Triakis octahedron

The triakis octahedron is one of the 13 Catalan solids. It has 24 isosceles triangles as faces, with 6 order-8 and 8 order-3 vertices. It is the dual of the uniform truncated cube.

Triakis octahedron
Rank3
TypeUniform dual
Notation
Bowers style acronymTikko
Coxeter diagramm4m3o ()
Conway notationkO
Elements
Faces24 isosceles triangles
Edges12+24
Vertices6+8
Vertex figure6 octagons, 8 triangles
Measures (edge length 1)
Dihedral angle${\displaystyle \arccos \left(-{\frac {3+8{\sqrt {2}}}{17}}\right)\approx 147.35010^{\circ }}$
Central density1
Number of external pieces24
Level of complexity3
Related polytopes
ArmyTikko
RegimentTikko
DualTruncated cube
ConjugateGreat triakis octahedron
Abstract & topological properties
Flag count144
Euler characteristic2
SurfaceSphere
OrientableYes
Genus0
Properties
SymmetryB3, order 48
Flag orbits3
ConvexYes
NatureTame

It can also be obtained as the convex hull of a cube and an octahedron, where the edges of the octahedron are ${\displaystyle {\frac {2+{\sqrt {2}}}{2}}\approx 1.70711}$ times the length of those of the cube. Using an octahedron that is any number more than ${\displaystyle {\sqrt {2}}\approx 1.41421}$ times the edge length of the cube gives a fully symmetric variant of this polyhedron. The upper limit is ${\displaystyle {\frac {3{\sqrt {2}}}{2}}}$, where the cube's vertices coincide with the face centers of the octahedron.

Each face of this polyhedron is an isosceles triangle with base side length ${\displaystyle {\frac {2+{\sqrt {2}}}{2}}\approx 1.70711}$ times those of the side edges. These triangles have apex angle ${\displaystyle \arccos \left({\frac {1-2{\sqrt {2}}}{4}}\right)\approx 117.20057^{\circ }}$ and base angles ${\displaystyle \arccos \left({\frac {2+{\sqrt {2}}}{4}}\right)\approx 31.39972^{\circ }}$.

Vertex coordinates

A triakis octahedron with dual edge length 1 has vertex coordinates given by all permutations of:

• ${\displaystyle \left(\pm \left(1+{\sqrt {2}}\right),\,0,\,0\right)}$ ,
• ${\displaystyle \left(\pm 1,\,\pm 1,\,\pm 1\right)}$ .