# Triakis octahedron

Triakis octahedron

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' 0"
$\arccos ( -\frac{3 + 8\sqrt{2}}{17} )$
Properties convex, face-transitive

Truncated cube
(dual polyhedron)

Net

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

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

## Orthogonal projections

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

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

## Related polyhedra

Spherical triakis octahedron

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.

Dimensional family of truncated spherical polyhedra and tilings: 3.2n.2n
Symmetry
*n32
[n,3]
Spherical Euclid. Compact hyperb. Paraco. Noncompact hyperbolic
*232
[2,3]
D3h
*332
[3,3]
Td
*432
[4,3]
Oh
*532
[5,3]
Ih
*632
[6,3]
P6m
*732
[7,3]

*832
[8,3]...

*∞32
[∞,3]

[12i,3] [9i,3] [6i,3] [3i,3]
Figures
Schläfli 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} t{3i,3}
Coxeter
Dual
figures
Triakis
figures

V3.4.4

V3.6.6

V3.8.8

V3.10.10

V3.12.12

V3.14.14

V3.16.16

V3.∞.∞
Coxeter

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.

Dimensional family of truncated polyhedra and tilings: n.8.8
Symmetry
*n42
[n,4]
Spherical Euclidean Compact hyperbolic Paracompact
*242
[2,4]
D4h
*342
[3,4]
Oh
*442
[4,4]
P4m
*542
[5,4]
*642
[6,4]
*742
[7,4]
*842
[8,4]...
*∞42
[∞,4]
Truncated
figures
2.8.8
3.8.8

4.8.8

5.8.8

6.8.8

7.8.8

8.8.8

∞.8.8
Coxeter
Schläfli

t{4,2}

t{4,3}

t{4,4}

t{4,5}

t{4,6}

t{4,7}

t{4,8}

t{4,∞}
Uniform dual figures
n-kis
figures

V2.8.8

V3.8.8

V4.8.8

V5.8.8

V6.8.8

V7.8.8

V8.8.8

V∞.8.8
Coxeter

## References

1. ^ Conway, Symmetries of things, p.284