geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ...

, stellation is the process of extending a

polygon In geometry, a polygon () is a plane figure that is described by a finite number of straight line segments connected to form a closed ''polygonal chain'' (or ''polygonal circuit''). The bounded plane region, the bounding circuit, or the two to ...

in two

dimension In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coor ...

polyhedron In geometry, a polyhedron (plural polyhedra or polyhedrons; ) is a three-dimensional shape with flat polygonal faces, straight edges and sharp corners or vertices. A convex polyhedron is the convex hull of finitely many points, not all on ...

in three dimensions, or, in general, a

polytope In elementary geometry, a polytope is a geometric object with flat sides ('' faces''). Polytopes are the generalization of three-dimensional polyhedra to any number of dimensions. Polytopes may exist in any general number of dimensions as an ...

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 Latin ''stella'', "star". Stellation is the reciprocal or dual process to ''

faceting 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 ...

''.

Kepler's definition

In 1619

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 o ...

defined stellation for polygons and polyhedra as the process of extending edges or faces until they meet to form a new polygon or polyhedron. He stellated the regular

dodecahedron In geometry, a dodecahedron (Greek , from ''dōdeka'' "twelve" + ''hédra'' "base", "seat" or "face") or duodecahedron is any polyhedron with twelve flat faces. The most familiar dodecahedron is the regular dodecahedron with regular pentagon ...

to obtain two regular star polyhedra, the

small stellated dodecahedron In geometry, the small stellated dodecahedron is a Kepler-Poinsot polyhedron, named by Arthur Cayley, and with Schläfli symbol . It is one of four nonconvex regular polyhedra. It is composed of 12 pentagrammic faces, with five pentagrams meeti ...

and

great stellated dodecahedron In geometry, the great stellated dodecahedron is a Kepler-Poinsot polyhedron, with Schläfli symbol . It is one of four nonconvex regular polyhedra. It is composed of 12 intersecting pentagrammic faces, with three pentagrams meeting at each ve ...

. He also stellated 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 ...

to obtain the

stella octangula 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 depict ...

, a regular compound of two tetrahedra.

Stellating polygons

Stellating a regular polygon symmetrically creates a regular star polygon or polygonal compound. These polygons are characterised by the number of times ''m'' that the polygonal boundary winds around the centre of the figure. Like all regular polygons, their vertices lie on a circle. ''m'' also corresponds to the number of vertices around the circle to get from one end of a given edge to the other, starting at 1. A regular star polygon is represented by its Schläfli symbol , where ''n'' is the number of vertices, ''m'' is the ''step'' used in sequencing the edges around it, and ''m'' and ''n'' are

coprime In mathematics, two integers and are coprime, relatively prime or mutually prime if the only positive integer that is a divisor of both of them is 1. Consequently, any prime number that divides does not divide , and vice versa. This is equivale ...

(have no common

factor Factor, a Latin word meaning "who/which acts", may refer to: Commerce * Factor (agent), a person who acts for, notably a mercantile and colonial agent * Factor (Scotland), a person or firm managing a Scottish estate * Factors of production, suc ...

). The case ''m'' = 1 gives the convex polygon . ''m'' also must be less than half of ''n''; otherwise the lines will either be parallel or diverge, preventing the figure from ever closing. If ''n'' and ''m'' do have a common factor, then the figure is a regular compound. For example is the regular compound of two triangles or hexagram, while is a compound of two pentagrams . Some authors use the Schläfli symbol for such regular compounds. Others regard the symbol as indicating a single path which is wound ''m'' times around vertex points, such that one edge is superimposed upon another and each vertex point is visited ''m'' times. In this case a modified symbol may be used for the compound, for example 2 for the hexagram and 2 for the regular compound of two pentagrams. A regular ''n''-gon has stellations if ''n'' is even (assuming compounds of multiple degenerate

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 visu ...

s are not considered), and stellations if ''n'' is

odd Odd means unpaired, occasional, strange or unusual, or a person who is viewed as eccentric. Odd may also refer to: Acronym * ODD (Text Encoding Initiative) ("One Document Does it all"), an abstracted literate-programming format for describing X ...

. Like the

heptagon In geometry, a heptagon or septagon is a seven-sided polygon or 7-gon. The heptagon is sometimes referred to as the septagon, using "sept-" (an elision of ''septua-'', a Latin-derived numerical prefix, rather than '' hepta-'', a Greek-derived nu ...

, the octagon also has two

octagram In geometry, an octagram is an eight-angled star polygon. The name ''octagram'' combine a Greek numeral prefix, '' octa-'', with the Greek suffix '' -gram''. The ''-gram'' suffix derives from γραμμή (''grammḗ'') meaning "line". Deta ...

mic stellations, one, being a star polygon, and the other, , being the compound of two

squares In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90- degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-length a ...

Stellating polyhedra

A polyhedron is stellated by extending the edges or face planes of a polyhedron until they meet again to form a new polyhedron or compound. The interior of the new polyhedron is divided by the faces into a number of cells. The face planes of a polyhedron may divide space into many such cells, and as the stellation process continues then more of these cells will be enclosed. For a symmetrical polyhedron, these cells will fall into groups, or sets, of congruent cells – we say that the cells in such a congruent set are of the same type. A common method of finding stellations involves selecting one or more cell types. This can lead to a huge number of possible forms, so further criteria are often imposed to reduce the set to those stellations that are significant and unique in some way. A set of cells forming a closed layer around its core is called a shell. For a symmetrical polyhedron, a shell may be made up of one or more cell types. Based on such ideas, several restrictive categories of interest have been identified. * Main-line stellations. Adding successive shells to the core polyhedron leads to the set of main-line stellations. * Fully supported stellations. The underside faces of a cell can appear externally as an "overhang." In a fully supported stellation there are no such overhangs, and all visible parts of a face are seen from the same side. * Monoacral stellations. Literally "single-peaked." Where there is only one kind of peak, or vertex, in a stellation (i.e. all vertices are congruent within a single symmetry orbit), the stellation is monoacral. All such stellations are fully supported. * Primary stellations. Where a polyhedron has planes of mirror symmetry, edges falling in these planes are said to lie in primary lines. If all edges lie in primary lines, the stellation is primary. All primary stellations are fully supported. * Miller stellations. In "The Fifty-Nine Icosahedra"

Coxeter Harold Scott MacDonald "Donald" Coxeter, (9 February 1907 – 31 March 2003) was a British and later also Canadian geometer. He is regarded as one of the greatest geometers of the 20th century. Biography Coxeter was born in Kensington to ...

, Du Val, Flather and Petrie record five rules suggested by

Miller A miller is a person who operates a mill, a machine to grind a grain (for example corn or wheat) to make flour. Milling is among the oldest of human occupations. "Miller", "Milne" and other variants are common surnames, as are their equivalent ...

. Although these rules refer specifically to the icosahedron's geometry, they have been adapted to work for arbitrary polyhedra. They ensure, among other things, that the rotational symmetry of the original polyhedron is preserved, and that each stellation is different in outward appearance. The four kinds of stellation just defined are all subsets of the Miller stellations. We can also identify some other categories: *A partial stellation is one where not all elements of a given dimensionality are extended. *A sub-symmetric stellation is one where not all elements are extended symmetrically. The

Archimedean solids In geometry, an Archimedean solid is one of the 13 solids first enumerated by Archimedes. They are the convex uniform polyhedra composed of regular polygons meeting in identical vertices, excluding the five Platonic solids (which are compose ...

and their duals can also be stellated. Here we usually add the rule that all of the original face planes must be present in the stellation, i.e. we do not consider partial stellations. For example the cube is not usually considered a stellation of the

cuboctahedron A cuboctahedron is a polyhedron with 8 triangular faces and 6 square faces. A cuboctahedron has 12 identical vertices, with 2 triangles and 2 squares meeting at each, and 24 identical edges, each separating a triangle from a square. As such, it ...

. Generalising Miller's rules there are: * 4 stellations of the

rhombic dodecahedron In geometry, the rhombic dodecahedron is a convex polyhedron with 12 congruent rhombic faces. It has 24 edges, and 14 vertices of 2 types. It is a Catalan solid, and the dual polyhedron of the cuboctahedron. Properties The rhombic dodecahed ...

* 187 stellations of the

triakis tetrahedron In geometry, a triakis tetrahedron (or kistetrahedron) is a Catalan solid with 12 faces. Each Catalan solid is the dual of an Archimedean solid. The dual of the triakis tetrahedron is the truncated tetrahedron. The triakis tetrahedron can be se ...

* 358,833,097 stellations of the

rhombic triacontahedron In geometry, the rhombic triacontahedron, sometimes simply called the triacontahedron as it is the most common thirty-faced polyhedron, is a convex polyhedron with 30 rhombic faces. It has 60 edges and 32 vertices of two types. It is a Ca ...

* 17 stellations of the

(4 are shown in Wenninger's ''Polyhedron Models'') * An unknown number of stellations of the

icosidodecahedron In geometry, an icosidodecahedron is a polyhedron with twenty (''icosi'') triangular faces and twelve (''dodeca'') pentagonal faces. An icosidodecahedron has 30 identical vertices, with two triangles and two pentagons meeting at each, and 60 i ...

; there are 7071671 non-

chiral Chirality is a property of asymmetry important in several branches of science. The word ''chirality'' is derived from the Greek (''kheir''), "hand", a familiar chiral object. An object or a system is ''chiral'' if it is distinguishable from i ...

stellations, but the number of chiral stellations is unknown. (20 are shown in Wenninger's ''Polyhedron Models'') Seventeen of the nonconvex uniform polyhedra are stellations of Archimedean solids.

Miller's rules

In the book ''

The Fifty-Nine Icosahedra ''The Fifty-Nine Icosahedra'' is a book written and illustrated by H. S. M. Coxeter, P. Du Val, H. T. Flather and J. F. Petrie. It enumerates certain stellations of the regular convex or Platonic icosahedron, according to a set of rules put forw ...

'', J.C.P. Miller proposed a set of rules for defining which stellation forms should be considered "properly significant and distinct". These rules have been adapted for use with stellations of many other polyhedra. Under Miller's rules we find: * There are no stellations of the

tetrahedron 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 th ...

, because all faces are adjacent * There are no stellations of the cube, because non-adjacent faces are parallel and thus cannot be extended to meet in new edges * There is 1 stellation of the

, the

* There are 3 stellations of the

: the

, the

great dodecahedron In geometry, the great dodecahedron is a Kepler–Poinsot polyhedron, with Schläfli symbol and Coxeter–Dynkin diagram of . It is one of four nonconvex regular polyhedra. It is composed of 12 pentagonal faces (six pairs of parallel pentagon ...

and the

, all of which are Kepler–Poinsot polyhedra. * There are 58 stellations of the icosahedron, including the

great icosahedron In geometry, the great icosahedron is one of four Kepler–Poinsot polyhedra (nonconvex regular polyhedra), with Schläfli symbol and Coxeter-Dynkin diagram of . It is composed of 20 intersecting triangular faces, having five triangles meeti ...

(one of the Kepler–Poinsot polyhedra), and the second and

final Final, Finals or The Final may refer to: * Final (competition), the last or championship round of a sporting competition, match, game, or other contest which decides a winner for an event ** Another term for playoffs, describing a sequence of con ...

stellations of the icosahedron. The 59th model in ''The fifty nine icosahedra'' is the original icosahedron itself. Many "Miller stellations" cannot be obtained directly by using Kepler's method. For example many have hollow centres where the original faces and edges of the core polyhedron are entirely missing: there is nothing left to be stellated. On the other hand, Kepler's method also yields stellations which are forbidden by Miller's rules since their cells are edge- or vertex-connected, even though their faces are single polygons. This discrepancy received no real attention until Inchbald (2002).

Other rules for stellation

Miller's rules by no means represent the "correct" way to enumerate stellations. They are based on combining parts within the stellation diagram in certain ways, and don't take into account the topology of the resulting faces. As such there are some quite reasonable stellations of the icosahedron that are not part of their list – one was identified by James Bridge in 1974, while some "Miller stellations" are questionable as to whether they should be regarded as stellations at all – one of the icosahedral set comprises several quite disconnected cells floating symmetrically in space. As yet an alternative set of rules that takes this into account has not been fully developed. Most progress has been made based on the notion that stellation is the reciprocal or dual process to

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 ...

, whereby parts are removed from a polyhedron without creating any new vertices. For every stellation of some polyhedron, there is a dual facetting of the

dual polyhedron In geometry, every polyhedron is associated with a second dual structure, where the vertices of one correspond to the faces of the other, and the edges between pairs of vertices of one correspond to the edges between pairs of faces of the other. ...

, and vice versa. By studying facettings of the dual, we gain insights into the stellations of the original. Bridge found his new stellation of the icosahedron by studying the facettings of its dual, the dodecahedron. Some polyhedronists take the view that stellation is a two-way process, such that any two polyhedra sharing the same face planes are stellations of each other. This is understandable if one is devising a general algorithm suitable for use in a computer program, but is otherwise not particularly helpful. Many examples of stellations can be found in the list of Wenninger's stellation models.

Stellating polytopes

The stellation process can be applied to higher dimensional polytopes as well. A stellation diagram of an ''n''-polytope exists in an (''n'' − 1)-dimensional hyperplane of a given

facet Facets () are flat faces on geometric shapes. The organization of naturally occurring facets was key to early developments in crystallography, since they reflect the underlying symmetry of the crystal structure. Gemstones commonly have facets cut ...

. For example, in 4-space, the

great grand stellated 120-cell In geometry, the great grand stellated 120-cell or great grand stellated polydodecahedron is a regular star 4-polytope with Schläfli symbol , one of 10 regular Schläfli-Hess 4-polytopes. It is unique among the 10 for having 600 vertices, and ...

is the final stellation of the

regular 4-polytope In mathematics, a regular 4-polytope is a regular four-dimensional polytope. They are the four-dimensional analogues of the regular polyhedra in three dimensions and the regular polygons in two dimensions. There are six convex and ten star reg ...

120-cell In geometry, the 120-cell is the convex regular 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol . It is also called a C120, dodecaplex (short for "dodecahedral complex"), hyperdodecahedron, polydodecahedron, hec ...

Naming stellations

The first systematic naming of stellated polyhedra was Cayley's naming of the regular star polyhedra (nowadays known as the Kepler–Poinsot polyhedra). This system was widely, but not always systematically, adopted for other polyhedra and higher polytopes.

John Conway John Horton Conway (26 December 1937 – 11 April 2020) was an English mathematician active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory. He also made contributions to many branches ...

devised a terminology for stellated

polyhedra In geometry, a polyhedron (plural polyhedra or polyhedrons; ) is a three-dimensional shape with flat polygonal faces, straight edges and sharp corners or vertices. A convex polyhedron is the convex hull of finitely many points, not all on ...

and polychora (Coxeter 1974). In this system the process of extending edges to create a new figure is called ''stellation'', that of extending faces is called ''greatening'' and that of extending cells is called ''aggrandizement'' (this last does not apply to polyhedra). This allows a systematic use of words such as 'stellated', 'great', and 'grand' in devising names for the resulting figures. For example Conway proposed some minor variations to the names of the Kepler–Poinsot polyhedra.

Stellation to infinity

Wenninger noticed that some polyhedra, such as the cube, do not have any finite stellations. However stellation cells can be constructed as prisms which extend to infinity. The figure comprising these prisms may be called a stellation to infinity. By most definitions of a polyhedron, however, these stellations are not strictly polyhedra. Wenninger's figures occurred as duals of the uniform hemipolyhedra, where the faces that pass through the center are sent to vertices "at infinity".

From mathematics to art

Alongside from his contributions to mathematics,

Magnus Wenninger Father Magnus J. Wenninger OSB (October 31, 1919Banchoff (2002)– February 17, 2017) was an American mathematician who worked on constructing polyhedron models, and wrote the first book on their construction. Early life and education Born to Ge ...

is described in the context of the relationship of

mathematics and art Mathematics and art are related in a variety of ways. Mathematics has itself been described as an art mathematical beauty, motivated by beauty. Mathematics can be discerned in arts such as Music and mathematics, music, dance, painting, Mathema ...

as making "especially beautiful" models of complex stellated polyhedra. Marble floor mosaic Basilica of St Mark Vencice

The

Italian Renaissance The Italian Renaissance ( it, Rinascimento ) was a period in Italian history covering the 15th and 16th centuries. The period is known for the initial development of the broader Renaissance culture that spread across Europe and marked the trans ...

artist

Paolo Uccello Paolo Uccello ( , ; 1397 – 10 December 1475), born Paolo di Dono, was an Italian (Florentine) painter and mathematician who was notable for his pioneering work on visual perspective in art. In his book ''Lives of the Most Excellent Painters, S ...

created a floor mosaic showing a small stellated dodecahedron in the Basilica of St Mark, Venice, c. 1430. Uccello's depiction was used as the symbol for the

Venice Biennale The Venice Biennale (; it, La Biennale di Venezia) is an international cultural exhibition hosted annually in Venice, Italy by the Biennale Foundation. The biennale has been organised every year since 1895, which makes it the oldest of ...

in 1986 on the topic of "Art and Science". The same stellation is central to two lithographs by M. C. Escher: ''Contrast (Order and Chaos)'', 1950, and '' Gravitation'', 1952.

References

* Bridge, N. J.; Facetting the dodecahedron, ''Acta Crystallographica'' A30 (1974), pp. 548–552. *

, H.S.M.; ''Regular complex polytopes'' (1974). *

, H.S.M.; Du Val, P.; Flather, H. T.; and Petrie, J. F. ''The Fifty-Nine Icosahedra'', 3rd Edition. Stradbroke, England: Tarquin Publications (1999). * Inchbald, G.; In search of the lost icosahedra, ''The Mathematical Gazette'' 86 (2002), pp. 208-215. * Messer, P.; Stellations of the rhombic triacontahedron and beyond, ''Symmetry: culture and science'', 11 (2000), pp. 201–230. * *

External links

* {{mathworld , urlname = Stellation , title = Stellation
Stellating the Icosahedron and Facetting the Dodecahedron

Stella: Polyhedron Navigator
– Software for exploring polyhedra and printing nets for their physical construction. Includes uniform polyhedra, stellations, compounds, Johnson solids, etc.
Enumeration of stellations

Vladimir Bulatov ''Polyhedra Stellation.''

Vladimir Bulatov's Polyhedra Stellations Applet packaged as an OS X application

Stellation Applet

An Interactive Creation of Polyhedra Stellations with Various Symmetries

Further Stellations of the Uniform Polyhedra, John Lawrence Hudson
The Mathematical Intelligencer, Volume 31, Number 4, 2009 Polygons Polyhedra Polytopes