A cuboctahedron, also called the heptaparallelohedron or dymaxion (the latter according to Buckminster Fuller; Rawles 1997), is Archimedean solid with faces . It is one of the two convex quasiregular polyhedra. It is uniform polyhedron and Wenninger model . It has Schläfli symbol and Wythoff symbol .

A cuboctahedron appears in the lower left as one of the polyhedral "stars" in M. C. Escher's 1948 wood engraving "Stars" (Forty 2003, Plate 43), as well is in the mezzotint "Crystal" (Bool et al. 1982, p. 293).

It is implemented in Mathematica as PolyhedronData["Cuboctahedron"].

It is shown above in a number of symmetric projections.

The dual polyhedron is the rhombic dodecahedron. The cuboctahedron has the octahedral group of symmetries. According to Heron, Archimedes ascribed the cuboctahedron to Plato (Heath 1981; Coxeter 1973, p. 30). The polyhedron vertices of a cuboctahedron with polyhedron edge length of are , , and .

The mineral argentite () forms cuboctahedral crystals (Steinhaus 1999, p. 203).

In the Season 2 Star Trek episode "By Any Other Name" (1968), aliens known as Kelvins reduce crewmembers Shae and Yeoman Thompson to two small gray cuboctahedron which are purported to contain their essences. Rojan, the Kelvin leader, then crushes Thompson's polyhedron as a warning to Captain Kirk (William Shatner), thus killing her, but restores Shae to human form.

The inradius of the dual, midradius of the solid and dual, and circumradius of the solid for are

			(1)
			(2)
			(3)

The distances from the center of the solid to the centroids of the triangular and square faces are

			(4)
			(5)

The dihedral angle between triangular and square faces is

			(6)
			(7)

The surface area and volume are

			(8)
			(9)

Faceted versions of the cuboctahedron include the cubohemioctahedron and octahemioctahedron.

The solid common to both the cube and octahedron (left figure) in a cube-octahedron compound is a cuboctahedron (right figure; Ball and Coxeter 1987).

The cuboctahedron can be inscribed in the rhombic dodecahedron (left figure; Steinhaus 1999, p. 206). The centers of the square faces determine an octahedron (right figure; Ball and Coxeter 1987, p. 143).

Wenninger (1989) lists four of the possible stellations of the cuboctahedron: the cube-octahedron compound, a truncated form of the stella octangula, a sort of compound of six intersecting square pyramids, and an attractive concave solid formed of rhombi meeting four at a time.

If a cuboctahedron 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 Johnson solid , the triangular orthobicupola.

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 a cuboctahedron (Steinhaus 1999, pp. 203-207).

Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recreations and Essays, 13th ed. New York: Dover, p. 137, 1987.

Bool, F. H.; Kist, J. R.; Locher, J. L.; and Wierda, F. M. C. Escher: His Life and Complete Graphic Work. New York: Abrams, 1982.

Coxeter, H. S. M. Regular Polytopes, 3rd ed. New York: Dover, 1973.

Cundy, H. and Rollett, A. "Cuboctahedron. ." §3.7.2 in Mathematical Models, 3rd ed. Stradbroke, England: Tarquin Pub., p. 102, 1989.

Escher, M. C. "Stars." Wood engraving. 1948. http://www.mcescher.com/Gallery/back-bmp/LW359.jpg.

Forty, S. M.C. Escher. Cobham, England: TAJ Books, 2003.

Geometry Technologies. "Cubeoctahedron [sic]." http://www.scienceu.com/geometry/facts/solids/cubeocta.html.

Ghyka, M. The Geometry of Art and Life. New York: Dover, p. 54, 1977.

Heath, T. L. A History of Greek Mathematics, Vol. 1: From Thales to Euclid. New York: Dover, 1981.

Kasahara, K. Origami Omnibus: Paper-Folding for Everyone. Tokyo: Japan Publications, p. 206, 1988.

Rawles, B. Sacred Geometry Design Sourcebook: Universal Dimensional Patterns. Nevada City, CA: Elysian Pub., p. 207, 1997.

Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, pp. 203-205, 1999.

Wenninger, M. J. "The Cuboctahedron." Model 11 in Polyhedron Models. Cambridge, England: Cambridge University Press, p. 25, 1989.

Wenninger, M. J. "Commentary on the Stellation of the Archimedean Solids." In Polyhedron Models. New York: Cambridge University Press, pp. 66-72, 1989.

CITE THIS AS:

Weisstein, Eric W. "Cuboctahedron." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/Cuboctahedron.html