Triakis Tetrahedron

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


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"The" triakis tetrahedron is the dual polyhedron of the truncated tetrahedron (Holden 1971, p. 55) It can be constructed by augmentation of a unit edge-length tetrahedron by a pyramid with height sqrt(6)/15. It is illustrated above together with a wireframe version and a net that can be used for its construction.

It is Wenninger dual W_6.


The triakis tetrahedron is the convex hull of the equilateral augmented dodecahedron.

Tetrahedra inscriptable in a triakis tetrahedron

Five tetrahedra of unit edge length (corresponding to a central tetrahedron and its regular augmentation) and one tetrahedron of edge length 5/3 can be inscribed in the vertices of the unit triakis tetrahedron, forming the configurations illustrated above.

The triakis tetrahedron formed by taking the dual of a truncated tetrahedron with unit edge lengths has side lengths


Normalizing so that s_1=1 gives surface area and volume


See also

Archimedean Dual, Archimedean Solid, Augmentation, Augmented Tetrahedron, Triakis Tetrahedral Graph, Triakis Tetrahedron Stellations, Triakis Truncated Tetrahedron, Truncated Tetrahedron

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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.Holden, A. Shapes, Space, and Symmetry. New York: Columbia University Press, p. 55, 1971.Wenninger, M. J. Dual Models. Cambridge, England: Cambridge University Press, pp. 14-15 and 33, 1983.

Cite this as:

Weisstein, Eric W. "Triakis Tetrahedron." From MathWorld--A Wolfram Web Resource.

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