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Cube 10-Compound


Cube10-Compound

A cube 10-compound by be obtained by beginning with an initial cube and rotating it by an angle theta=sin^(-1)(sqrt(3/8)) about the (1,1,1) axis, then adding a second cube obtained by rotating the first by an angle 2pi/5 about the (0,1,phi) axis, where phi is the golden ratio.

It is implemented in the Wolfram Language as PolyhedronData["CubeTenCompound"].

Cube10-CompoundConstruction

The angle theta places corresponding faces of first two cubes in a symmetrical position relative to one another, and makes each of these faces cut the other in an isosceles right triangle. The remaining eight cubes of the compound are then generating by adding four more pairs of cubes rotated by angles -2npi/5 about the axis (1,phi,0) (the same rotations used to construct the cube 5-compound) for n=1, 2, 3, 4.

Cube10-CompoundNet

A net for constructing the solid is illustrated above. The edge lengths are given by

s_1=1/(31)sqrt(1/(10)(10529-5252sqrt(2)-2566sqrt(5)+834sqrt(10)))
(1)
s_2=1/2sqrt(3/4(46-28sqrt(2)-sqrt(5(489-340sqrt(2)))))
(2)
s_3=1/2sqrt((75)/2-26sqrt(2)+11sqrt(5)-8sqrt(10))
(3)
s_4=1/2(1+sqrt(2)-sqrt(5))
(4)
s_5=(961x^8-9322390x^6+96805765x^4-251177850x^2+3294225)_5
(5)
s_6=(1296x^8-12566016x^6+37141816x^4-7875936x^2+136161)_5
(6)
s_7=1/2sqrt(9693-(13693)/2sqrt(2)-(6121)/2sqrt(10)+(8647)/2sqrt(5))
(7)
s_8=sqrt(54+37sqrt(2)-(47)/2sqrt(5)-17sqrt(10))
(8)
s_9=3/2sqrt(1/(10)(39-22sqrt(2)-sqrt(5(97-60sqrt(2)))))
(9)
s_(10)=1/2sqrt((51)/2-4sqrt(2)+5/2sqrt(5)-8sqrt(10))
(10)
s_(11)=1/(124)(-47+122sqrt(2)+13sqrt(5)-41sqrt(10))
(11)
s_(12)=1/6(-1-4sqrt(2)+sqrt(5)+2sqrt(10))
(12)
s_(13)=1/4(4-sqrt(2)-2sqrt(5)+sqrt(10)),
(13)

where (P(z))_n indicated a polynomial root.

The surface area of the solid is

S=1/(62)(18901sqrt(10)+30155sqrt(2)-12410sqrt(5)-74030)
(14)
 approx 10.26.
(15)

See also

Cube, Cube 2-Compound, Cube 3-Compound, Cube 4-Compound, Cube 5-Compound, Cube 6-Compound, Cube 20-Compound, Octahedron 10-Compound, Polyhedron Compound

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References

Hart, G. "Cube 10-Compound A." http://www.georgehart.com/virtual-polyhedra/vrml/cubes_A5_D3_a.wrl.Verheyen, H. F. Symmetry Orbits. Boston, MA: Birkhäuser, 2007.

Cite this as:

Weisstein, Eric W. "Cube 10-Compound." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Cube10-Compound.html

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