# Great triakis octahedron

Great triakis octahedron
Rank3
TypeUniform dual
SpaceSpherical
Notation
Coxeter diagramm4/3m3o ()
Elements
Faces24 isosceles triangles
Edges12+24
Vertices8+6
Vertex figures8 triangles
6 octagrams
Measures (edge length 1)
Inradius${\displaystyle \frac{\sqrt{17(23−16\sqrt2)}}{17} ≈ 0.14804}$
Dihedral angle${\displaystyle \arccos\left(\frac{8\sqrt2-3}{17}\right) ≈ 60.72239^\circ}$
Central density7
Number of external pieces120
Related polytopes
DualQuasitruncated hexahedron
ConjugateTriakis octahedron
Abstract & topological properties
Flag count144
Euler characteristic2
OrientableYes
Genus0
Properties
SymmetryB3, order 48
ConvexNo
NatureTame

The great triakis octahedron is a uniform dual polyhedron. It consists of 24 isosceles triangles.

If its dual, the quasitruncated hexahedron, has an edge length of 1, then the short edges of the triangles will measure ${\displaystyle 2-\sqrt2 ≈ 0.58579}$, and the long edges will have a length of 2. The triangles have two interior angles of ${\displaystyle \arccos\left(\frac12-\frac{\sqrt2}{4}\right) ≈ 81.57894^\circ}$, and one of ${\displaystyle \arccos\left(\frac14+\frac{\sqrt2}{2}\right) ≈ 16.84212^\circ}$.

## Vertex coordinates

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

• ${\displaystyle \left(±\left(\sqrt2-1\right),\,0,\,0\right),}$
• ${\displaystyle \left(±1,\,±1,\,±1\right).}$