Hyperbolic Octahedron

The hyperbolic octahedron is a hyperbolic version of the Euclidean octahedron, which is a special case of the astroidal ellipsoid with .

It is given by the parametric equations

			(1)
			(2)
			(3)

for and .

It is an algebraic surface of degree 18 with complicated terms. However, it has 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)

The volume is given by

(13)

an exact expression for which is apparently not known.

The Gaussian curvature is

(14)

while the mean curvature is given by a complicated expression.