Great triakis octahedron

Great triakis octahedron
(3D model) (OFF file)
Rank	3
Type	Uniform dual
Space	Spherical
Notation
Coxeter diagram	m4/3m3o ()
Elements
Faces	24 isosceles triangles
Edges	12+24
Vertices	8+6
Vertex figures	8 triangles
	6 octagrams
Measures (edge length 1)
Inradius	$\frac{\sqrt{17(23−16\sqrt2)}}{17} ≈ 0.14804$
Dihedral angle	$\arccos\left(\frac{8\sqrt2-3}{17}\right) ≈ 60.72239^\circ$
Central density	7
Number of external pieces	120
Related polytopes
Dual	Quasitruncated hexahedron
Conjugate	Triakis octahedron
Abstract & topological properties
Flag count	144
Euler characteristic	2
Orientable	Yes
Genus	0
Properties
Symmetry	B3, order 48
Convex	No
Nature	Tame

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 $2-\sqrt2 ≈ 0.58579$, and the long edges will have a length of 2. The triangles have two interior angles of $\arccos\left(\frac12-\frac{\sqrt2}{4}\right) ≈ 81.57894^\circ$, and one of $\arccos\left(\frac14+\frac{\sqrt2}{2}\right) ≈ 16.84212^\circ$.

Vertex coordinates

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

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

External links

• Wikipedia Contributors. "Great triakis octahedron".
• McCooey, David. "Great Triakis Octahedron"