Stellated octahedron

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.{{citation | doi-access = free

The stellated octahedron is a faceting of the cube, meaning removing part of the polygonal faces without creating new vertices of a cube.{{citation

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.{{citation

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.{{citation | editor-first = Jean-François | editor-last = Gabriel | contribution-url = https://books.google.com/books?id=FkM0945nFV8C&pg=PA233 | editor1-last = Emmer | editor1-first = Michele | editor2-last = Abate | editor2-first = Marco | hdl-access = free

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

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).{{citation

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.

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.

References

References

1. (1996). "The Book of Numbers". Springer.
2. Hart, George W.. (1996). "The Polyhedra of M.C. Escher". Virtual Polyhedra.
3. "Obelisco". Ayuntamiento de Zaragoza.
4. Dannelley, Richard. (1995). "Sedona: Beyond the Vortex: Activating the Planetary Ascension Program with Sacred Geometry, the Vortex, and the Merkaba". Light Technology Publishing.
5. Melchizedek, Drunvalo. (2000). "The Ancient Secret of the Flower of Life: An Edited Transcript of the Flower of Life Workshop Presented Live to Mother Earth from 1985 to 1994 -, Volume 1". Light Technology Publishing.
6. (26 Dec 1994). "The Teaching On Spherical Breathing (Merkaba Meditation)".
7. Marar, Ton. (May 20, 2022). "A Ludic Journey into Geometric Topology". [[Springer Nature]].
8. (2010). "Pocket Dictionary of Biblical Studies: Over 300 Terms Clearly & Concisely Defined". InterVarsity Press.
9. Brisson, David W.. (1978). "Hypergraphics: visualizing complex relationships in art, science, and technology". Westview Press for the American Association for the Advancement of Science.