geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...

, stellation is the process of extending a

polygon In geometry, a polygon () is a plane figure made up of line segments connected to form a closed polygonal chain. The segments of a closed polygonal chain are called its '' edges'' or ''sides''. The points where two edges meet are the polygon ...

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 coo ...

s, a

polyhedron In geometry, a polyhedron (: polyhedra or polyhedrons; ) is a three-dimensional figure with flat polygonal Face (geometry), faces, straight Edge (geometry), edges and sharp corners or Vertex (geometry), vertices. The term "polyhedron" may refer ...

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 the 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 Vertex (geometry), vertices. New edges of a faceted po ...

''.

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 of p ...

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 (; ) or duodecahedron is any polyhedron with twelve flat faces. The most familiar dodecahedron is the regular dodecahedron with regular pentagons as faces, which is a Platonic solid. There are also three Kepler–Po ...

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 List of regular polytopes#Non-convex 2, regular polyhedra. It is composed of 12 pentag ...

and the

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 eac ...

. He also stellated the regular

octahedron In geometry, an octahedron (: octahedra or octahedrons) is any 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. Many types of i ...

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 depicte ...

, a regular compound of two tetrahedra.

Stellating polygons

Stellating a regular polygon symmetrically creates a regular

star polygon In geometry, a star polygon is a type of non-convex polygon. Regular star polygons have been studied in depth; while star polygons in general appear not to have been formally defined, Decagram (geometry)#Related figures, certain notable ones can ...

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 In geometry, the Schläfli symbol is a notation of the form \ that defines List of regular polytopes and compounds, regular polytopes and tessellations. The Schläfli symbol is named after the 19th-century Swiss mathematician Ludwig Schläfli, wh ...

, 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 number theory, 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 equiv ...

(have no common factor). 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 , can be seen as a compound polygon, 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 l ...

, 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 bigon, digon, or a ''2''-gon, is a polygon with two sides (edge (geometry), edges) and two Vertex (geometry), vertices. Its construction is Degeneracy (mathematics), degenerate in a Euclidean plane because either the two sides wou ...

s are not considered), and stellations if ''n'' is odd. 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 ''Wikt:septa-, septa-'' (an elision of ''Wikt:septua-, septua-''), a Latin-derived numerical prefix, rather than ...

, the

octagon In geometry, an octagon () is an eight-sided polygon or 8-gon. A '' regular octagon'' has Schläfli symbol and can also be constructed as a quasiregular truncated square, t, which alternates two types of edges. A truncated octagon, t is a ...

also has two

octagram In geometry, an octagram is an eight-angled star polygon. The name ''octagram'' combine a Greek numeral prefix, ''wikt:octa-, octa-'', with the Greek language, Greek suffix ''wikt:-gram, -gram''. The ''-gram'' suffix derives from γραμμή ...

mic stellations, one, being a

, and the other, , being the compound of two

squares In geometry, a square is a regular polygon, regular quadrilateral. It has four straight sides of equal length and four equal angles. Squares are special cases of rectangles, which have four equal angles, and of rhombuses, which have four equal si ...

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, 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 equivalents ...

. 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 The Archimedean solids are a set of thirteen convex polyhedra whose faces are regular polygon and are vertex-transitive, although they aren't face-transitive. The solids were named after Archimedes, although he did not claim credit for them. They ...

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 A cube or regular hexahedron is a three-dimensional space, three-dimensional solid object in geometry, which is bounded by six congruent square (geometry), square faces, a type of polyhedron. It has twelve congruent edges and eight vertices. It i ...

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 vertex (geometry), vertices, with 2 triangles and 2 squares meeting at each, and 24 identical edge (geometry), edges, each separating a tr ...

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

rhombic dodecahedron In geometry, the rhombic dodecahedron is a Polyhedron#Convex_polyhedra, convex polyhedron with 12 congruence (geometry), congruent rhombus, rhombic face (geometry), faces. It has 24 edge (geometry), edges, and 14 vertex (geometry), vertices of 2 ...

* 187 stellations of the

triakis tetrahedron In geometry, a triakis tetrahedron (or tristetrahedron, or kistetrahedron) is a solid constructed by attaching four triangular pyramids onto the triangular faces of a regular tetrahedron, a Kleetope of a tetrahedron. This replaces the equilateral ...

* 358,833,097 stellations of the

rhombic triacontahedron The rhombic triacontahedron, sometimes simply called the triacontahedron as it is the most common thirty-faced polyhedron, is a convex polyhedron with 30 rhombus, rhombic face (geometry), faces. It has 60 edge (geometry), edges and 32 vertex ...

* 17 stellations of the

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

icosidodecahedron In geometry, an icosidodecahedron or pentagonal gyrobirotunda is a polyhedron with twenty (''icosi-'') triangular faces and twelve (''dodeca-'') pentagonal faces. An icosidodecahedron has 30 identical Vertex (geometry), vertices, with two triang ...

; there are 7,071,671 non-

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

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 Harold Scott MacDonald Coxeter, H. S. M. Coxeter, Patrick du Val, P. Du Val, H. T. Flather and J. F. Petrie. It enumerates certain stellations of the regular convex or Platonic re ...

'', 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 (: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular Face (geometry), faces, six straight Edge (geometry), edges, and four vertex (geometry), vertices. The tet ...

, because all faces are adjacent * There are no stellations of the

, 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 one of four Kepler–Poinsot polyhedra. It is composed of 12 pentagonal faces (six pairs of parallel pentagons), intersecting each other making a pentagrammic path, with five pentagons meeting at each vert ...

and the

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

icosahedron In geometry, an icosahedron ( or ) is a polyhedron with 20 faces. The name comes . The plural can be either "icosahedra" () or "icosahedrons". There are infinitely many non- similar shapes of icosahedra, some of them being more symmetrical tha ...

, including the great icosahedron (one of the Kepler–Poinsot polyhedra), and the

second The second (symbol: s) is a unit of time derived from the division of the day first into 24 hours, then to 60 minutes, and finally to 60 seconds each (24 × 60 × 60 = 86400). The current and formal definition in the International System of U ...

and

final Final, Finals or The Final may refer to: *Final examination or finals, a test given at the end of a course of study or training *Final (competition), the last or championship round of a sporting competition, match, game, or other contest which d ...

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 Vertex (geometry), vertices. New edges of a faceted po ...

, 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 In geometry, a hyperplane is a generalization of a two-dimensional plane in three-dimensional space to mathematical spaces of arbitrary dimension. Like a plane in space, a hyperplane is a flat hypersurface, a subspace whose dimension is ...

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 cu ...

. For example, in 4-space, the great grand stellated 120-cell is the final stellation of the

regular 4-polytope In mathematics, a regular 4-polytope or regular polychoron is a regular polytope, regular 4-polytope, four-dimensional polytope. They are the four-dimensional analogues of the Regular polyhedron, regular polyhedra in three dimensions and the regul ...

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 devised a terminology for stellated

polyhedra In geometry, a polyhedron (: polyhedra or polyhedrons; ) is a three-dimensional figure with flat polygonal faces, straight edges and sharp corners or vertices. The term "polyhedron" may refer either to a solid figure or to its boundary su ...

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 ...

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 ( ) was a period in History of Italy, Italian history between the 14th and 16th centuries. The period is known for the initial development of the broader Renaissance culture that spread across Western Europe and marked t ...

artist

Paolo Uccello Paolo Uccello ( , ; 1397 – 10 December 1475), born Paolo di Dono, was an Italian Renaissance painter and mathematician from Florence who was notable for his pioneering work on visual Perspective (graphical), perspective in art. In his book ''Liv ...

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 ( ; ) is an international cultural exhibition hosted annually in Venice, Italy. There are two main components of the festival, known as the Art Biennale () and the Venice Biennale of Architecture, Architecture Biennale (), ...

in 1986 on the topic of "Art and Science". The same stellation is central to two

lithograph Lithography () is a planographic method of printing originally based on the miscibility, immiscibility of oil and water. The printing is from a stone (lithographic limestone) or a metal plate with a smooth surface. It was invented in 1796 by ...

s by

M. C. Escher Maurits Cornelis Escher (; ; 17 June 1898 – 27 March 1972) was a Dutch graphic artist who made woodcuts, lithography, lithographs, and mezzotints, many of which were Mathematics and art, inspired by mathematics. Despite wide popular int ...

: ''Contrast (Order and Chaos)'', 1950, and ''

Gravitation In physics, gravity (), also known as gravitation or a gravitational interaction, is a fundamental interaction, a mutual attraction between all massive particles. On Earth, gravity takes a slightly different meaning: the observed force b ...

'', 1952.

References

* Bridge, N. J.; Facetting the dodecahedron, ''Acta Crystallographica'' A30 (1974), pp. 548–552. * Coxeter, H.S.M.; ''Regular complex polytopes'' (1974). * Coxeter, 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

*
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

{{Webarchive, url=https://web.archive.org/web/20191228054914/http://members.ozemail.com.au/~llan/i59.html , date=2019-12-28

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

Kepler's definition

Stellating polygons

Stellating polyhedra

Miller's rules

Other rules for stellation

Stellating polytopes

Naming stellations

Stellation to infinity

From mathematics to art

See also

References

External links