Triakis octahedron

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.

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	Oh, 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

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:

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

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.

Uniform octahedral polyhedra
Symmetry: [4,3], (*432)							[4,3]+ (432)	[1+,4,3] = [3,3] (*332)		[3+,4] (3*2)
{4,3}	t{4,3}	r{4,3} r{31,1}	t{3,4} t{31,1}	{3,4} {31,1}	rr{4,3} s2{3,4}	tr{4,3}	sr{4,3}	h{4,3} {3,3}	h2{4,3} t{3,3}	s{3,4} s{31,1}
		=	=	=				= or	= or	=
Duals to uniform polyhedra
V43	V3.82	V(3.4)2	V4.62	V34	V3.43	V4.6.8	V34.4	V33	V3.62	V35

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

*n32 symmetry mutation of truncated tilings: t{n,3} v t e
Symmetry *n32 [n,3]	Spherical				Euclid.	Compact hyperb.		Paraco.	Noncompact hyperbolic
	*232 [2,3]	*332 [3,3]	*432 [4,3]	*532 [5,3]	*632 [6,3]	*732 [7,3]	*832 [8,3]...	*∞32 [∞,3]	[12i,3]	[9i,3]	[6i,3]
Truncated figures
Symbol	t{2,3}	t{3,3}	t{4,3}	t{5,3}	t{6,3}	t{7,3}	t{8,3}	t{∞,3}	t{12i,3}	t{9i,3}	t{6i,3}
Triakis figures
Config.	V3.4.4	V3.6.6	V3.8.8	V3.10.10	V3.12.12	V3.14.14	V3.16.16	V3.∞.∞

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.

*n42 symmetry mutation of truncated tilings: n.8.8 v t e
Symmetry *n42 [n,4]	Spherical		Euclidean	Compact hyperbolic				Paracompact
	*242 [2,4]	*342 [3,4]	*442 [4,4]	*542 [5,4]	*642 [6,4]	*742 [7,4]	*842 [8,4]...	*∞42 [∞,4]
Truncated figures
Config.	2.8.8	3.8.8	4.8.8	5.8.8	6.8.8	7.8.8	8.8.8	∞.8.8
n-kis figures
Config.	V2.8.8	V3.8.8	V4.8.8	V5.8.8	V6.8.8	V7.8.8	V8.8.8	V∞.8.8