**1.** Supercube is sometimes used as a synonym for hypercube or tesseract, that is to say, a cube extended into the fourth dimension.

**2.** A (three dimensional) shape generated by replacing the exponent ^{2} in the equation *x*^{2} + y^{2} + z^{2} = r^{2}. with any other number.

*x*^{2} + y^{2} + z^{2} = r^{2}. is the formula for a sphere, but when you replace the exponents with any other number, you will get something like a cube, although it will usually have rounded corners.

If the exponent is changed to a number higher than two, the shape will approach a cube (with rounded corners); the higher the number, the closer to a cube the resulting shape will be. If the exponent is infinity, you will get a perfect cube.

If the exponent is changed to a number lower than two you get various things; 1 will get you an octahedron, 0.5 will get you a cube with concave sides.

There’s a lot more to supercubes, but I am not math-literate enough to understand more than this.

http://www.melbpc.org.au/pcupdate/9605/9605article8.htm

and

http://www.melbpc.org.au/pcupdate/9903/9903article4.htm

Have both pictures and more info.