![Academ_Stellated_dodecagon](https://upload.wikimedia.org/wikipedia/commons/7/79/Academ_Stellated_dodecagon.svg)
In
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 c ...
, 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 toge ...
in two
dimension
In physics and mathematics, the dimension of a Space (mathematics), mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any Point (geometry), point within it. Thus, a Line (geometry), lin ...
s,
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 th ...
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 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 pol ...
''.
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 to obtain two regular star polyhedra, the
small stellated dodecahedron and
great stellated dodecahedron. He also stellated the regular
octahedron to obtain the
stella octangula, 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, certain notable ones can arise through truncation operations ...
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 regular polytopes and tessellations.
The Schläfli symbol is named after the 19th-century Swiss mathematician Ludwig Schläfli, who generalized Euclidean geometry to more ...
, where ''n'' is the number of vertices, ''m'' is the ''step'' used in sequencing the edges around it, and ''m'' and ''n'' are
coprime (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 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 ...
, 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
Even may refer to:
General
* Even (given name), a Norwegian male personal name
* Even (surname)
* Even (people), an ethnic group from Siberia and Russian Far East
** Even language, a language spoken by the Evens
* Odd and Even, a solitaire game w ...
(assuming compounds of multiple degenerate
digons 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, the
octagon
In geometry, an octagon (from the Greek ὀκτάγωνον ''oktágōnon'', "eight angles") 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, whi ...
also has two
octagrammic stellations, one, being a
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, certain notable ones can arise through truncation operations ...
, 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 adj ...
.
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. 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 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
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 ...
is not usually considered a stellation of the
cuboctahedron.
Generalising Miller's rules there are:
* 4 stellations of the
rhombic dodecahedron
* 187 stellations of the
triakis tetrahedron
* 358,833,097 stellations of the
rhombic triacontahedron
* 17 stellations of the
cuboctahedron (4 are shown in
Wenninger's ''Polyhedron Models'')
* An unknown number of stellations of the
icosidodecahedron; there are 7071671 non-
chiral 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'', 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 the o ...
, because all faces are adjacent
* There are no stellations 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 ...
, because non-adjacent faces are parallel and thus cannot be extended to meet in new edges
* There is 1 stellation of the
octahedron, the
stella octangula
* There are 3 stellations of the
dodecahedron: the
small stellated dodecahedron, the
great dodecahedron and the
great stellated dodecahedron, 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 and . The plural can be either "icosahedra" () or "icosahedrons".
There are infinitely many non- similar shapes of icosahedra, some of them being more symmetrica ...
, including the
great icosahedron (one of the Kepler–Poinsot polyhedra), and the
second
The second (symbol: s) is the unit of time in the International System of Units (SI), historically defined as of a day – this factor derived from the division of the day first into 24 hours, then to 60 minutes and finally to 60 seconds ...
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 cont ...
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 geometry, a stellation diagram or stellation pattern is a two-dimensional diagram in the plane of some face of a polyhedron, showing lines where other face planes intersect with this one. The lines cause 2D space to be divided up into regions. ...
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, whereby parts are removed from a polyhedron without creating any new vertices. For every stellation of some polyhedron, there is a
dual
Dual or Duals may refer to:
Paired/two things
* Dual (mathematics), a notion of paired concepts that mirror one another
** Dual (category theory), a formalization of mathematical duality
*** see more cases in :Duality theories
* Dual (grammatical ...
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
In geometry, a stellation diagram or stellation pattern is a two-dimensional diagram in the plane of some face of a polyhedron, showing lines where other face planes intersect with this one. The lines cause 2D space to be divided up into regions. ...
of an ''n''-polytope exists in an (''n'' − 1)-dimensional
hyperplane
In geometry, a hyperplane is a subspace whose dimension is one less than that of its ''ambient space''. For example, if a space is 3-dimensional then its hyperplanes are the 2-dimensional planes, while if the space is 2-dimensional, its hyper ...
of a given
facet.
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 ha ...
is the final stellation of the
regular 4-polytope 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, heca ...
.
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 o ...
devised a terminology for stellated
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 toge ...
s,
polyhedra 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
![Magnus Wenninger polyhedral models](https://upload.wikimedia.org/wikipedia/commons/b/bb/Magnus_Wenninger_polyhedral_models.jpg)
Alongside from his contributions to mathematics,
Magnus Wenninger is described in the context of the relationship of
mathematics and art as making "especially beautiful" models of complex stellated polyhedra.
![Marble floor mosaic Basilica of St Mark Vencice](https://upload.wikimedia.org/wikipedia/commons/9/95/Marble_floor_mosaic_Basilica_of_St_Mark_Vencice.jpg)
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 Italians, Italian (Florentine) Florentine painting, painter and mathematician who was notable for his pioneering work on visual Perspective (graphical), perspective in art. ...
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
lithograph
Lithography () is a planographic method of printing originally based on the 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 the German a ...
s by
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 ...
: ''Contrast (Order and Chaos)'', 1950, and ''
Gravitation
In physics, gravity () is a fundamental interaction which causes mutual attraction between all things with mass or energy. Gravity is, by far, the weakest of the four fundamental interactions, approximately 1038 times weaker than the stron ...
'', 1952.
See also
* ''
The Fifty-Nine Icosahedra''
*
List of Wenninger polyhedron models
This is an indexed list of the uniform and stellated polyhedra from the book ''Polyhedron Models'', by Magnus Wenninger.
The book was written as a guide book to building polyhedra as physical models. It includes templates of face elements for cons ...
Includes 44 stellated forms of the octahedron, dodecahedron, icosahedron, and icosidodecahedron, enumerated the 1974 book "Polyhedron Models" by Magnus Wenninger
*
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 ...
Includes 5 regular compounds and 4 dual regular compounds.
*
List of polyhedral stellations
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
* {{mathworld , urlname = Stellation , title = Stellation
Stellating the Icosahedron and Facetting the DodecahedronStella: Polyhedron Navigator– Software for exploring polyhedra and printing nets for their physical construction. Includes uniform polyhedra, stellations, compounds, Johnson solids, etc.
Enumeration of stellationsVladimir Bulatov ''Polyhedra Stellation.''Vladimir Bulatov's Polyhedra Stellations Applet packaged as an OS X applicationStellation AppletAn Interactive Creation of Polyhedra Stellations with Various SymmetriesFurther Stellations of the Uniform Polyhedra, John Lawrence HudsonThe Mathematical Intelligencer, Volume 31, Number 4, 2009
Polygons
Polyhedra
Polytopes