[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 [invent]ed/[discover]ed in 1959 by a [Denmark|Danish] [inventor]/[author] [Piet Hein]. He was [comission]ed 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 [efficient]ly than an [ellipse]. [Piet Hein] came up with the following [formula] for a class of [shape]s:

            |  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 [graph]s, 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, [architect]s 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 [rotate]d 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 [superegg]s: They are [easy|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.

For more [information], see [Martin Gardner|Gardner's] bibliographic entries:

J. Allard, "Note on [Square]s and [Cube]s," Mathematics Magazine, Vol. 37, September 1964, p. 210-14.
N.T. Gridgeman, "Lame Ovals," [Mathematical Gazette], Vol. 54, February 1970, pp. 31-37.
A. Chamberlin, 'King of Supershape," Esquire, January 1967, p. 112ff.
J. Hicks, "[Piet Hein], Bestrides [Art] and [Science]," [Life Magazine|Life], Octover 14, 1966, pp. 55-66.

superellipse (thing)