Stellated octahedron

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Stellated octahedron
Compound of two tetrahedra.png
Type Regular compound
Schläfli symbol {{3,3}} = a{4,3}
Coxeter diagram CDel nodes 10ru.pngCDel split2.pngCDel node.png + CDel nodes 01rd.pngCDel split2.pngCDel node.png = CDel node h3.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png
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 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).

Construction[edit]

The stellated octahedron 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.

Related concepts[edit]

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[edit]

The 4D8 Puzzle is a puzzle similar to the Rubik's Cube, in the shape of a truncated stellated octahedron.

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

Gallery[edit]

Stella-octangula-in-cube.png
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.svg Stellated octahedron persp 2.svg Stellated octahedron persp 4.svg Stellated octahedron persp 3.svg

References[edit]

External links[edit]