# Stellated octahedron

(Redirected from Stella octangula)
Stellated octahedron
Type Regular compound
Coxeter symbol {4,3}[2{3,3}]{3,4}[1]
Schläfli symbol {{3,3}} = a{4,3}
Coxeter diagram =
Stellation core Octahedron
Convex hull Cube
Index UC4, W19
Polyhedra 2 tetrahedra
Faces 8 triangles
Edges 12
Vertices 8
Dual Self-dual
Symmetry group
Coxeter 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 a 3D extension of the hexagram: the hexagram is a two-dimensional shape formed from two overlapping equilateral triangles, centrally symmetric to each other, and in the same way the stellated octahedron can be formed from two centrally symmetric overlapping tetrahedra. It can also be seen as one of the stages in the construction of a 3D Koch Snowflake, a fractal shape formed by repeated attachment of smaller tetrahedra to each triangular face of a larger figure. The first stage of the construction of the Koch Snowflake is a single central tetrahedron, and the second stage, formed by adding four smaller tetrahedra to the faces of the central tetrahedron, is the stellated octahedron.

## Construction

The stellated octahedron can be constructed in several ways:

## Related concepts

As a spherical tiling, the combined edges in the compound of two tetrahedra form a rhombic dodecahedron.

A compound of two spherical tetrahedra can be constructed, as illustrated.

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.

The stella octangula numbers are figurate numbers that count the number of balls that can be arranged into the shape of a stellated octahedron. They are

0, 1, 14, 51, 124, 245, 426, 679, 1016, 1449, 1990, .... (sequence A007588 in OEIS)

## In popular culture

The Stellated octahedron appears with several other polyhedra and polyhedral compounds in M. C. Escher's print "Stars",[2] and provides the central form in Escher's Double Planetoid (1949).[3]

## Gallery

 It is the only fully symmetric facetting of the cube

## References

1. ^ Regular polytopes, pp.48-50, p.98
2. ^ Hart, George W. (1996), "The Polyhedra of M.C. Escher", Virtual Polyhedra.
3. ^ Coxeter, H. S. M. (1985), "A special book review: M. C. Escher: His life and complete graphic work", The Mathematical Intelligencer 7 (1): 59–69, doi:10.1007/BF03023010. See in particular p. 61.