TOPICS
Search

Small Triakis Octahedron


In general, a triakis octahedron is a non-regular icositetrahedron that can be constructed as a positive augmentation of regular octahedron. Such a solid is also known as a trisoctahedron, especially to mineralogists (Correns 1949, p. 41; Berry and Mason 1959, p. 127). While the resulting icositetrahedron is not regular, its faces are all identical.

SmallTriakisOctahedronSolidWireframeNet

Make Your Own Small Triakis Octahedron

Print & fold
Print in 3D

The small triakis octahedron, called simply the triakis octahedron by Holden (1971, p. 55), is the 24-faced dual polyhedron of the truncated cube and is Wenninger dual W_8. The addition of the word "small" is necessary to distinguish it from the great triakis octahedron, which is the dual of the stellated truncated hexahedron. It is illustrated above together with a wireframe version and a net that can be used for its construction.

The small triakis octahedron can be constructed by augmentation of a unit edge-length octahedron by a pyramid with height sqrt(3)-2/3sqrt(6).

A small triakis octahedron appears in the middle right as one of the polyhedral "stars" in M. C. Escher's 1948 wood engraving "Stars" (Forty 2003, Plate 43).

It is implemented in the Wolfram Language as PolyhedronData["SmallTriakisOctahedron"].

SmallTriakisOctahedronConvexHulls

The small triakis octahedron is the convex hull of the equilateral augmented cube.

Small triakis octahedron inscribed solids

An octahedron and stella octangula can be inscribed on the vertices of the small triakis octahedron (E. Weisstein, Dec. 25, 2009).

SmallTriakisOctahedronAndDual

The dual polyhedron of the small triakis octahedron is the truncated cube, both of which illustrated above together with their common midsphere. For a truncated cube of unit side length the dual has edges of lengths

s_1=2
(1)
s_2=2+sqrt(2).
(2)

Normalizing so that s_1=1, the resulting small triakis octahedron has surface area and volume

S=3sqrt(7+4sqrt(2))
(3)
V=1/2(3+2sqrt(2)).
(4)

See also

Archimedean Dual, Archimedean Solid, Great Triakis Octahedron, Icositetrahedron, Small Triakis Octahedral Graph, Small Triakis Octahedron Stellations, Truncated Cube

Explore with Wolfram|Alpha

References

Berry, L. G. and Mason, B. Mineralogy: Concepts, Descriptions, Determinations. San Francisco, CA: W. H. Freeman, 1959.Correns, C. W. Einführung in die Mineralogie (Kristallographie und Petrologie). Berlin: Springer-Verlag, 1949.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.Holden, A. Shapes, Space, and Symmetry. New York: Columbia University Press, p. 55, 1971.Wenninger, M. J. Dual Models. Cambridge, England: Cambridge University Press, p. 7, 1983.

Cite this as:

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

Subject classifications