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Truncated Octahedron


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The truncated octahedron is the 14-faced Archimedean solid with faces 8{6}+6{4}. It is also the uniform polyhedron with Maeder index 8 (Maeder 1997), Wenninger index 7 (Wenninger 1989), Coxeter index 20 (Coxeter et al. 1954), and Har'El index 13 (Har'El 1993). It has Schläfli symbol t{3,4} and Wythoff symbol 24|3. It was called the "mecon" by Buckminster Fuller (Rawles 1997). It is illustrated above together with a wireframe version and a net that can be used for its construction.

TruncatedOctProjections

Some symmetric projections of the truncated octahedron are illustrated above.

The truncated octahedron has the O_h octahedral group of symmetries. The form of the fluorite (CaF_2) resembles the truncated octahedron (Steinhaus 1999, pp. 207-208).

The truncated octahedron is a space-filling polyhedron (Steinhaus 1999, pp. 187-190 and 207) and therefore has a Dehn invariant of 0.

It is implemented in the Wolfram Language as PolyhedronData["TruncatedOctahedron"] or UniformPolyhedron["TruncatedTetrahedron"]. Precomputed properties are available as PolyhedronData["TruncatedTetrahedron", prop].

TruncatedOctahedronConvexHulls

The truncated octahedron is the convex hull of the tetragonal disphenoid 6-compound.

TruncatedOctahedronPyramid

The solid of edge length a can be formed from an octahedron of edge length 3a via truncation by removing six square pyramids, each with edge slant height s=1, base a=1 on a side, and height h. The height and base area of the square pyramid are then

h=sqrt(s^2-1/4a^2csc^2(pi/n))
(1)
=1/2sqrt(2)a
(2)
A_b=a^2
(3)

and its volume is

V_(square pyramid)=1/3A_bh
(4)
=1/6sqrt(2)a^3.
(5)

The volume of the truncated octahedron is then given by the volume of the octahedron

V_(octahedron)=1/3sqrt(2)(3a)^3
(6)
=9sqrt(2)a^3
(7)

minus six times the volume of the square pyramid,

V=V_(octahedron)-6V_(square pyramid)
(8)
=8sqrt(2)a^3.
(9)

The surface area of the truncated octahedron is

 S=(6+12sqrt(3))a^2.
(10)
TruncatedOctahedronAndDual

The dual polyhedron of the truncated octahedron is the tetrakis hexahedron, both of which are illustrated above together with their common midsphere.

The inradius r of the dual, midradius rho of the solid and dual, and circumradius R of the solid for a=1 are

r=9/(20)sqrt(10) approx 1.42302
(11)
rho=3/2=1.5
(12)
R=1/2sqrt(10) approx 1.58114.
(13)

The distances from the center of the solid to the centroids of the square and hexagonal faces are given by

r_4=sqrt(2)
(14)
r_6=1/2sqrt(6).
(15)

See also

Archimedean Solid, Equilateral Zonohedron, Icositetrahedron, Kelvin's Conjecture, Octahedron, Rhombic Dodecahedron Stellations, Square Pyramid, Truncation

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References

Coxeter, H. S. M. Regular Polytopes, 3rd ed. New York: Dover, pp. 29-30 and 257, 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. "Truncated Octahedron. 4.6^2." §3.7.4 in Mathematical Models, 3rd ed. Stradbroke, England: Tarquin Pub., p. 104, 1989.Geometry Technologies. "Truncated Octahedron." http://www.scienceu.com/geometry/facts/solids/tr_octa.html.Har'El, Z. "Uniform Solution for Uniform Polyhedra." Geometriae Dedicata 47, 57-110, 1993.Kasahara, K. "Three More Semiregular Polyhedrons Become Possible." Origami Omnibus: Paper-Folding for Everyone. Tokyo: Japan Publications, p. 225, 1988.Maeder, R. E. "08: Truncated Octahedron." 1997. https://www.mathconsult.ch/static/unipoly/08.html.Rawles, B. Sacred Geometry Design Sourcebook: Universal Dimensional Patterns. Nevada City, CA: Elysian Pub., p. 208, 1997.Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, 1999.Wenninger, M. J. "Truncated Octahedron." Model 7 in Polyhedron Models. Cambridge, England: Cambridge University Press, p. 21, 1989.

Cite this as:

Weisstein, Eric W. "Truncated Octahedron." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/TruncatedOctahedron.html

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