Triakis octahedron

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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

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.

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:

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