Triakis Octahedron
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Triakis Octahedron
Triakis octahedron
(Click here for rotating model)
Type Catalan solid
Coxeter diagram CDel node f1.pngCDel 4.pngCDel node f1.pngCDel 3.pngCDel node.png
Conway notation kO
Face type V3.8.8
DU09 facets.png

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?
Properties convex, face-transitive
Truncated hexahedron.png
Truncated cube
(dual polyhedron)
Triakis octahedron Net

In geometry, a triakis octahedron (or trigonal trisoctahedron[1] or kisoctahedron[2]) 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:

Cartesian coordinates

Put , then the 14 points and , and are the vertices of a triakis octahedron centered at the origin.

The length of the long edges equals , and that of the short edges .

The faces are isosceles triangles with one obtuse and two acute angles. The obtuse angle equals and the acute ones equal .

Orthogonal projections

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

Orthogonal projections
[2] [4] [6]
Dual truncated cube t01 e88.png Dual truncated cube t01 B2.png Dual truncated cube t01.png
Cube t01 e88.png 3-cube t01 B2.svg 3-cube t01.svg

Cultural references

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.

3D model of a triakis octahedron
Animation of triakis octahedron and other related polyhedra
Spherical triakis octahedron

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.


  1. ^ "Clipart tagged: 'forms'".
  2. ^ Conway, Symmetries of things, p.284

External links

  This article uses material from the Wikipedia page available here. It is released under the Creative Commons Attribution-Share-Alike License 3.0.



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