Stellated octahedron

Stellated octahedron
Stellated octahedron
Compound of two tetrahedra.png
Type Regular compound
Coxeter-Dynkin diagram CDel node.pngCDel split1.pngCDel nodes 10l.png + CDel node.pngCDel split1.pngCDel nodes 01l.png
Schläfli symbol {{3,3}}
Stellation core Octahedron
Convex hull Cube
Index UC4, W19
Polyhedra 2 tetrahedra
Faces 8 triangles
Edges 12
Vertices 8
Dual Self-dual
Symmetry group octahedral (Oh), [4,3] or [[3,3]]
Subgroup restricting to one constituent tetrahedral (Td)

The stellated octahedron, or stella octangula, is the only stellation of the octahedron. It was named by Johannes Kepler in 1609, though it was known to earlier geometers. It was depicted in Pacioli's Divina Proportione, 1509.

It is the simplest of five regular polyhedral compounds.

It can be seen as either a polyhedron compound or a stellation:

As a compound, it is constructed as the union of two tetrahedra (a tetrahedron and its dual tetrahedron). The vertex arrangement of the two tetrahedra is shared by a cube. The intersection of the two tetrahedra form an inner octahedron, which shares the same face-planes as the compound.

It can be seen as an octahedron with tetrahedral pyramids on each face. It has the same topology as the convex Catalan solid, the triakis octahedron, which has much shorter pyramids.

The stellation facets are very simple: Stellation of octahedron facets.png (See Wenninger model W19.)

It can also be constructed from eight of the 20 vertices of the dodecahedron.


Artistic references

The Stellated octahedron and other polyhedra appear in M. C. Escher's print "Stars".


It is the only fully symmetric facetting of the cube
Stellated octahedron persp 7.svg Stellated octahedron persp 1.svg Stellated octahedron persp 6.svg
CubeAndStel.gif Stellated octahedron persp 2.svg Stellated octahedron persp 4.svg Stellated octahedron persp 3.svg