|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 a 3D extension of the Star of David (as it is two tetrahedra overlapping each other), and as a stage in the construction of a 3D Koch Snowflake (as it is four small tetrahedra attached to a central, larger one).
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.
The stellation facets are very simple: (See Wenninger model W19.)
It can also be constructed from eight of the 20 vertices of the dodecahedron.
The two tetrahedra of the compound view of the stellated octahedron are "desmic", meaning that (when interpreted as a line in projective space) each edge of one tetrahedron crosses two opposite edges of the other tetrahedron. One of these two crossings is visible in the stellated octahedron; the other crossing occurs at a point at infinity of the projective space, between two parallel edges of the two tetrahedra. These two tetrahedra can be completed to a desmic system of three tetrahedra, where the third tetrahedron has as its four vertices the three crossing points at infinity and the centroid of the two finite tetrahedra. The same twelve tetrahedron vertices also form the points of Reye's configuration.
In popular culture
It is the only fully symmetric facetting of the cube
- Peter R. Cromwell, Polyhedra, Cambridge University Press (1997) Polyhedra
- Luca Pacioli, De Divina Proportione, 1509.
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