The stellated octahedron is the only

stellation In geometry, stellation is the process of extending a polygon in two dimensions, 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 el ...

of the

octahedron In geometry, an octahedron (plural: octahedra, octahedrons) is a polyhedron with eight faces. The term is most commonly used to refer to the regular octahedron, a Platonic solid composed of eight equilateral triangles, four of which meet at ea ...

. It is also called the stella octangula (Latin for "eight-pointed star"), a name given to it by

Johannes Kepler Johannes Kepler (; ; 27 December 1571 – 15 November 1630) was a German astronomer, mathematician, astrologer, natural philosopher and writer on music. He is a key figure in the 17th-century Scientific Revolution, best known for his laws ...

in 1609, though it was known to earlier

geometer A geometer is a mathematician whose area of study is geometry. Some notable geometers and their main fields of work, chronologically listed, are: 1000 BCE to 1 BCE * Baudhayana (fl. c. 800 BC) – Euclidean geometry, geometric algebra * ...

s. It was depicted in Pacioli's ''De Divina Proportione,'' 1509. It is the simplest of five regular

polyhedral compound 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. The outer vertices of a compound can be connected ...

s, and the only regular

compound of two tetrahedra In geometry, a compound of two tetrahedra is constructed by two overlapping tetrahedra, usually implied as regular tetrahedra. Stellated octahedron There is only one uniform polyhedral compound, the stellated octahedron, which has octahedral ...

. 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 , can be seen as a compound composed of an upwards (blue here) and downwards (pink) facing equilateral triangle, with their intersection as a regular hexagon (in green). A hexagram ( Greek language, Greek) or sexagram (Latin) is a six-pointed ...

: the hexagram is a two-dimensional shape formed from two overlapping equilateral triangles,

centrally symmetric In geometry, a point reflection (point inversion, central inversion, or inversion through a point) is a type of isometry of Euclidean space. An object that is invariant under a point reflection is said to possess point symmetry; if it is invari ...

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 In geometry, the 5-cell is the convex 4-polytope with Schläfli symbol . It is a 5-vertex four-dimensional object bounded by five tetrahedral cells. It is also known as a C5, pentachoron, pentatope, pentahedroid, or tetrahedral pyramid. It is ...

. It can also be seen as one of the stages in the construction of a 3D

Koch snowflake The Koch snowflake (also known as the Koch curve, Koch star, or Koch island) is a fractal curve and one of the earliest fractals to have been described. It is based on the Koch curve, which appeared in a 1904 paper titled "On a Continuous Curv ...

, 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: (±1/2, ±1/2, 0) (0, 0, ±1/√2) (±1, 0, ±1/√2) (0, ±1, ±1/√2) The stellated octahedron can be constructed in several ways: *It is a

of the

regular octahedron In geometry, an octahedron (plural: octahedra, octahedrons) is a polyhedron with eight faces. The term is most commonly used to refer to the regular octahedron, a Platonic solid composed of eight equilateral triangles, four of which meet at ea ...

, sharing the same face planes. (See Wenninger model W₁₉.) *It is also a regular

polyhedron compound 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. The outer vertices of a compound can be connected ...

, when constructed as the union of two regular

tetrahedra In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all the o ...

(a regular tetrahedron and its dual tetrahedron). *It can be obtained as an augmentation of the regular

, by adding tetrahedral

pyramids A pyramid (from el, πυραμίς ') is a structure whose outer surfaces are triangular and converge to a single step at the top, making the shape roughly a pyramid in the geometric sense. The base of a pyramid can be trilateral, quadrilate ...

on each face. In this construction it has the same topology as the convex

Catalan solid In mathematics, a Catalan solid, or Archimedean dual, is a dual polyhedron to an Archimedean solid. There are 13 Catalan solids. They are named for the Belgian mathematician Eugène Catalan, who first described them in 1865. The Catalan sol ...

, the

triakis octahedron In geometry, a triakis octahedron (or trigonal trisoctahedron or kisoctahedronConway, Symmetries of things, p. 284) is an Archimedean dual solid, or a Catalan solid. Its dual is the truncated cube. It can be seen as an octahedron with triangular ...

, which has much shorter pyramids. *It is a

facetting Stella octangula as a faceting of the cube In geometry, faceting (also spelled facetting) is the process of removing parts of a polygon, polyhedron or polytope, without creating any new vertices. New edges of a faceted polyhedron may be cre ...

of the

cube In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. Viewed from a corner it is a hexagon and its net is usually depicted as a cross. The cube is the only r ...

, sharing the

vertex arrangement In geometry, a vertex arrangement is a set of points in space described by their relative positions. They can be described by their use in polytopes. For example, a ''square vertex arrangement'' is understood to mean four points in a plane, equ ...

. *It can be seen as a

antiprism In geometry, an antiprism or is a polyhedron composed of two parallel direct copies (not mirror images) of an polygon, connected by an alternating band of triangles. They are represented by the Conway notation . Antiprisms are a subclass o ...

; with being a tetragram, a compound of two dual

digon In geometry, a digon is a polygon with two sides (edges) and two vertices. Its construction is degenerate in a Euclidean plane because either the two sides would coincide or one or both would have to be curved; however, it can be easily visua ...

s, 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 Net or net may refer to: Mathematics and physics * Net (mathematics), a filter-like topological generalization of a sequence * Net, a linear system of divisors of dimension 2 * Net (polyhedron), an arrangement of polygons that can be folded up ...

of a four-dimensional octahedral pyramid, consisting of a central octahedron surrounded by eight tetrahedra.

Related concepts

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 In mathematics, the concept of a projective space originated from the visual effect of perspective, where parallel lines seem to meet ''at infinity''. A projective space may thus be viewed as the extension of a Euclidean space, or, more generally ...

) 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, between two parallel edges of the two tetrahedra. 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 number In mathematics, a stella octangula number is a figurate number based on the stella octangula, of the form .. The sequence of stella octangula numbers is :0, 1, 14, 51, 124, 245, 426, 679, 1016, 1449, 1990, ... Only two of these numbers are squar ...

s are

figurate number The term figurate number is used by different writers for members of different sets of numbers, generalizing from triangular numbers to different shapes (polygonal numbers) and different dimensions (polyhedral numbers). The term can mean * polygon ...

s 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, ....

In popular culture

The stellated octahedron appears with several other polyhedra and polyhedral compounds in

M. C. Escher Maurits Cornelis Escher (; 17 June 1898 – 27 March 1972) was a Dutch graphic artist who made mathematically inspired woodcuts, lithographs, and mezzotints. Despite wide popular interest, Escher was for most of his life neglected in t ...

's print "

Stars A star is an astronomical object comprising a luminous spheroid of plasma held together by its gravity. The nearest star to Earth is the Sun. Many other stars are visible to the naked eye at night, but their immense distances from Earth ma ...

", and provides the central form in Escher's '' Double Planetoid'' (1949). The obelisk in the center of the in

Zaragoza Zaragoza, also known in English as Saragossa,''Encyclopædia Britannica'"Zaragoza (conventional Saragossa)" is the capital city of the Zaragoza Province and of the autonomous community of Aragon, Spain. It lies by the Ebro river and its tributari ...

Spain , image_flag = Bandera de España.svg , image_coat = Escudo de España (mazonado).svg , national_motto = ''Plus ultra'' (Latin)(English: "Further Beyond") , national_anthem = (English: "Royal March") , i ...

, is surrounded by twelve stellated octahedral lamppposts, shaped to form a three-dimensional version of the

Flag of Europe The Flag of Europe or European Flag consists of twelve golden stars forming a circle on a blue field. It was designed and adopted in 1955 by the Council of Europe (CoE) as a symbol for the whole of Europe. Since 1985, the flag has also been ...

. Some modern mystics have associated this shape with the "merkaba", which according to them is a "counter-rotating energy field" named from an ancient Egyptian word. However, the word "merkaba" is actually

Hebrew Hebrew (; ; ) is a Northwest Semitic language of the Afroasiatic language family. Historically, it is one of the spoken languages of the Israelites and their longest-surviving descendants, the Jews and Samaritans. It was largely preserved ...

, and more properly refers to a

chariot A chariot is a type of cart driven by a charioteer, usually using horses to provide rapid motive power. The oldest known chariots have been found in burials of the Sintashta culture in modern-day Chelyabinsk Oblast, Russia, dated to c. 2000&nbs ...

in the visions of

Ezekiel Ezekiel (; he, יְחֶזְקֵאל ''Yəḥezqēʾl'' ; in the Septuagint written in grc-koi, Ἰεζεκιήλ ) is the central protagonist of the Book of Ezekiel in the Hebrew Bible. In Judaism, Christianity, and Islam, Ezekiel is acknow ...

. The resemblance between this shape and the two-dimensional

star of David The Star of David (). is a generally recognized symbol of both Jewish identity and Judaism. Its shape is that of a hexagram: the compound of two equilateral triangles. A derivation of the ''seal of Solomon'', which was used for decorative ...

has also been frequently noted.

References

External links

* *{{KlitzingPolytopes, ../incmats/so.htm, 3D compound, mode=cs2 Polyhedral stellation Polyhedral compounds