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}, B_{3}, [4,3], (*432) 
Rotation group  O, [4,3]^{+}, (432) 
Dihedral angle  147°21′00″ arccos(−3 + 8√2/17) 
Properties  convex, facetransitive 
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:
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°.
The triakis octahedron has three symmetry positions, two located on vertices, and one midedge:
Projective symmetry 
[2]  [4]  [6] 

Triakis octahedron 

Truncated cube 
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{3^{1,1}} 
t{3,4} t{3^{1,1}} 
{3,4} {3^{1,1}} 
rr{4,3} s_{2}{3,4} 
tr{4,3}  sr{4,3}  h{4,3} {3,3} 
h_{2}{4,3} t{3,3} 
s{3,4} s{3^{1,1}} 
= 
= 
= 
= or 
= or 
=  




 
Duals to uniform polyhedra  
V4^{3}  V3.8^{2}  V(3.4)^{2}  V4.6^{2}  V3^{4}  V3.4^{3}  V4.6.8  V3^{4}.4  V3^{3}  V3.6^{2}  V3^{5} 
The triakis octahedron is a part of a sequence of polyhedra and tilings, extending into the hyperbolic plane. These facetransitive figures have (*n32) reflectional symmetry.
*n32 symmetry mutation of truncated tilings: t{n,3}  

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 facetransitive figures have (*n42) reflectional symmetry.
*n42 symmetry mutation of truncated tilings: n.8.8  

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  
nkis 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 