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The triangular orthobicupola is Johnson solid 
, consisting of eight equilateral
triangles and six squares. If a triangular orthobicupola
is oriented with triangles on top and bottom, the two
halves may be rotated one sixth of a turn with respect to each other to obtain the
cuboctahedron.
The triangular orthobicupola has volume
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(1)
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and Dehn invariant
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(2)
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(3)
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where the first expression uses the basis of Conway et al. (1999). It can be dissected into the cuboctahedron, from which it differs only by the relative rotation of the top and bottom cupolas.
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In hexagonal close packing, layers of spheres are packed so that spheres
in alternating layers overlie one another. As in cubic
close packing, each sphere is surrounded by 12 other
spheres. Taking a collection of 13 such spheres
gives the cluster illustrated above. Connecting the centers of the external 12 spheres gives 
(Steinhaus 1999, pp. 203-205).
While it is not a space-filling polyhedron, it fills space when combined with octahedra, as illustrated above (photo courtesy of Ed Pegg, Jr., pers. comm., Sept. 23, 2004).
