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Cuboctahedron


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The cuboctahedron, also called the heptaparallelohedron or dymaxion (the latter according to Buckminster Fuller; Rawles 1997), is the Archimedean solid with faces 8{3}+6{4}. It is illustrated above together with a wireframe version and a net that can be used for its construction.

The cuboctahedron is one of the two convex quasiregular polyhedra. It is also the uniform polyhedron with Maeder index 7 (Maeder 1997), Wenninger index 11 (Wenninger 1989), Coxeter index 19 (Coxeter et al. 1954), and Har'El index 12 (Har'El 1993). It has Schläfli symbol {3; 4} and Wythoff symbol 2|34.

CuboctahedronProjections

It is shown above in a number of symmetric projections.

The cuboctahedron is implemented in the Wolfram Language as PolyhedronData["Cuboctahedron"] or UniformPolyhedron["Cuboctahedron"]. Precomputed properties are available as PolyhedronData["Cuboctahedron", prop].

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).

CuboctahedronConvexHulls

The cuboctahedron is the convex hull of Escher's solid (together with the first rhombic dodecahedron stellation and square dipyramid 3-compound which shares its outer hull) as well as the cubohemioctahedron and octahemioctahedron uniform polyhedra.

CuboctahedronAndDual

The dual polyhedron of the cuboctahedron is the rhombic dodecahedron, both of which are illustrated above together with their common midsphere.

The cuboctahedron has the O_h 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 sqrt(2) are (0,+/-1,+/-1), (+/-1,0,+/-1), and (+/-1,+/-1,0).

The mineral argentite (Ag_2S) 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 r of the dual, midradius rho of the solid and dual, and circumradius R of the solid for a=1 are

r=3/4=0.75
(1)
rho=1/2sqrt(3) approx 0.86602
(2)
R=1.
(3)

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

r_3=1/3sqrt(6)
(4)
r_4=1/2sqrt(2).
(5)

The dihedral angle between triangular and square faces is

alpha=sec^(-1)(-sqrt(3))
(6)
=125.26... degrees.
(7)

The surface area and volume are

S=6+2sqrt(3)
(8)
V=5/3sqrt(2).
(9)

The cuboctahedron has Dehn invariant

D=-24<3>_2
(10)
=-24tan^(-1)(sqrt(2)),
(11)

where the first expression uses the basis of Conway et al. (1999). It can be dissected into the triangular orthobicupola, from which it differs only by the relative rotation of the top and bottom cupolas.

Faceted versions of the cuboctahedron include the cubohemioctahedron and octahemioctahedron.

CubeOctahedronPointsCubeOctahedronCommon

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.

CuboctahedronJ27

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 J_(27), the triangular orthobicupola.

CubicClosePackingClusterCubicClosePackingCuboct

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).


See also

Archimedean Solid, Cube, Cube-Octahedron Compound, Cubic Close Packing, Cuboctahedral Graph, Cubohemioctahedron, Equilateral Zonohedron, Octahedron, Octahemioctahedron, Quasiregular Polyhedron, Rhombic Dodecahedron, Rhombic Dodecahedron Stellations, Rhombus, Space-Filling Polyhedron, Sphere Packing, Stellation, Triangular Orthobicupola

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References

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.Conway, J. H.; Radin, C.; and Sadun, L. "On Angles Whose Squared Trigonometric Functions Are Rational." Discr. Computat. Geom. 22, 321-332, 1999.Coxeter, H. S. M. Regular Polytopes, 3rd ed. New York: Dover, 1973.Coxeter, H. S. M.; Longuet-Higgins, M. S.; and Miller, J. C. P. "Uniform Polyhedra." Phil. Trans. Roy. Soc. London Ser. A 246, 401-450, 1954.Cundy, H. and Rollett, A. "Cuboctahedron. (3.4)^2." §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.Har'El, Z. "Uniform Solution for Uniform Polyhedra." Geometriae Dedicata 47, 57-110, 1993.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.Maeder, R. E. "07: Cuboctahedron." 1997. https://www.mathconsult.ch/static/unipoly/07.html.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. https://mathworld.wolfram.com/Cuboctahedron.html

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