# Great triakis octahedron

Great triakis octahedron
Rank3
TypeUniform dual
Notation
Coxeter diagramm4/3m3o ()
Elements
Faces24 isosceles triangles
Edges12+24
Vertices6+8
Vertex figures8 triangles
6 octagrams
Measures (edge length 1)
Inradius${\displaystyle {\frac {\sqrt {17(23-16{\sqrt {2}})}}{17}}\approx 0.14804}$
Dihedral angle${\displaystyle \arccos \left({\frac {8{\sqrt {2}}-3}{17}}\right)\approx 60.72239^{\circ }}$
Central density7
Number of external pieces120
Related polytopes
DualQuasitruncated hexahedron
ConjugateTriakis octahedron
Convex coreDeltoidal icositetrahedron
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 base edges of the triangles will measure ${\displaystyle 2-{\sqrt {2}}\approx 0.58579}$, and the lateral edges will have a length of 2. The triangles have two interior angles of ${\displaystyle \arccos \left({\frac {2-{\sqrt {2}}}{4}}\right)\approx 81.57894^{\circ }}$, and one of ${\displaystyle \arccos \left({\frac {1+2{\sqrt {2}}}{4}}\right)\approx 16.84212^{\circ }}$.

## Vertex coordinates

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

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