Stellated octahedron

Stellated octahedron
File:Dual compound 4 max.png
Type	Regular compound Polyhedral compound UC4 W19
Faces	8 triangles
Edges	12
Vertices	8
Schläfli symbol	{{3,3}} a{4,3} ß{2,4} ßr{2,2}
Coxeter diagram	{4,3}[2{3,3}]{3,4}
Symmetry group	octahedral symmetry, pyritohedral symmetry
Dual polyhedron	self-dual

File:3D model of a Stellated Octahedron.stl

3D model of stellated octahedron.

The stellated octahedron is the only stellation of the octahedron. It is also called the stella octangula (Latin for "eight-pointed star"), a name given to it by Johannes Kepler in 1609, though it was known to earlier geometers. It was depicted in Pacioli's 1509 De Divina Proportione.

It is the simplest of the five regular polyhedral compounds, and the only regular polyhedral compound composed of only two polyhedra.

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. This can be generalized to any desired number of higher dimensions; the four-dimensional equivalent construction is the compound of two 5-cells.

Construction and properties

The stellated octahedron is constructed by a stellation of the regular octahedron. In other words, it extends to form equilateral triangles on each regular octahedron's faces. It is an example of a non-convex deltahedron. Magnus Wenninger's Polyhedron Models denote this model as nineteenth W₁₉.

The stellated octahedron is a faceting of the cube, meaning removing part of the polygonal faces without creating new vertices of a cube. It has the same three-dimensional point group symmetry as the cube, an octahedral symmetry.

File:Stellation of octahedron facets.svg

Stellation plane of a stellated octahedron

File:CubeAndStel.svg

Stellated octahedron as a cube faceting

The stellated octahedron is also a regular polyhedron compound, when constructed as the union of two regular tetrahedra. Hence, the stellated octahedron is also called "compound of two tetrahedra". The two tetrahedra share a common intersphere in the centre, making the compound self-dual. There exist compositions of all symmetries of tetrahedra reflected about the cube's center, so the stellated octahedron may also have pyritohedral symmetry.

The stellated octahedron can be obtained as an augmentation of the regular octahedron, by adding tetrahedral pyramids on each face. This results in its volume being the sum of eight tetrahedra's and one regular octahedron's volume, $\frac{3}{2}$ times the side length. However, this construction is topologically similar as the Catalan solid of a triakis octahedron with much shorter pyramids, known as the Kleetope of an octahedron.

It can be seen as a {4/2} antiprism; with {4/2} being a tetragram, a compound of two dual digons, and the tetrahedron seen as a digonal antiprism, this can be seen as a compound of two digonal antiprisms.

It can be seen as a net of a four-dimensional octahedral pyramid, consisting of a central octahedron surrounded by eight tetrahedra.

Related concepts

File:Spherical compound of two tetrahedra.png

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, where each edge of one tetrahedron crosses the parallel edge of the other tetrahedron. 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. These numbers are the form of $n(2n^2-1)$ for $n$ being the positive integers; the first ten such numbers are:

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

In popular culture

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

File:Zaragoza - Plaza de Europa 1.jpg

One of the stellated octahedra in the Plaza de Europa, Zaragoza

The obelisk in the center of the Plaza de Europa in Zaragoza, Spain, is surrounded by twelve stellated octahedral lampposts, shaped to form a three-dimensional version of the Flag of Europe.

The stellated octahedron was used as the logo for Joe Hawley of Tally Hall's side project, Miracle Musical. The only album released under the pseudonym being Hawaii Part II, released on December 12, 2012. Track 3 off of the album, Black Rainbows, mentions the aforementioned shape in it's chorus, sung by Madi Diaz.

Mysticism

Some modern mystics have associated this shape with the "merkaba": a "counterrotating field of light" that "transport[s] body and soul to other dimensions." New Age authors have attributed the merkaba to ancient Egyptian origins — traditionally, "mer" stood for pyramid, "ka" for soul, and "ba" for personality or spiritual essence that guides the soul. In a different tradition, Jewish "Merkabah" mysticism details a living chariot in the visions of Ezekiel (in Hebrew, chariot is written מֶרְכָּבָה and pronounced merkābâ, where "rakab" means "to ride" or "to be carried"), used by higher angels for motility.

The resemblance between this shape and the two-dimensional star of David has also been frequently noted.

External links

• Weisstein, Eric W., "Stella Octangula" ("Compound of two tetrahedra") at MathWorld.