Hyperbolic Octahedron

The hyperbolic octahedron is the special case of the astroidal ellipsoid with . It is a surface in Euclidean space. The name does not denote a hyperbolic polyhedron in hyperbolic space, whose faces lie in hyperbolic planes (Virto 2010, pp. 3-4).

It is given by the parametric equations

			(1)
			(2)
			(3)

for and .

It is an algebraic surface of degree 18 with the simple Cartesian equation

(4)

where is taken to mean . Cross sections through the , , or planes are therefore astroids.

The first fundamental form coefficients are

			(5)
			(6)
			(7)

the second fundamental form coefficients are

			(8)
			(9)
			(10)

The area element is

(11)

giving the surface area as

(12)

For , the cross section at height is an astroid with scale . Using the area of an astroid of scale gives the volume

			(13)
			(14)
			(15)
			(16)

where is the beta function.

On regular patches, the Gaussian curvature is

(17)

while the mean curvature is given by a complicated expression.