Triakis octahedron

Triakis octahedron
File:Triakisoctahedron.jpg (Click here for rotating model)
Type	Catalan solid
Coxeter diagram	Template:CDD
Conway notation	kO
Face type	V3.8.8 File: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″ arccos(−Template:Sfrac)
Properties	convex, face-transitive
File:Truncated hexahedron.png Truncated cube (dual polyhedron)	Triakis octahedron Net 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 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 at different relative distances from the center.

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

$\begin{array}{rl}A& =3\sqrt{7+4\sqrt{2}}\\ V& =\frac{3+2\sqrt{2}}{2}\end{array}$

File:Triakis octahedron.stl

3D model of a triakis octahedron

Cartesian coordinates

Let α =

1. REDIRECT Template:Radic

Template:Rcat shell − 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

1. REDIRECT Template:Radic

Template:Rcat shell, and that of the short edges 2

1. REDIRECT Template:Radic

Template:Rcat shell − 2.

The faces are isosceles triangles with one obtuse and two acute angles. The obtuse angle equals arccos(Template:Sfrac − Template:Sfrac) ≈ ° and the acute ones equal arccos(Template:Sfrac + Template:Sfrac) ≈ °.

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	File:Dual truncated cube t01 e88.png	File:Dual truncated cube t01 B2.svg	File:Dual truncated cube t01.svg
Truncated cube	File:Cube t01 e88.svg	File:3-cube t01 B2.svg	File:3-cube t01.svg

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:Kleetope of octahedron.gif

Animation of triakis octahedron and other related polyhedra

File:Spherical triakis octahedron.png

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. Template:Truncated figure4 table

Further reading

• Template:The Geometrical Foundation of Natural Structure (book) (Section 3-9)
• (The thirteen semiregular convex polyhedra and their duals, Page 17, Triakisoctahedron)
• The Symmetries of Things 2008, John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, Template:Isbn [1] (Chapter 21, Naming the Archimedean and Catalan polyhedra and tilings, page 284, Triakis octahedron)

External links

• Template:Mathworld2
• Triakis Octahedron – Interactive Polyhedron Model
• Virtual Reality Polyhedra www.georgehart.com: The Encyclopedia of Polyhedra
  • VRML model Template:Webarchive
  • Conway Notation for Polyhedra Try: "dtC"