Simply put, the superellipse is a shape that mediates between the orthogonal and the round. That is, it is more rectilinear than an ellipse, but rounder than a rectangle. If this seems unusual, that is because it is.

The superellipse was invented/discovered in 1959 by a Danish inventor/author Piet Hein. He was comissioned by the Stockholm city-planning department to find a shape for a new city center in what is now called Sergel's Square. The problem was that the area, a rectangle, needed to be filled with some ovloid shape that nested well, and yet filled the space more efficiently than an ellipse. Piet Hein came up with the following formula for a class of shapes:

```            |  X  |n    |  Y  |n
| --- |  +  | --- |    =  1
|  a  |     |  b  |
```

Let a = b for now. As n tends from zero to infinity, the Cartesian graph of this shape warps from crossed lines, when n = 0, to a diamond when n = 1, to a circle when n = 2, finally to a square when n = infinity. Piet Hein actually examined a different set of graphs, namely, with a and b equal to the semi-axes of the rectanglular area in Stockholm. What he found, after an exhaustive search, was that when n = 2.5, the resultant shape, which he dubbed the superellipse combined the best, most aesthetically pleasing features of both the rectangle and the ellipse; exactly what the Stockholm planning committee needed! When a = b, the shape is called a supercircle.

Since Piet Hein's discovery, architects have found a multitude of uses for the superellipse. By varying parameters a, b, and n, one can create an enormous variety of possible shapes. Some have suggested ratios such as a/b = the Golden Ratio and n = e = 2.71828...!

When a superellipse is rotated on its long axis, to create a solid with a circular cross-section, the result is a superegg! This shape is, in turn, a special case of the superellipsoid with formula:

```            |  X  |n    |  Y  |n    |  Z  |n
| --- |  +  | --- |  +  | --- |  =  1
|  a  |     |  b  |     |  c  |
```

An interesting fact about supereggs: They are easily balanced on their ends with little or no effort. It is possible to balance any superegg in this way, regardless of its height or width.

All information from:
Gardner, Martin. "Piet Hein's Superellipse." Reprinted in The Collosal Book of Mathematics. New York: W.W. Norton & Co., 2001.