The super eggThe super egg was designed by the Danish poet and scientist Piet Hein. in 1965. It is obtained by rotating a super ellipse around one of its axes.What is the Super Egg? Lets first look at Super ellipses. 
The Squircle and other Super ellipsesA super ellipse x/a^{e}+y/b^{e} = 1 around an xaxes. The number e is called the exponent of the super ellipse. For e=2, one obtains the usual ellipse. Super ellipses have been introduced by Gabriel Lamé. For exponents e smaller than 2, the super ellipse is called a hypoellipse, an example being e=2/3, the asteroid. For exponents e larger than 2, the super ellipse is called a hyperellipse an example being e=5/2, the Piet Hein ellipse. For a=b=1, we get "super circles". For e=4, in particular, one gets the squircle. Here is how one can plot a super ellipse with Mathematica:a=5;b=6;e=5/2; ContourPlot[Abs[x/a]^e+Abs[y/b]^e==1, {x,7,7}, {y,7,7}]For more information on super ellipses, consult 2dcurves.com or Wikipedia. 
The Super egg and other SupersA super ellipsoid is a surface with super ellipses as traces. It is defined implicitly as(x^e + y^e)^f/e + z^f = 1 .It becomes the usual ellipsoid for e=f=2. You see that the xtrace as well as the ytraces are super ellipses with parameter f. The ztrace is a super ellipse with parameter e. Here is how we can draw a super ellipsoid with e=6/5 and f=5/2: e=6/5; f=5/2; ContourPlot3D[ (Abs[x]^e + Abs[y]^e)^(f/e) + Abs[z]^f == 1, {x,2,2},{y,2,2},{z,2,2}]Of course, we can always deform and look more generally at surfaces of the type (x/a^e + y/b^e)^f/e + z/c^f = 1 .These are the general super ellipsoids. For e=5/2,f=2, a/b = 6/5 and c=1 the super ellipsoid is called the super egg e=5/2; f=5/2; a=6/5; b=1; c=1; ContourPlot3D[ (Abs[x/a]^e + Abs[y/b]^e)^(f/e) + Abs[z/c]^f == 1, {x,2,2},{y,2,2},{z,2,2}] 
Super quadricsRelated to supper ellipsoids are super quadrics:x/a^e +y/b^f + z/c^g = 1But the class of supper ellipsoids and super quadrics are different. The sets have intersections although Super ellipsoids with e=f are super quadrics with e=f. The super egg especially is a super quadric. Here is how one can draw super quadrics: e=6/5; f=5/2; g=3; S = ContourPlot3D[ Abs[x]^e + Abs[y]^f + Abs[z]^g == 1, {x,2,2},{y,2,2},{z,2,2}] 

Is it useful?It can be just beautiful. It can be seen as an art object. But the superegg is also sold as an ice cube for drinks. Why not just take a sphere? Well, ice cubes aren't cubes neither. And also, it would just not be so super any more. And sometimes, you just need a pickup line: "Do you know that the icecube in your drink is a very special surface? By the way, ..."? You get the idea. 