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A polyhedron compound of two cubes. The compound at left (also called the double cube) is obtained by allowing two cubes
to share opposite polyhedron vertices, then
rotating one a sixth of a turn about a 
axis (Holden 1991, p. 34). The compound at right combines
two cubes, one rotated 
with respect to the other along a 
axis, producing an octagrammic prism.
The 
compound appears twice (in the lower left as a beveled wireframe and in the lower
center as a solid) as polyhedral "stars" in M. C. Escher's 1948
wood engraving "Stars" (Forty 2003, Plate 43).
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The left illustration above shows an origami cube 
2-compound (Brill 1996, pp. 90-92).
The right illustration shows the net of one pyramid of the compound. Each pyramidal
portion is composed of two doms (1-2 right triangles) and
one isosceles right triangle. If the original cube has edge lengths 1, then edge
lengths of the net are given by
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(1)
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(2)
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(3)
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(4)
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The surface area of this compound is
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(5)
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compared to 
for each of the two original cubes. Rather surprisingly, the surface area of the
compound is therefore a rational number.
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The solid common to the two cubes in the 
compound is a hexagonal
dipyramid (left figure), and the convex hull is
an elongated hexagonal dipyramid (right figure).
If a second cube is rotated about a 
axis with respect to a fixed cube, then the edges indicated
in black above remain intersecting throughout the entire 1/3 turn. The 
-position of the intersection as a function of rotation angle
is given by the complicated expression
![x=(12costheta-3cos(2theta)-sqrt(3)[4sintheta+sin(2theta)])/(2[6+2costheta+cos(2theta)-2sqrt(3)(costheta-1)sintheta]).](/images/equations/Cube2-Compound/NumberedEquation2.svg) |
(6)
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