 Triakis Octahedron
Get Triakis Octahedron essential facts below. View Videos or join the Triakis Octahedron discussion. Add Triakis Octahedron to your PopFlock.com topic list for future reference or share this resource on social media.
Triakis Octahedron
Triakis octahedron Type Catalan solid
Coxeter diagram     Conway notation kO
Face type V3.8.8
isosceles triangle
Faces 24
Edges 36
Vertices 14
Vertices by type 8{3}+6{8}
Symmetry group Oh, B3, [4,3], (*432)
Rotation group O, [4,3]+, (432)
Dihedral angle 147°21?00?
arccos(-)
Properties convex, face-transitive Truncated cube
(dual polyhedron) Net

In geometry, a triakis octahedron (or trigonal trisoctahedron or kisoctahedron) is an Archimedean dual solid, or a Catalan solid. Its dual is the truncated cube.

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 the fact 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:

{\begin{aligned}A&=3{\sqrt {7+4{\sqrt {2}}}}\\V&={\frac {3+2{\sqrt {2}}}{2}}\end{aligned}} ## Cartesian coordinates

Put $\alpha ={\sqrt {2}}-1$ , then the 14 points $(\pm \alpha ,\pm \alpha ,\pm \alpha )$ and $(\pm 1,0,0)$ , $(0,\pm 1,0)$ and $(0,0,\pm 1)$ are the vertices of a triakis octahedron centered at the origin.

The length of the long edges equals ${\sqrt {2}}$ , and that of the short edges $2{\sqrt {2}}-2$ .

The faces are isosceles triangles with one obtuse and two acute angles. The obtuse angle equals $\arccos({\frac {1}{4}}-{\frac {1}{2}}{\sqrt {2}})\approx 117.200\,570\,380\,16^{\circ }$ and the acute ones equal $\arccos({\frac {1}{2}}+{\frac {1}{4}}{\sqrt {2}})\approx 31.399\,714\,809\,92^{\circ }$ .

## Orthogonal projections

The triakis octahedron has three symmetry positions, two located on vertices, and one mid-edge:

## Related polyhedra

The triakis octahedron is one of a family of duals to the uniform polyhedra related to the cube and regular 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.

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.