Supersphere

The supersphere is the algebraic surface that is the special case of the superellipse with . It has equation

(1)

or

(2)

for radius and exponent .

Special cases are summarized in the following table, together with their volumes.

	surface	volume with
1	regular octahedron
2	sphere
4
6	Hauser's "cube"
	cube	8

The surface area is given by

(3)

(Trott 2006, p. 301), where .

The volume enclosed is given by

			(4)
			(5)

As , the solid becomes a cube, so

(6)

as it must. This is a special case of the integral 3.2.2.2

(7)

in Prudnikov et al. (1986, p. 583). The cases and appear to be the only integers whose corresponding solids have simple moment of inertia tensors, given by

			(8)
			(9)