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 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''.
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
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. 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 , 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 equival ...
(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, su ...
). 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
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 wh ...
(assuming compounds of multiple degenerate
digons are not considered), and stellations if ''n'' is
odd.
Like the
heptagon, the
octagon 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.
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
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 composed ...
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
* 187 stellations of the
triakis tetrahedron
* 358,833,097 stellations of the
rhombic triacontahedron
* 17 stellations 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 ...
(4 are shown in
Wenninger's ''Polyhedron Models'')
* An unknown number of stellations of the
icosidodecahedron; 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'', 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, 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
octahedron, the
stella octangula
* There are 3 stellations of the
dodecahedron: 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 ...
, the
great dodecahedron and the
great stellated dodecahedron, all of which are Kepler–Poinsot polyhedra.
* There are 58 stellations of the
icosahedron, including the
great icosahedron (one of the Kepler–Poinsot polyhedra), and the
second and
final 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
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
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 hyperp ...
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, he ...
.
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
polygons,
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
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 as making "especially beautiful" models of complex stellated polyhedra.
The
Italian Renaissance artist
Paolo Uccello 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 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.
See also
* ''
The Fifty-Nine Icosahedra''
*
List of Wenninger polyhedron models Includes 44 stellated forms of the octahedron, dodecahedron, icosahedron, and icosidodecahedron, enumerated the 1974 book "Polyhedron Models" by Magnus Wenninger
*
Polyhedral compound 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