It can be seen as an octahedron with triangular pyramids added to each face; that is, it is the Kleetope of the octahedron. It is also sometimes called a trisoctahedron, or, more fully, trigonal trisoctahedron. Both names reflect that it has three triangular faces for every face of an octahedron. The tetragonal trisoctahedron is another name for the deltoidal icositetrahedron, a different polyhedron with three quadrilateral faces for every face of an octahedron.
This convex polyhedron is topologically similar to the concave stellated octahedron. They have the same face connectivity, but the vertices are in different relative distances from the center.
If its shorter edges have length 1, its surface area and volume are:
Cartesian coordinates
Let α = √2 − 1, then the 14 points (±α, ±α, ±α) and (±1, 0, 0), (0, ±1, 0) and (0, 0, ±1) are the vertices of a triakis octahedron centered at the origin.
The length of the long edges equals √2, and that of the short edges 2√2 − 2.
The faces are isosceles triangles with one obtuse and two acute angles. The obtuse angle equals arccos(1/4 − √2/2) ≈ 117.20057038016° and the acute ones equal arccos(1/2 + √2/4) ≈ 31.39971480992°.
Orthogonal projections
The triakis octahedron has three symmetry positions, two located on vertices, and one mid-edge:
More information Projective symmetry, Triakis octahedron ...
The triakis octahedron is a part of a sequence of polyhedra and tilings, extending into the hyperbolic plane. These face-transitive figures have (*n32) reflectional symmetry.
3D model of a triakis octahedronAnimation of triakis octahedron and other related polyhedraSpherical triakis octahedron
More information Symmetry*n32 [n,3], Spherical ...
*n32 symmetry mutation of truncated tilings: t{n,3}
The triakis octahedron is also a part of a sequence of polyhedra and tilings, extending into the hyperbolic plane. These face-transitive figures have (*n42) reflectional symmetry.
More information Symmetry*n42 [n,4], Spherical ...
*n42 symmetry mutation of truncated tilings: n.8.8
Williams, Robert (1979). The Geometrical Foundation of Natural Structure: A Source Book of Design. Dover Publications, Inc. ISBN0-486-23729-X. (Section 3-9)
The Symmetries of Things 2008, John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, ISBN978-1-56881-220-5 (Chapter 21, Naming the Archimedean and Catalan polyhedra and tilings, page 284, Triakis octahedron)
This article uses material from the Wikipedia article Triakis_octahedron, and is written by contributors.
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