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)
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(2)
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(3)
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for and .
It is an algebraic surface of degree 18 with complicated terms. However, it has the simple Cartesian equation
(4)
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where is taken to mean . Cross sections through the , , or planes are therefore astroids.
The first fundamental form coefficients are
(5)
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(6)
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(7)
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the second fundamental form coefficients are
(8)
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(9)
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(10)
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The area element is
(11)
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giving the surface area as
(12)
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The volume is given by
(13)
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an exact expression for which is apparently not known.
The Gaussian curvature is
(14)
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while the mean curvature is given by a complicated expression.