Stellated octahedron
| Stellated octahedron | |
|---|---|
 | |
| 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 |
The stellated octahedron is a shape made from two regular tetrahedra crossing each other. 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]
Two variations of this shape have been considered, one consisting of the two tetrahedra themselves and another consisting of their union, with interior boundaries removed. The two tetrahedra form the simplest of the five regular polyhedral compounds, and the only regular polyhedral compound composed of only two polyhedra. Their union forms the only stellation of the octahedron.
This shape can be seen as a 3D extension of the hexagram: the hexagram is a two-dimensional shape formed from two crossing equilateral triangles, centrally symmetric to each other, and in the same way, the stellated octahedron can be formed from two centrally symmetric crossing 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
[edit]The stellated octahedron is constructed by a stellation of the regular octahedron. In a stellation, the faces of the underlying polyhedron are extended within the same planes to enclose a different volume. Here, the extension in each plane consists of three equilateral triangles, surrounding the original triangular face of the octahedron and having the same size as it.[3] The resulting polyhedron, with 24 equilateral-triangle faces, 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]
The stellated octahedron is also a regular polyhedron compound, when constructed as a system of two regular tetrahedra. Hence, the stellated octahedron is also called "compound of two tetrahedra".[3] The two tetrahedra share a common midsphere, making the compound self-dual.[8] Other compounds of two tetrahedra can be formed by rotating one tetrahedron in other ways around a common center; these have smaller symmetry groups which may include 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] This construction produces a non-convex polyhedron with the same combinatorial structure as the convex triakis octahedron, a Catalan solid of with much shorter pyramids. Any of these pyramid attachments, convex or non-convex, may be known as the Kleetope of an octahedron.[11]
Related concepts
[edit]
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]
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.[13] The same twelve tetrahedron vertices also form the points of Reye's configuration.[14]
In popular culture
[edit]The stellated octahedron appears with several other polyhedra and polyhedral compounds in M. C. Escher's print "Stars",[15] and provides the central form in Escher's Double Planetoid (1949).[16]
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.[17]
The single decoration on the tomb of André Breton is a stellated octahedron.[18]
Mysticism
[edit]Some modern mystics have associated this shape with the "merkaba":[19] a "counterrotating field of light"[20][a] that "transport[s] body and soul to other dimensions."[22] New Age authors have attributed the merkaba to ancient Egyptian origins[20] — 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.[23]
The resemblance between this shape and the two-dimensional star of David has also been frequently noted.[24]
Notes
[edit]References
[edit]- ^ Coxeter, Harold (1973), "The five regular compounds", Regular Polytopes (3rd ed.), Dover Publications, pp. 47–50, 96–104, ISBN 0-486-61480-8
- ^ 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
- ^ a b Cromwell, Peter R. (1997), Polyhedra, Cambridge University Press, pp. 171, 261, ISBN 978-0-521-55432-9
- ^ 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
- ^ Wenninger, Magnus J. (1971), Polyhedron Models, Cambridge University Press, p. 37
- ^ Inchbald, Guy (2006), "Facetting Diagrams", The Mathematical Gazette, 90 (518): 253–261, doi:10.1017/S0025557200179653, JSTOR 40378613
- ^ Coxeter (1973), p. 49.
- ^ Pugh, Anthony (1976), Polyhedra: A Visual Approach, University of California Press, p. 88, ISBN 9780520030565
- ^ Smith, James (2000), Methods of Geometry, John Wiley & Sons, p. 403–404, ISBN 978-1-118-03103-2
- ^ 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
- ^ 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
- ^ Conway, John; Guy, Richard (1996), The Book of Numbers, Springer, p. 51, ISBN 978-0-387-97993-9
- ^ Hudson, R. W. H. T. (1905), Kummer's quartic surface, Cambridge University Press, pp. 1–2
- ^ Hilbert, David; Cohn-Vossen, Stephan (1952), "22. Reye's configuration", Geometry and the Imagination (2nd ed.), New York: Chelsea, pp. 134–143
- ^ Hart, George W. (1996), "The Polyhedra of M.C. Escher", Virtual Polyhedra.
- ^ 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.
- ^ "Obelisco" [Obelisk], Zaragoza es Cultura (in Spanish), Ayuntamiento de Zaragoza, retrieved 2021-10-19
- ^ Artnet, January 2020, "Philippe Decrauzat on artnet" "the stellated tetrahedron (stella octangula) which decorates André Breton's gravestone in the Cimetière des Batignolles in Paris".
- ^ 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
- ^ a b 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. 4,5, ISBN 9781891824173
- ^ Melchizedek, Drunvalo (26 Dec 1994), "The Teaching On Spherical Breathing (Merkaba Meditation)", Internet Archive
- ^ Marar, Ton (May 20, 2022), "Ancient Greek Big Bang Theory", A Ludic Journey into Geometric Topology, Cham: Springer Nature, p. 25, doi:10.1007/978-3-031-07442-4, MR 4485665
- ^ 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
- ^ 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
External links
[edit]
- Weisstein, Eric W., "Stella Octangula" ("Compound of two tetrahedra") at MathWorld.
- Klitzing, Richard, "3D compound"
{{cite web}}: CS1 maint: overridden setting (link)