Stellated octahedron

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Stellated octahedron
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} [1]
Symmetry group octahedral symmetry, pyritohedral symmetry
Dual polyhedron self-dual
3D model of stellated octahedron. 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 . [2]

Contents

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. [3] It is an example of a non-convex deltahedron. [4] Magnus Wenninger's Polyhedron Models denote this model as nineteenth W19. [5]

The stellated octahedron is a faceting of the cube, meaning removing part of the polygonal faces without creating new vertices of a cube. [6] It has the same three-dimensional point group symmetry as the cube, an octahedral symmetry. [7]

Stellation of octahedron facets.svg
Stellation plane of a stellated octahedron
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". [3] The two tetrahedra share a common intersphere in the centre, making the compound self-dual. [8] There exist compositions of all symmetries of tetrahedra reflected about the cube's center, so the stellated octahedron may also have pyritohedral symmetry. [9]

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, times the side length. [10] 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. [11]

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.

As a spherical tiling, the combined edges in the compound of two tetrahedra form a rhombic dodecahedron. 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 for being the positive integers; the first ten such numbers are: [12]

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

The stellated octahedron appears with several other polyhedra and polyhedral compounds in M. C. Escher's print "Stars", [13] and provides the central form in Escher's Double Planetoid (1949). [14]

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  [ es ] in Zaragoza, Spain, is surrounded by twelve stellated octahedral lampposts, shaped to form a three-dimensional version of the Flag of Europe. [15]

Some modern mystics have associated this shape with the "merkaba", [16] which according to them is a "counter-rotating energy field" named from an ancient Egyptian word. [17] However, the word "merkaba" is actually Hebrew, and more properly refers to a chariot in the visions of Ezekiel. [18] The resemblance between this shape and the two-dimensional star of David has also been frequently noted. [19]

The musical project "Miracle Musical" (often stylized as ミラクルミュージカル [20] ), spearheaded by Tally Hall member Joe Hawley, [21] uses the stellated octahedron in the artwork and lyrics of the song "Black Rainbows" [22] a song in the only album of the project called, “HAWAII: PART II” released on 12/12/12 at 12:12. [23] [20] [22] [24]

References

  1. ↑ Coxeter, Harold (1973), "The five regular compounds", Regular Polytopes (3rd ed.), Dover Publications, pp. 47–50, 96–104, ISBN   0-486-61480-8
  2. ↑ Barnes, John (2009), "Shapes and Solids", Gems of Geometry, Springer, pp. 25–56, doi:10.1007/978-3-642-05092-3_2, ISBN   978-3-642-05091-6
  3. 1 2 Cromwell, Peter R. (1997), Polyhedra, Cambridge University Press, p. 171, 261, ISBN   978-0-521-55432-9
  4. ↑ Pedersen, M. C.; Hyde, S. T. (2018), "Polyhedra and packings from hyperbolic honeycombs", Proceedings of the National Academy of Sciences , 115 (27): 6905–6910, Bibcode:2018PNAS..115.6905P, doi: 10.1073/pnas.1720307115 , PMC   6142264 , PMID   29925600
  5. ↑ Wenninger, Magnus J. (1971), Polyhedron Models, Cambridge University Press, p. 37
  6. ↑ Inchbald, Guy (2006), "Facetting Diagrams", The Mathematical Gazette , 90 (518): 253–261, doi:10.1017/S0025557200179653, JSTOR   40378613
  7. ↑ Coxeter (1973), p.  49.
  8. ↑ Pugh, Anthony (1976), Polyhedra: A Visual Approach, University of California Press, p. 88, ISBN   9780520030565
  9. ↑ Smith, James (2000), Methods of Geometry, John Wiley & Sons, p. 403–404, ISBN   978-1-118-03103-2
  10. ↑ Loeb, Arthur (1997), "Deconstruction of the Cube", in Gabriel, Jean-François (ed.), Beyond the Cube: The Architecture of Space Frames and Polyhedra, John Wiley & Sons, p. 233
  11. ↑ Brigaglia, Aldo; Palladino, Nicla; Vaccaro, Maria Alessandra (2018), "Historical notes on star geometry in mathematics, art and nature", in Emmer, Michele; Abate, Marco (eds.), Imagine Math 6: Between Culture and Mathematics, Springer International Publishing, pp. 197–211, doi:10.1007/978-3-319-93949-0_17, hdl: 10447/325250 , ISBN   978-3-319-93948-3
  12. ↑ Conway, John; Guy, Richard (1996), The Book of Numbers, Springer, p. 51, ISBN   978-0-387-97993-9
  13. ↑ Hart, George W. (1996), "The Polyhedra of M.C. Escher", Virtual Polyhedra.
  14. ↑ Coxeter, H. S. M. (1985), "A special book review: M. C. Escher: His life and complete graphic work", The Mathematical Intelligencer, 7 (1): 59–69, doi:10.1007/BF03023010, S2CID   189887063 . See in particular p. 61.
  15. ↑ "Obelisco" [Obelisk], Zaragoza es Cultura (in Spanish), Ayuntamiento de Zaragoza, retrieved 2021-10-19
  16. ↑ Dannelley, Richard (1995), Sedona: Beyond the Vortex: Activating the Planetary Ascension Program with Sacred Geometry, the Vortex, and the Merkaba, Light Technology Publishing, p. 14, ISBN   9781622336708
  17. ↑ 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, p. 4, ISBN   9781891824173
  18. ↑ Patzia, Arthur G.; Petrotta, Anthony J. (2010), Pocket Dictionary of Biblical Studies: Over 300 Terms Clearly & Concisely Defined, The IVP Pocket Reference Series, InterVarsity Press, p. 78, ISBN   9780830867028
  19. ↑ Brisson, David W. (1978), Hypergraphics: visualizing complex relationships in art, science, and technology, Westview Press for the American Association for the Advancement of Science, p. 220, The Stella octangula is the 3-d analog of the Star of David
  20. 1 2 "ミラクルミュージカル". ミラクルミュージカル. Retrieved 2025-10-10.
  21. ↑ Resident Emma (2021-04-09). band. Interview - Joe Hawley (2019) . Retrieved 2025-10-10– via YouTube.
  22. 1 2 Miracle Musical (Ft. Madi Diaz) – Black Rainbows , retrieved 2025-10-10
  23. ↑ Musicboard, Hawaii: Part II by Miracle Musical - Musicboard , retrieved 2025-10-10
  24. ↑ "Miracle Musical Store". Miracle Musical. Retrieved 2025-10-10.
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