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
(3D model) (OFF file)
Rank	3
Type	Uniform dual
Notation
Bowers style acronym	Tikko
Coxeter diagram	m4m3o ()
Conway notation	kO
Elements
Faces	24 isosceles triangles
Edges	12+24
Vertices	6+8
Vertex figure	6 octagons, 8 triangles
Measures (edge length 1)
Dihedral angle	$\arccos \left(-{\frac {3+8{\sqrt {2}}}{17}}\right)\approx 147.35010^{\circ }$
Central density	1
Number of external pieces	24
Level of complexity	3
Related polytopes
Army	Tikko
Regiment	Tikko
Dual	Truncated cube
Conjugate	Great triakis octahedron
Abstract & topological properties
Flag count	144
Euler characteristic	2
Surface	Sphere
Orientable	Yes
Genus	0
Properties
Symmetry	B3, order 48
Flag orbits	3
Convex	Yes
Nature	Tame

It can also be obtained as the convex hull of a cube and an octahedron, where the edges of the octahedron are ${\frac {2+{\sqrt {2}}}{2}}\approx 1.70711$ times the length of those of the cube. Using an octahedron that is any number more than ${\sqrt {2}}\approx 1.41421$ times the edge length of the cube gives a fully symmetric variant of this polyhedron. The upper limit is ${\frac {3{\sqrt {2}}}{2}}$, 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 ${\frac {2+{\sqrt {2}}}{2}}\approx 1.70711$ times those of the side edges. These triangles have apex angle $\arccos \left({\frac {1-2{\sqrt {2}}}{4}}\right)\approx 117.20057^{\circ }$ and base angles $\arccos \left({\frac {2+{\sqrt {2}}}{4}}\right)\approx 31.39972^{\circ }$.

Vertex coordinates

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

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

External links

• Klitzing, Richard. "tikko".
• Wikipedia contributors. "Triakis octahedron".
• McCooey, David. "Triakis Octahedron"
• Quickfur. "The Triakis Octahedron".