Stellated octahedron

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. 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==

, 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, .... ==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). The obelisk in the center of the in Zaragoza, Spain, is surrounded by twelve stellated octahedral lampposts, shaped to form a three-dimensional version of the Flag of Europe. The single decoration on the tomb of André Breton is a stellated octahedron. 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 The resemblance between this shape and the two-dimensional star of David has also been frequently noted. == References ==