Stellated octahedron

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Stellated octahedron
Compound of two tetrahedra.png
Seen as a compound of two regular tetrahedra (red and yellow)
Type Regular compound
Coxeter symbol{4,3}[2{3,3}]{3,4} [1]
Schläfli symbols {{3,3}}
a{4,3}
ß{2,4}
ßr{2,2}
Coxeter diagrams CDel nodes 10ru.pngCDel split2.pngCDel node.pngCDel nodes 01rd.pngCDel split2.pngCDel node.png
CDel node h3.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png
CDel node h3.pngCDel 2x.pngCDel node h3.pngCDel 4.pngCDel node.png
CDel node h3.pngCDel 2x.pngCDel node h3.pngCDel 2x.pngCDel node h3.png
Stellation core Octahedron
Convex hull Cube
IndexUC4, W19
Polyhedra2 tetrahedra
Faces8 triangles
Edges12
Vertices8
DualSelf-dual
Symmetry group
Coxeter group
Oh, [4,3], order 48
D4h, [4,2], order 16
D2h, [2,2], order 8
D3d, [2+,6], order 12
Subgroup restricting
to one constituent
Td, [3,3], order 24
D2d, [2+,4], order 8
D2, [2,2]+, order 4
C3v, [3], order 6
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 De Divina Proportione, 1509. [2]

Contents

It is the simplest of five regular polyhedral compounds, and the only regular compound of two tetrahedra. It is also the least dense of the regular polyhedral compounds, having a density of 2.

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. It can also be seen as one of the stages in the construction of a 3D Koch snowflake, a fractal shape formed by repeated attachment of smaller tetrahedra to each triangular face of a larger figure. The first stage of the construction of the Koch Snowflake is a single central tetrahedron, and the second stage, formed by adding four smaller tetrahedra to the faces of the central tetrahedron, is the stellated octahedron.

Construction

The Cartesian coordinates of the stellated octahedron are as follows:

The stellated octahedron can be constructed in several ways:

Stellated octahedron stellation plane.png
In perspective
Stellation of octahedron facets.png
Stellation plane
The only stellation of a regular octahedron, with one stellation plane in yellow.
CubeAndStel.svg
Facetting of a cube
Diagonal facet of cube.png
A single diagonal triangle facetting in red
The stellated octahedron is the first iteration of the 3D analogue of a Koch snowflake. Koch cube.gif
The stellated octahedron is the first iteration of the 3D analogue of a Koch snowflake.

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. They are

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

The stellated octahedron appears with several other polyhedra and polyhedral compounds in M. C. Escher's print "Stars", [3] and provides the central form in Escher's Double Planetoid (1949). [4]

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. [5]

Some modern mystics have associated this shape with the "merkaba", [6] which according to them is a "counter-rotating energy field" named from an ancient Egyptian word. [7] However, the word "merkaba" is actually Hebrew, and more properly refers to a chariot in the visions of Ezekiel. [8] The resemblance between this shape and the two-dimensional star of David has also been frequently noted. [9]

The musical project "Miracle Musical" (often stylized in its original Japanese title ミラクルミュージカル, pronounced "mirakuru myujikaru" [10] ), spearheaded by Tally Hall member Joe Hawley along with bandmate Ross Federman and honorary bandmate Bora Karaca, makes multiple references towards the stellated octahedron as the stella octangula. The shape is shown on the main website of the project, as well as the merchandise store. [10] [11] The third song on their first and only studio album, "Hawaii: Part II", "Black Rainbows" features a lyric sung by Madi Diaz which simply says "Stella octangula". [12]

Related Research Articles

In geometry, an octahedron is a polyhedron with eight faces. One special case is the regular octahedron, a Platonic solid composed of eight equilateral triangles, four of which meet at each vertex. Regular octahedra occur in nature as crystal structures. Many types of irregular octahedra also exist, including both convex and non-convex shapes.

In geometry, a polyhedral compound is a figure that is composed of several polyhedra sharing a common centre. They are the three-dimensional analogs of polygonal compounds such as the hexagram.

In geometry, a Platonic solid is a convex, regular polyhedron in three-dimensional Euclidean space. Being a regular polyhedron means that the faces are congruent regular polygons, and the same number of faces meet at each vertex. There are only five such polyhedra:

In geometry, a tetrahedron, also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertices. The tetrahedron is the simplest of all the ordinary convex polyhedra.

<span class="mw-page-title-main">Stellation</span> Extending the elements of a polytope to form a new figure

In geometry, stellation is the process of extending a polygon in two dimensions, a polyhedron in three dimensions, or, in general, a polytope in n dimensions to form a new figure. Starting with an original figure, the process extends specific elements such as its edges or face planes, usually in a symmetrical way, until they meet each other again to form the closed boundary of a new figure. The new figure is a stellation of the original. The word stellation comes from the Latin stellātus, "starred", which in turn comes from the Latin stella, "star". Stellation is the reciprocal or dual process to faceting.

<span class="mw-page-title-main">Tetrakis hexahedron</span> Catalan solid with 24 faces

In geometry, a tetrakis hexahedron is a Catalan solid. Its dual is the truncated octahedron, an Archimedean solid.

<i>Stars</i> (M. C. Escher) Wood engraving print by M. C. Escher

Stars is a wood engraving print created by the Dutch artist M. C. Escher in 1948, depicting two chameleons in a polyhedral cage floating through space.

<span class="mw-page-title-main">Great icosahedron</span> Kepler-Poinsot polyhedron with 20 faces

In geometry, the great icosahedron is one of four Kepler–Poinsot polyhedra, with Schläfli symbol {3,52} and Coxeter-Dynkin diagram of . It is composed of 20 intersecting triangular faces, having five triangles meeting at each vertex in a pentagrammic sequence.

<span class="mw-page-title-main">Alternation (geometry)</span> Removal of alternate vertices

In geometry, an alternation or partial truncation, is an operation on a polygon, polyhedron, tiling, or higher dimensional polytope that removes alternate vertices.

<span class="mw-page-title-main">Midsphere</span> Sphere tangent to every edge of a polyhedron

In geometry, the midsphere or intersphere of a convex polyhedron is a sphere which is tangent to every edge of the polyhedron. Not every polyhedron has a midsphere, but the uniform polyhedra, including the regular, quasiregular and semiregular polyhedra and their duals all have midspheres. The radius of the midsphere is called the midradius. A polyhedron that has a midsphere is said to be midscribed about this sphere.

<span class="mw-page-title-main">Regular dodecahedron</span> Convex polyhedron with 12 regular pentagonal faces

A regular dodecahedron or pentagonal dodecahedron is a dodecahedron composed of regular pentagonal faces, three meeting at each vertex. It is an example of Platonic solids, described as cosmic stellation by Plato in his dialogues, and it was used as part of Solar System proposed by Johannes Kepler. However, the regular dodecahedron, including the other Platonic solids, has already been described by other philosophers since antiquity.

<span class="mw-page-title-main">Compound of ten tetrahedra</span> Polyhedral compound

The compound of ten tetrahedra is one of the five regular polyhedral compounds. This polyhedron can be seen as either a stellation of the icosahedron or a compound. This compound was first described by Edmund Hess in 1876.

<span class="mw-page-title-main">Compound of five cubes</span> Polyhedral compound

The compound of five cubes is one of the five regular polyhedral compounds. It was first described by Edmund Hess in 1876.

<span class="mw-page-title-main">Compound of two tetrahedra</span> Polyhedral compound

In geometry, a compound of two tetrahedra is constructed by two overlapping tetrahedra, usually implied as regular tetrahedra.

<span class="mw-page-title-main">Compound of six tetrahedra</span> Polyhedral compound

The compound of six tetrahedra is a uniform polyhedron compound. It's composed of a symmetric arrangement of 6 tetrahedra. It can be constructed by inscribing a stella octangula within each cube in the compound of three cubes, or by stellating each octahedron in the compound of three octahedra.

<span class="mw-page-title-main">Compound of twelve tetrahedra with rotational freedom</span> Polyhedral compound

This uniform polyhedron compound is a symmetric arrangement of 12 tetrahedra, considered as antiprisms. It can be constructed by superimposing six identical copies of the stella octangula, and then rotating them in pairs about the three axes that pass through the centres of two opposite cubic faces. Each stella octangula is rotated by an equal angle θ. Equivalently, a stella octangula may be inscribed within each cube in the compound of six cubes with rotational freedom, which has the same vertices as this compound.

<span class="mw-page-title-main">Compound of three octahedra</span> Polyhedral compound

In mathematics, the compound of three octahedra or octahedron 3-compound is a polyhedral compound formed from three regular octahedra, all sharing a common center but rotated with respect to each other. Although appearing earlier in the mathematical literature, it was rediscovered and popularized by M. C. Escher, who used it in the central image of his 1948 woodcut Stars.

<span class="mw-page-title-main">Double Planetoid</span> Escher print of a stellated octahedron shaped planetoid

Double Planetoid is a wood engraving print by the Dutch artist M. C. Escher, first printed in 1949.

References

  1. H.S.M. Coxeter, Regular Polytopes , (3rd edition, 1973), Dover edition, ISBN   0-486-61480-8, 3.6 The five regular compounds, pp.47-50, 6.2 Stellating the Platonic solids, pp.96-104
  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. Hart, George W. (1996), "The Polyhedra of M.C. Escher", Virtual Polyhedra.
  4. 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.
  5. "Obelisco" [Obelisk], Zaragoza es Cultura (in Spanish), Ayuntamiento de Zaragoza, retrieved 2021-10-19
  6. 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
  7. 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
  8. 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
  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, p. 220, The Stella octangula is the 3-d analog of the Star of David
  10. 1 2 "ミラクルミュージカル". ミラクルミュージカル. Retrieved 2024-03-09.
  11. "Miracle Musical Store". Miracle Musical. Retrieved 2024-03-09.
  12. Miracle Musical (Ft. Joe Hawley & Madi Diaz) – Black Rainbows , retrieved 2024-03-09