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 De Divina Proportione, 1509.^[2]

Stellated octahedron

Type	Regular compound
Coxeter symbol	{4,3}[2{3,3}]{3,4}[1]
Schläfli symbols	{{3,3}} a{4,3} ß{2,4} ßr{2,2}
Coxeter diagrams	∪
Stellation core	regular octahedron
Convex hull	cube
Index	UC4, W19
Polyhedra	two tetrahedra
Faces	8 triangles
Edges	12
Vertices	8
Dual polyhedron	self-dual
Symmetry group	octahedral symmetry, pyritohedral symmetry

3D model of stellated octahedron.

It is the simplest of 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 amount 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.^[3] Magnus Wenninger's Polyhedron Models denote this model as nineteenth W₁₉.^[4]

The stellated octahedron is a faceting of the cube, meaning removing part of the polygonal faces without creating new vertices of a cube.^[5] The symmetry operation of a stellated octahedron has the same one as the cube. Hence, its three-dimensional point group symmetry is an octahedral symmetry.^[6]

Stellation plane of a stellated octahedron

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".^[3] The two tetrahedra share a common intersphere in the centre, making the compound self-dual.^[7] There exist compositions of all symmetries of tetrahedra reflected about the cube's center, so the stellated octahedron may also have pyritohedral symmetry.^[8]

The stellated octahedron can be obtained as an augmentation of the regular octahedron, by adding tetrahedral pyramids on each face. This results in that its volume is the sum of eight tetrahedrons' and one regular octahedron's volume, ${\textstyle {\frac {3}{2}}}$ times of the side length.^[9] 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.^[10]

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

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 ${\displaystyle n(2n^{2}-1)}$ for ${\displaystyle n}$ being the positive integers; the first ten such numbers are:^[11]

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",^[12] and provides the central form in Escher's Double Planetoid (1949).^[13]

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

The obelisk in the center of the Plaza de Europa [es] in Zaragoza, Spain, is surrounded by twelve stellated octahedral lampposts, shaped to form a three-dimensional version of the Flag of Europe.^[14]

Some modern mystics have associated this shape with the "merkaba",^[15] which according to them is a "counter-rotating energy field" named from an ancient Egyptian word.^[16] However, the word "merkaba" is actually Hebrew, and more properly refers to a chariot in the visions of Ezekiel.^[17] The resemblance between this shape and the two-dimensional star of David has also been frequently noted.^[18]