where does the superellipse fit into all of this? Before we can answer that question, we need to dive into the mathematics and geometry of Lamé curves.

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What is a Lamé Curve?

A French mathematician named Gabriel Lamé discovered a unique set of closed curves in the 19th century. These family of curves were named after him and they take the following mathematical form in the cartesian coordinate system:

|x/a|^n+|y/b|^n

Here ’n’, ‘a’, and ‘b’ are positive real numbers, and the vertical bars represent the absolute value of the terms they contain. ‘x’ and ‘y’ represent the set of all points on the Lamé curve that satisfy the above equation.

When 0 < n < 1, the Lamé curve appears to be a four-pointed star with concave edges. Below, you can see a special case that occurs at n = ½ and a = b, which is known as an Astroid. When n = 1, the Lamé curve becomes a parallelogram. Below, you can see a special case that occurs when a = b (rhombus).

When 1 < n < 2, the Lamé curve appears to be a rhombus/parallelogram with convex edges (subellipse). When n = 2, the Lamé curve transforms into an ellipse. Above, you can see a special case where the Lamé curve is a circle for a=b. Finally, when n > 2, the Lamé curve becomes what Danish polymath and scientist Piet Hein named as “the superellipse”.

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Piet Hein’s Superellipse

In1959, city planning officials in the Swedish city of Stockholm announced a design competition for the roundabout that was to be constructed through the town square (known as Sergels Torg).

The designers faced the challenge of incorporating both curves and straight lines into their designs. All of the solutions until a certain point were either traffic-friendly and looked sub-par or looked good and impeded traffic.

This was until Piet Hein came up with a brilliant solution. He famously used the concept of Lame curves to solve the challenge. Via trial and error and the help of a computer, he landed on ‘n = 2.5’ and ‘a/b = 6/5’ to create a superellipse design that won the competition.

As a result, the roundabout you see in the title image of this essay became a reality. After this event, the superellipse became quite famous in the design world in both Scandinavia and Europe.

The Influence of the Superellipse on Design

Following his successful outing in Stockholm, Piet Hein started getting consulting requests from industries in Denmark, Sweden, Norway, and Finland. He became famous for his solutions to orthogonal-versus-circular problems.

Not only Piet Hein, but other famous designers of the time also started adopting this style of beautifying generally straight and mundane products. The influence of the superellipse ranged from furniture and lamps to silverware and textile.

Piet Hein extended the concept to three dimensions. One interesting three-dimensional widget he came up with was the superegg, which was an affine concept derived from the superellipse. The specialty of the superegg was that it was stable on either polar end, that is, it could perfectly balance itself on its ends. It looked elegant and impressive.

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Final Remarks

In Piet Hein’s own words:

“The superellipse has the same convincing unity as the circle and ellipse, but it is less obvious and less banal.”

— Piet Hein

When it comes to mathematical elegance and beauty, the superellipse receives stiff competition from notions such as the golden ratio, the silver ratio, fractals, etc.

Is the superellipse the unity of mathematical beauty? In my opinion, it is not. But it certainly has its place in the world of mathematical elegance and beauty!

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Reference: Piet Hein and Martin Gardner.

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Further reading that might interest you: Logarithms: The Long Forgotten Story Of Scientific Progress and A Technical Investigation Into The Rule Of 72.

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