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Hyperbolic Octahedron


HyperbolicOctahedron

The hyperbolic octahedron is a hyperbolic version of the Euclidean octahedron, which is a special case of the astroidal ellipsoid with a=b=c=1.

It is given by the parametric equations

x=(cosucosv)^3
(1)
y=(sinucosv)^3
(2)
z=sin^3v
(3)

for u in [-pi/2,pi/2] and v in [-pi,pi].

It is an algebraic surface of degree 18 with complicated terms. However, it has the simple Cartesian equation

 x^(2/3)+y^(2/3)+z^(2/3)=1,
(4)

where c^(2/3) is taken to mean (c^2)^(1/3)=|c|^(2/3). Cross sections through the x=0, y=0, or z=0 planes are therefore astroids.

The first fundamental form coefficients are

E=9cos^2usin^2ucos^6v
(5)
F=9/4cos^5vsinvsin(4u)
(6)
G=9cos^2vsin^2v[cos^2v(cos^6u+sin^6u)+sin^2v],
(7)

the second fundamental form coefficients are

e=(24|csc(2u)cscv|cos^2ucos^3vsin^2usin^2v)/(sqrt(9-2cos(4u)cos^2v-7cos(2v)))
(8)
f=0
(9)
g=(24|csc(2u)cscv|cos^2ucosvsin^2usin^2v)/(sqrt(9-2cos(4u)cos^2v-7cos(2v))).
(10)

The area element is

 dA=9/4cos^4vcosu|sinusinv| 
 ×sqrt(9-2cos(4u)cos^2v-7cos(2v)),
(11)

giving the surface area as

 A=(17)/(12)pi.
(12)

The volume is given by

 V approx 0.359038,
(13)

an exact expression for which is apparently not known.

The Gaussian curvature is

 K=(sec^4v)/((cos^2ucos^vsin^2u+sin^2v)^2),
(14)

while the mean curvature is given by a complicated expression.


See also

Astroidal Ellipsoid, Hyperbolic Cube, Hyperbolic Dodecahedron, Hyperbolic Icosahedron, Hyperbolic Tetrahedron

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References

Gray, A. Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 396-398, 1997.Nordstrand, T. "Astroidal Ellipsoid." http://jalape.no/math/asttxt. Rivin, I. "Hyperbolic Polyhedra Graphics." http://library.wolfram.com/infocenter/Demos/4558/.Trott, M. "The Cover Image: Hyperbolic Platonic Bodies." §8.3.10 in The Mathematica GuideBook for Graphics. New York: Springer-Verlag, 2004. http://www.mathematicaguidebooks.org/.

Referenced on Wolfram|Alpha

Hyperbolic Octahedron

Cite this as:

Weisstein, Eric W. "Hyperbolic Octahedron." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/HyperbolicOctahedron.html

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