Triakis octahedron

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Triakis octahedron
Triakis octahedron
(Click here for rotating model)
Type Catalan solid
Face type isosceles triangle
Faces 24
Edges 36
Vertices 14
Vertices by type 8{3}+6{8}
Face configuration V3.8.8
Symmetry group Oh, [4,3], *432
Dihedral angle 147°21'0"
 \arccos ( -\frac{3 + 8\sqrt{2}}{17} )
Properties convex, face-transitive
Truncated hexahedron.png
Truncated cube
(dual polyhedron)
Triakis octahedron Net
Net

In geometry, a triakis octahedron 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.

It is also called the small triakis octahedron, so as to differentiate it from the great triakis octahedron, the dual of the stellated truncated hexahedron.

If its shorter edges have length 1, its surface area and volume are:

A=3\sqrt{7+4\sqrt{2}}
V=\frac{1}{2}(3+2\sqrt{2}).

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