Stellated octahedron

{{Short description|Polyhedral compound}}

{{infobox

| title = Stellated octahedron

| image = 200px

| label1 = Type

| data1 = Regular compound

| label2 = Coxeter symbol

| data2 = {4,3}[2{3,3}]{3,4}{{citation

| last = Coxeter | first = Harold | author-link = Harold Scott MacDonald Coxeter

| title-link = Regular Polytopes (book)

| title = Regular Polytopes

| edition = 3rd

| year = 1973

| publisher = Dover Publications

| isbn = 0-486-61480-8

| chapter = The five regular compounds

| pages = 47–50, 96–104

}}

| label3 = Schläfli symbols

| data3 = {{3,3}}
a{4,3}
ß{2,4}
ßr{2,2}

| label4 = Coxeter diagrams

| data4 = {{CDD|nodes_10ru|split2|node}} ∪ {{CDD|nodes_01rd|split2|node}}
{{CDD|node_h3|4|node|3|node}}
{{CDD|node_h3|2x|node_h3|4|node}}
{{CDD|node_h3|2x|node_h3|2x|node_h3}}

| label5 = Stellation core | data5 = regular octahedron

| label6 = Convex hull | data6 = cube

| label7 = Index | data7 = UC4, W19

| label8 = Polyhedra | data8 = two tetrahedra

| label9 = Faces | data9 = 8 triangles

| label10 = Edges | data10 = 12

| label11 = Vertices | data11 = 8

| label12 = Dual polyhedron | data12 = self-dual

| label13 = Symmetry group | data13 = octahedral symmetry, pyritohedral symmetry

}}

File:3D model of a Stellated Octahedron.stl

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 De Divina Proportione, 1509.{{citation

| last = Barnes | first = John

| contribution = Shapes and Solids

| doi = 10.1007/978-3-642-05092-3_2

| pages = 25–56

| publisher = Springer

| title = Gems of Geometry

| year = 2009| isbn = 978-3-642-05091-6

}}.

It is the simplest of 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 amount 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.{{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, \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.

Related concepts

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}}

References

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