Look around you. What are the shapes you can see? The chances are that most of the shapes can be classed as either orthogonal1 or round2 - particularly with man-made items. This dichotomy pervades the modern world; few things are made that do not include right-angled corners or circles. Although your brain can readily interpret round shapes as being circles, what you are actually seeing tends to be ellipses (squashed or stretched circles).
Where a compromise exists it is frequently a mathematically inelegant hybrid3. The most common example is where the corners of a rectangle have been replaced with quarter-circles; for example, mouse mats and playing cards.
However, there is a mathematically aesthetic compromise that can vary all the way between a square and an ellipse (and beyond), using a simple formula: the superellipse.
Squarer Than the Average Circle
Superellipses are the family of curves that lie between ellipses and rectangles (or circles and squares if the axes are equal).
They form a subset of the general family of Lamé curves first studied by the French physicist Gabriel Lamé early in the 19th Century. These all have the Cartesian formula:
The vertical straight lines enclosing the x and y variables indicate the absolute value (ie, the value ignoring whether it is positive or negative; also sometimes confusingly known as the modulus operator). The constants a and b define the height and width of the curve4. The exponent, n, is the power the other terms are raised to5 and its value determines the shape of the curve.
As n approaches zero the curve degenerates to two straight crossed lines along the axes.
When n=1 you get a diamond with vertices on the axes.
When n=2 you get an ellipse (or if a and b are equal a circle).
If the value of n is increased beyond two, you have a superellipse6.
As n approaches infinity the shape approaches that of a rectangle (this you may have to take on trust unless you have excellent mathematical intuition).
It is worth noting that even though superellipses can look as though they have straight sides joined by curves, they are actually curved all the way around. Even where a segment looks straight it is actually slightly curved, with a calculable centre of curvature.
For any arc of any curve, the centre of curvature is where you would put your compass point to draw that arc. A circle has a constant centre of curvature (ie, you can draw the whole thing without moving the compass point). A straight line can be considered as having a centre of curvature infinitely far away. The patchwork curve mentioned in the quote below was a set of arcs with fixed curvatures joined to make a curve. It was considered unaesthetic because there were points where the curvature 'jumped'. Superellipses have smoothly varying curvature all the way around.
Curvier Than the Average Square
Piet Hein coined the term 'superellipse'. It was his response to a design challenge that had arisen in 1959, for the redevelopment of an open space in Stockholm, Sergel's Torg7. This area lay where two broad arterial roads met in the middle of the city, and what was wanted was to have the traffic flowing around a central area, which had fountains and (below street level) proposed restaurants and shops. Piet Hein was looking for an aesthetic shape that could be nested, and also allow for traffic flow around it.
He hit on the superellipse with an exponent of 2.5 as ideal to fit the needs of the developers. He said of his plan for Sergel's Torg:
Man is the animal that draws lines which he himself then stumbles over. In the whole pattern of civilization there have been two tendencies, one toward straight lines and rectangular patterns and one toward circular lines.
There are reasons, mechanical and psychological, for both tendencies. Things made with straight lines fit well together and save space. And we can move easily - physically or mentally - around things made with round lines.
But we are in a straitjacket, having to accept one or the other, when often some intermediate form would be better. To draw something freehand - such as the patchwork traffic circle they tried in Stockholm - will not do. It isn't fixed, isn't definite like a circle or square. You don't know what it is. It isn't aesthetically satisfying.
The superellipse solved the problem. It is neither round nor rectangular, but in between. Yet it is fixed, it is definite - it has a unity.
Although Sergel's Torg, completed in 1967, was the first major use of Piet Hein's superellipse it was not the only one, superellipses becoming ubiquitous in Scandinavian design of that period. They appeared as tables, chairs, beds, dishes, and all manner of late 1960s lifestyle designs.
If you rotate the superellipse around the major axis you create a superegg8, which due to the flattened ends (known as flachpunkts to jargon fans) is, unlike a real egg, easily able to balance on its ends. In an accurately made superegg these areas are not perfectly flat, but are so little curved that they are practically flat. Obviously the closer one approaches to a standard ellipse, with n=2, the smaller the flat area, and the greater the chance of toppling. They often look like they should be unstable, but weirdly balance on their ends. As with the superelliptical furniture supereggs enjoyed a brief vogue as an executive toy.
Humpty Dumpty Had a Great Fall
Unfortunately, Sergel's Torg like many other examples of sixties architecture is now under threat of redevelopment, and is not the vibrant centre once envisaged, but a dead urban space.
Similarly, very little superelliptical design is still available in shops - although you should still be able to get a superelliptical table by trawling the net.
References and Further Reading
- Mathematical Carnival by Martin Gardner (now unfortunately out of print).
- This Danish article appears to be a translation of Martin Gardner's Superellipse article, but with the bonus of a photo of part of Sergel's Torg, and some supereggs.
- The more mathematically minded may want to read more about superellipses at Mathworld.
- Or even more about normal, boring ellipses.