Stellated octahedron

Stellated octahedron
Type	Regular compound
Coxeter-Dynkin diagram	+
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) [3,3]

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: (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".

Gallery

It is the only fully symmetric facetting of the cube

See also

• Polyhedron
• Polyhedron models
• Desmic tetrahedra
• 4D8 Puzzle
• Stella octangula number

References

• Peter R. Cromwell, Polyhedra, Cambridge University Press (1997) Polyhedra
• Luca Pacioli, De Divina Proportione, 1509.

External links

• VRML model: [1]
• Weisstein, Eric W., "Stella Octangula" from MathWorld.