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Triangular Orthobicupola


J27
J27Net

The triangular orthobicupola is Johnson solid J_(27), 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

 V=5/3sqrt(2)
(1)

and Dehn invariant

D=-24<3>_2
(2)
=-24tan^(-1)(sqrt(2)),
(3)

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.

HexagonalClosePackingClus
HexagonalClosePackingSolid

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 J_(27) (Steinhaus 1999, pp. 203-205).

Space filling

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).


See also

Cuboctahedron, Johnson Solid, Space-Filling Polyhedron, Sphere Packing

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References

Conway, J. H.; Radin, C.; and Sadun, L. "On Angles Whose Squared Trigonometric Functions Are Rational." Discr. Computat. Geom. 22, 321-332, 1999.Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, pp. 203-205, 1999.

Cite this as:

Weisstein, Eric W. "Triangular Orthobicupola." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/TriangularOrthobicupola.html

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