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.{{citation

| last = Cromwell | first = Peter R.

| title = Polyhedra

| year = 1997

| url = https://archive.org/details/polyhedra0000crom/page/261

| publisher = Cambridge University Press

| page = 171, 261

| isbn = 978-0-521-55432-9

}} Magnus Wenninger's Polyhedron Models denote this model as nineteenth W₁₉.{{citation

| last = Wenninger | first = Magnus J.

| year = 1971

| title = Polyhedron Models

| publisher = Cambridge University Press

| page = 37

}}

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

| last = Inchbald | first = Guy

| year = 2006

| title = Facetting Diagrams

| journal = The Mathematical Gazette

| volume = 90 | issue = 518 | pages = 253–261

| doi = 10.1017/S0025557200179653

| jstor = 40378613

}} 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.{{sfnp|Coxeter|1973|p=[http://books.google.com/books?id=2ee7AQAAQBAJ&pg=PA49 49]}}

{{multiple image

| image1 = Stellation of octahedron facets.png

| caption1 = Stellation plane of a stellated octahedron

| image2 = CubeAndStel.svg

| caption2 = Stellated octahedron as a cube faceting

| total_width = 400

| align = center

}}

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".{{r|cromwell}} The two tetrahedra share a common intersphere in the centre, making the compound self-dual.{{citation

| last = Pugh | first = Anthony

| isbn = 9780520030565

| page = 88

| publisher = University of California Press

| title = Polyhedra: A Visual Approach

| year = 1976

| url = https://books.google.com/books?id=IDDxpYQTR7kC&pg=PA88

}} There exist compositions of all symmetries of tetrahedra reflected about the cube's center, so the stellated octahedron may also have pyritohedral symmetry.{{citation

| last = Smith | first = James

| year = 2000

| title = Methods of Geometry

| url = https://books.google.com/books?id=B0khWEZmOlwC&pg=PA403

| page = 403–404

| publisher = John Wiley & Sons

| isbn = 978-1-118-03103-2

}}

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, $\backslash frac\{3\}\{2\}$ times of the side length.{{citation

| last = Loeb | first = Arthur

| editor-first = Jean-François | editor-last = Gabriel

| title = Beyond the Cube: The Architecture of Space Frames and Polyhedra

| contribution = Deconstruction of the Cube

| contribution-url = https://books.google.com/books?id=FkM0945nFV8C&pg=PA233

| page = 233

| publisher = John Wiley & Sons

| year = 1997

}} 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.{{citation

| last1 = Brigaglia | first1 = Aldo

| last2 = Palladino | first2 = Nicla

| last3 = Vaccaro | first3 = Maria Alessandra

| editor1-last = Emmer | editor1-first = Michele

| editor2-last = Abate | editor2-first = Marco

| contribution = Historical notes on star geometry in mathematics, art and nature

| doi = 10.1007/978-3-319-93949-0_17

| pages = 197–211

| publisher = Springer International Publishing

| title = Imagine Math 6: Between Culture and Mathematics

| year = 2018| hdl = 10447/325250

| isbn = 978-3-319-93948-3

}}

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.

File:Spherical compound of two tetrahedra.png, 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:{{citation|title=The Book of Numbers|page=51|url=https://books.google.com/books?id=0--3rcO7dMYC&pg=PA51|first1=John|last1=Conway|author1-link=John Horton Conway|first2=Richard|last2=Guy|author2-link=Richard K. Guy|publisher=Springer|year=1996|isbn=978-0-387-97993-9}}

:0, 1, 14, 51, 124, 245, 426, 679, 1016, 1449, 1990, .... {{OEIS|A007588}}

The stellated octahedron appears with several other polyhedra and polyhedral compounds in M. C. Escher's print "Stars",{{citation|url=http://www.georgehart.com/virtual-polyhedra/escher.html|last=Hart|first=George W.|authorlink=George W. Hart|title=The Polyhedra of M.C. Escher|work=Virtual Polyhedra|year=1996}}. and provides the central form in Escher's Double Planetoid (1949).{{citation

| last = Coxeter | first = H. S. M. | author-link = Harold Scott MacDonald Coxeter

| doi = 10.1007/BF03023010

| issue = 1

| journal = The Mathematical Intelligencer

| pages = 59–69

| title = A special book review: M. C. Escher: His life and complete graphic work

| volume = 7

| year = 1985| s2cid = 189887063 }}. See in particular p. 61.

{{CSS image crop|Location=right|Description=One of the stellated octahedra in the Plaza de Europa, Zaragoza|Image=Zaragoza - Plaza de Europa 1.jpg|bSize=800|cWidth=180|cHeight=180||oLeft=44|oTop=54}}

The obelisk in the center of the {{ill|Plaza de Europa (Zaragoza)|es|lt=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.{{citation|url=https://www.zaragoza.es/sede/servicio/arte-publico/33|title=Obelisco|language=es|trans-title=Obelisk|website=Zaragoza es Cultura|publisher=Ayuntamiento de Zaragoza|access-date=2021-10-19}}

Some modern mystics have associated this shape with the "merkaba",{{citation|title=Sedona: Beyond the Vortex: Activating the Planetary Ascension Program with Sacred Geometry, the Vortex, and the Merkaba|first=Richard|last=Dannelley|publisher=Light Technology Publishing|year=1995|isbn=9781622336708|url=https://books.google.com/books?id=eMDBS8SNUp8C&pg=PT14|page=14}} which according to them is a "counter-rotating energy field" named from an ancient Egyptian word.{{citation|title=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|first=Drunvalo|last=Melchizedek|publisher=Light Technology Publishing|year=2000|isbn=9781891824173|page=4|url=https://books.google.com/books?id=E0FEcoK0ds4C&pg=PA4}} However, the word "merkaba" is actually Hebrew, and more properly refers to a chariot in the visions of Ezekiel.{{citation|title=Pocket Dictionary of Biblical Studies: Over 300 Terms Clearly & Concisely Defined|series=The IVP Pocket Reference Series|first1=Arthur G.|last1=Patzia|first2=Anthony J.|last2=Petrotta|publisher=InterVarsity Press|year=2010|isbn=9780830867028|page=78|url=https://books.google.com/books?id=btEJAwAAQBAJ&pg=PA78}}

The resemblance between this shape and the two-dimensional star of David has also been frequently noted.{{citation|title=Hypergraphics: visualizing complex relationships in art, science, and technology|first=David W.|last=Brisson|publisher=Westview Press for the American Association for the Advancement of Science|year=1978|page=220|quote=The Stella octangula is the 3-d analog of the Star of David}}

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{{commons category}}

• {{MathWorld2|urlname=StellaOctangula|title=Stella Octangula|urlname2=Tetrahedron2-Compound|title2=Compound of two tetrahedra}}
• {{KlitzingPolytopes|../incmats/so.htm|3D compound|mode=cs2}}

Category:Polyhedral compounds

Category:Polyhedral stellation

Category:Sacred geometry