Triakis octahedron

Short description: Catalan solid with 24 faces

Triakis octahedron
(Click here for rotating model)
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	O_h, B₃, [4,3], (*432)
Rotation group	O, [4,3]⁺, (432)
Dihedral angle	147°21′00″ arccos(−3 + 8√2/17)
Properties	convex, face-transitive
Truncated cube (dual polyhedron)	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 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:

[math]\displaystyle{ \begin{align} A &= 3\sqrt{7+4\sqrt{2}} \\ V &= \frac{3+2\sqrt{2}}{2} \end{align} }[/math]

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:

Orthogonal projections
Projective symmetry	[2]	[4]	[6]
Triakis octahedron
Truncated cube

Cultural references

A triakis octahedron is a vital element in the plot of cult author Hugh Cook's novel The Wishstone and the Wonderworkers.

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. File:Triakis octahedron.stl

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.

References

↑ "Clipart tagged: ‘forms’". etc.usf.edu. https://etc.usf.edu/clipart/keyword/forms.
↑ Conway, Symmetries of things, p. 284

Williams, Robert (1979). The Geometrical Foundation of Natural Structure: A Source Book of Design. Dover Publications, Inc. ISBN 0-486-23729-X. (Section 3-9)
Wenninger, Magnus (1983), Dual Models, Cambridge University Press, doi:10.1017/CBO9780511569371, ISBN 978-0-521-54325-5 (The thirteen semiregular convex polyhedra and their duals, Page 17, Triakisoctahedron)
The Symmetries of Things 2008, John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, ISBN 978-1-56881-220-5 [1] (Chapter 21, Naming the Archimedean and Catalan polyhedra and tilings, page 284, Triakis octahedron)