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Stellation



 
 
Stellation is a process of constructing new polygon
Polygon

In geometry a polygon is traditionally a plane Shape that is bounded by a closed curve path or circuit, composed of a finite sequence of straight line segments ....
s (in two dimension
Dimension

In mathematics, the dimension of a space is roughly defined as the minimum number of coordinates needed to specify every point within it. For example: a point on the unit circle in the plane can be specified by two Cartesian coordinates but one can make do with a single coordinate , so the circle is 1-dimensional even though it exists in...
s), new polyhedra
Polyhedron

|}A polyhedron is often defined as a geometry object with flat faces and straight edges .This definition of a polyhedron is not very precise, and to a modern mathematician is quite unsatisfactory....
 in three dimensions, or, in general, new polytope
Polytope

In geometry, polytope is a generic term that can refer to a two-dimensional polygon, a three-dimensional polyhedron, or any of the various generalizations thereof, including generalizations to higher dimensions and other abstractions ....
s in n dimensions. The process consists of extending elements such as edges or face planes, usually in a symmetrical way, until they meet each other again. The new figure is a stellation of the original.

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






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Stellation is a process of constructing new polygon
Polygon

In geometry a polygon is traditionally a plane Shape that is bounded by a closed curve path or circuit, composed of a finite sequence of straight line segments ....
s (in two dimension
Dimension

In mathematics, the dimension of a space is roughly defined as the minimum number of coordinates needed to specify every point within it. For example: a point on the unit circle in the plane can be specified by two Cartesian coordinates but one can make do with a single coordinate , so the circle is 1-dimensional even though it exists in...
s), new polyhedra
Polyhedron

|}A polyhedron is often defined as a geometry object with flat faces and straight edges .This definition of a polyhedron is not very precise, and to a modern mathematician is quite unsatisfactory....
 in three dimensions, or, in general, new polytope
Polytope

In geometry, polytope is a generic term that can refer to a two-dimensional polygon, a three-dimensional polyhedron, or any of the various generalizations thereof, including generalizations to higher dimensions and other abstractions ....
s in n dimensions. The process consists of extending elements such as edges or face planes, usually in a symmetrical way, until they meet each other again. The new figure is a stellation of the original.

Kepler's definition

In 1619 Kepler 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 dodecahedron
Dodecahedron

A dodecahedron is any polyhedron with twelve faces, but usually a regular dodecahedron is meant: a Platonic solid composed of twelve regular pentagonal faces, with three meeting at each vertex....
 to obtain two of the regular star polyhedra (two of the Kepler-Poinsot polyhedra).

Stellated polygons

A stellation of a regular polygon is a regular star polygon
Star polygon

A star polygon is a non-convex polygon which looks in some way like a star. Only the regular ones have been studied in any depth; star polygons in general have never been formally defined....
 or polygonal compound
Star polygon

A star polygon is a non-convex polygon which looks in some way like a star. Only the regular ones have been studied in any depth; star polygons in general have never been formally defined....
.

A regular star polygon is represented by its Schläfli symbol , where n is the number of vertices, and m is the step used in sequencing the edges around it. If m is one, it is just the convex regular convex . The (m-1)st stellation is therefore .

A polygon compound appears if n and m have a common divisor, and the full stellation requires multiple cyclic paths to complete it. For example a hexagram is made of 2 triangles , and is made of 2 pentagrams . Some authors use the Schläfli symbol for such regular compounds, while others regard the symbol as indicating a single path which is wound m times around n/m vertex points, such that one edge is superimposed upon another and each vertex point is visited m times.

A regular n-gon has (n-4)/2 stellations if n is even, and (n-3)/2 stellations if n is odd.

Pentagram Green

The pentagram
Pentagram

A pentagram is the shape of a five-pointed star drawn with five straight strokes. The word pentagram comes from the Greek language word pe?t???a???? , a noun form of pe?t???a???? or pe?t???a???? , a word meaning roughly "five-lined" or "five lines"....
, , is the only stellation of a pentagon
Pentagon

In geometry, a pentagon is any five-sided polygon. A pentagon may be simple or self-intersecting. The internal angles in a simple pentagon total 540?....
Hexagram

The hexagram
Hexagram

A hexagram is a six-pointed geometric star figure, or 2, the compound of two equilateral triangle s. The intersection is a regular hexagon.While generally recognized as a symbol of Jewish identity it is used also in other historical, religious and cultural contexts, for example in #Use of the Star by Arabs and Muslims, and #Occurrence in...
, , the stellation of a hexagon
Hexagon

In geometry, a hexagon is a polygon with six edges and six Vertex . A regular hexagon has Schl?fli symbol ....
 and a compound of two triangles.

The enneagon
Enneagon

In geometry, a nonagon is a nine-sided polygon.The name "nonagon" is a hybrid word, from Latin , used equivalently, attested already in the 16th century in French nonogone and in English from the 17th century....
 (nonagon) has 3 enneagram
Enneagram

In geometry, an enneagram is a nine-pointed geometric figure. The term derives from two ancient Greek words: ennea and gramma ....
mic forms:
, , , with being 3 triangles.
Obtuse Heptagram
Acute Heptagram

The heptagon
Heptagon

In geometry, a heptagon is a polygon with seven sides and seven angles. In a regular polygon heptagon, in which all sides and all angles are equal, the sides meet at an angle of 5p/7 radians, 128.5714286 degree s....
 has two heptagram
Heptagram

A heptagram or septegram is a seven-pointed Star drawn with seven straight strokes....
mic forms:
,


Like the heptagon
Heptagon

In geometry, a heptagon is a polygon with seven sides and seven angles. In a regular polygon heptagon, in which all sides and all angles are equal, the sides meet at an angle of 5p/7 radians, 128.5714286 degree s....
, the octagon
Octagon

In geometry, an octagon is a polygon that has 8 sides. A regular octagon is represented by the Schl?fli symbol ....
 also has two octagram
Octagram

In geometry, an octagram is an eight-sided star polygon....
mic stellations, one, being a star polygon
Star polygon

A star polygon is a non-convex polygon which looks in some way like a star. Only the regular ones have been studied in any depth; star polygons in general have never been formally defined....
, and the other, , being the compound of two squares
Square (geometry)

In Euclidean geometry, a square is a regular polygon with four equal sides and four equal angles . A square with vertices ABCD would be denoted ....
.

Stellated polyhedra


First Stellation of Octahedron
First Stellation of Dodecahedron
Second Stellation of Dodecahedron
Third Stellation of Dodecahedron
Sixteenth Stellation of Icosahedron
First Stellation of Icosahedron
Seventeenth Stellation of Icosahedron


The face planes of a polyhedron divide space into many discrete cells. 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
    J. C. P. Miller

    Jeffrey Charles Percy Miller was an England mathematician and computing pioneer. He worked in number theory and on geometry, particularly polyhedra, where Miller's monster refers to the Great dirhombicosidodecahedron....
    . Although these rules refer specifically to the icosahedron's geometry, they can easily be 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
Cube

A cube is a three-dimensional space solid object bounded by six square faces, facets or sides, with three meeting at each wikt:vertex. The cube can also be called a Regular polyhedron hexahedron and is one of the five Platonic solids....
 is not considered a stellation of the cuboctahedron
Cuboctahedron

In geometry, a cuboctahedron is a polyhedron with eight triangular faces and six square faces. A cuboctahedron has 12 identical vertices, with two triangles and two squares meeting at each, and 24 identical edges, each separating a triangle from a square....
. There are:

  • 4 stellations of the rhombic dodecahedron
    Rhombic dodecahedron

    The rhombic dodecahedron is a convex set polyhedron with 12 rhombus faces. It is an Archimedean solid solid, or a Catalan solid. Its dual is the cuboctahedron....
  • 187 stellations of the triakis tetrahedron
    Triakis tetrahedron

    A triakis tetrahedron is an Archimedean solid solid, or a Catalan solid. Its dual is the truncated tetrahedron.It can be seen as a tetrahedron with Tetrahedron added to each face....
  • 358,833,097 stellations of the rhombic triacontahedron
    Rhombic triacontahedron

    In geometry, the rhombic triacontahedron is a convex set polyhedron with 30 rhombus faces. It is an Archimedean solid solid, or a Catalan solid....
  • 17 stellations of the cuboctahedron
    Cuboctahedron

    In geometry, a cuboctahedron is a polyhedron with eight triangular faces and six square faces. A cuboctahedron has 12 identical vertices, with two triangles and two squares meeting at each, and 24 identical edges, each separating a triangle from a square....
     (4 are shown in Wenninger
    List of Wenninger polyhedron models

    This table contains 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....
    's "Polyhedron Models")
  • Unknown stellations of the icosidodecahedron
    Icosidodecahedron

    An icosidodecahedron is a polyhedron with twenty triangular faces and twelve pentagonal faces. An icosidodecahedron has 30 identical vertices, with two triangles and two pentagons meeting at each, and 60 identical edges, each separating a triangle from a pentagon....
    , but many more than above! (19 are shown in Wenninger
    List of Wenninger polyhedron models

    This table contains 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....
    '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

The fifty nine icosahedra is a book written and illustrated by Harold Scott MacDonald Coxeter, Patrick du Val, H. T. Flather and J. F. Petrie....
, J.C.P. Miller proposed a set of rules
The fifty nine icosahedra

The fifty nine icosahedra is a book written and illustrated by Harold Scott MacDonald Coxeter, Patrick du Val, H. T. Flather and J. F. Petrie....
 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
    Tetrahedron

    A tetrahedron is a polyhedron composed of four triangle faces, three of which meet at each vertex . A regular tetrahedron is one in which the four triangles are regular, or "equilateral", and is one of the Platonic solids....
    , because all faces are adjacent
  • There are no stellations of the cube
    Cube

    A cube is a three-dimensional space solid object bounded by six square faces, facets or sides, with three meeting at each wikt:vertex. The cube can also be called a Regular polyhedron hexahedron and is one of the five Platonic solids....
    , because non-adjacent faces are parallel and thus cannot be extended to meet in new edges
  • There is 1 stellation of the octahedron
    Octahedron

    An octahedron is a polyhedron with eight faces. A regular octahedron is a Platonic solid composed of eight equilateral triangles, four of which meet at each wikt:vertex....
    , the stella octangula
    Stella octangula

    The stella octangula, also known as the stellated octahedron, Star Tetrahedron, eight-pointed star, or 2D geometric model as the Star of David....
  • There are 3 stellations of the dodecahedron
    Dodecahedron

    A dodecahedron is any polyhedron with twelve faces, but usually a regular dodecahedron is meant: a Platonic solid composed of twelve regular pentagonal faces, with three meeting at each vertex....
    : the small stellated dodecahedron
    Small stellated dodecahedron

    In geometry, the small stellated dodecahedron is a Kepler-Poinsot polyhedra. It is one of four nonconvex regular polyhedra. It is composed of 12 pentagrammic faces, with five pentagrams meeting at each vertex....
    , the great dodecahedron
    Great dodecahedron

    In geometry, the great dodecahedron is a Kepler-Poinsot polyhedra. It is one of four nonconvex regular polyhedra. It is composed of 12 pentagonal faces , with five pentagons meeting at each vertex, intersecting each other making a pentagrammic path....
     and the great stellated dodecahedron
    Great stellated dodecahedron

    In geometry, the great stellated dodecahedron is a Kepler-Poinsot polyhedra. It is one of four nonconvex regular polyhedra.It is composed of 12 intersecting pentagrammic faces, with three pentagrams meeting at each vertex....
    , all of which are Kepler-Poinsot polyhedra.
  • There are 58 stellations of the icosahedron
    Icosahedron

    In geometry, an icosahedron isany polyhedron having 20 faces, but usually a regular icosahedron is implied, which has equilateral triangle s as faces....
    , including the great icosahedron
    Great icosahedron

    In geometry, the great icosahedron is a Kepler-Poinsot polyhedra. It is one of four nonconvex regular polyhedra. It is composed of 20 intersecting triangular faces, with five triangles meeting at each vertex in a pentagrammic sequence....
     (one of the Kepler-Poinsot polyhedra), and the second and final
    Final stellation of the icosahedron

    In geometry, the complete icosahedron is the complete or final stellation of the icosahedron. It is "complete" or "final" in the sense that it includes all of the finite cells into which the face planes of the regular icosahedron divide space, so it is the outermost stellation of the icosahedron....
     stellations of the icosahedron. The 59th model in "The fifty nine icosahedra
    The fifty nine icosahedra

    The fifty nine icosahedra is a book written and illustrated by Harold Scott MacDonald Coxeter, Patrick du Val, H. T. Flather and J. F. Petrie....
    " 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 process to facetting
Facetting

|}In geometry, facetting is the process of removing parts of a polygon, polyhedron or polytope, without creating any new vertices.Facetting is the reciprocal or dual process to stellation....
, whereby parts are removed from a polyhedron without creating any new vertices. For every stellation of some polyhedron, there is a dual
Dual

Dual may refer to:*a pair or a grouping of two:** dual basis, in mathematics, a basis that uniquely has a zero or unity inner product with a given basis...
 facetting of the dual polyhedron
Dual polyhedron

In geometry, polyhedron are associated into pairs called duals, where the wikt:vertex of one correspond to the face s of the other. The dual of the dual is the original polyhedron....
, 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
List of Wenninger polyhedron models

This table contains 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....
.

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

John Horton Conway is a prolific mathematician active in the theory of finite group , knot theory, number theory, combinatorial game theory and coding theory....
 devised a terminology for stellated polygon
Polygon

In geometry a polygon is traditionally a plane Shape that is bounded by a closed curve path or circuit, composed of a finite sequence of straight line segments ....
s, polyhedra
Polyhedron

|}A polyhedron is often defined as a geometry object with flat faces and straight edges .This definition of a polyhedron is not very precise, and to a modern mathematician is quite unsatisfactory....
 and polychora
Polychoron

In geometry, a four-dimensional polytope is sometimes called a polychoron , from the Greek language root poly, meaning "many", and choros meaning "room" or "space"....
 (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.

Stellated polytopes


The stellation process applies to higher dimensional polytopes as well. A stellation diagram
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....
 of an n-polytope exists in an (n-1)-dimensional hyperplane
Hyperplane

A hyperplane is a concept in geometry. It is a higher-dimensional generalization of the concepts of a line in the plane and a plane in 3-dimensional space....
 of a given facet.

For example, in 4-space, the great grand stellated 120-cell
Great grand stellated 120-cell

In geometry, the great grand stellated 120-cell is a star polychoron with Schl?fli symbol . It is one of 10 regular Schl?fli-Hess polychoron....
 is the final stellation of the Regular 4-polytope
Regular 4-polytope

In geometry a regular 4-polytope can mean either a convex or nonconvex 4-polytope.See:* Convex regular 4-polytope - There are six convex regular polychora....
 120-cell
120-cell

In geometry, the 120-cell is the convex regular 4-polytope with Schl?fli symbol .The boundary of the 120-cell is composed of 120 dodecahedral cell with 4 meeting at each vertex....
.

See also

  • The fifty nine icosahedra
    The fifty nine icosahedra

    The fifty nine icosahedra is a book written and illustrated by Harold Scott MacDonald Coxeter, Patrick du Val, H. T. Flather and J. F. Petrie....
  • List of Wenninger polyhedron models
    List of Wenninger polyhedron models

    This table contains 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....
     Includes 44 stellated forms of the octahedron, dodecahedron, icosahedron, and icosidodecahedron, enumerated the 1974 book "Polyhedron Models" by Magnus Wenninger
  • Polyhedral compound
    Polyhedral compound

    A polyhedral compound is a polyhedron that is itself composed of several other polyhedra sharing a common centre. They are the three-dimensional analogs of star polygon#Star figuress such as the hexagram....
     Includes 5 regular compounds and 4 dual regular compounds.


External links

  • - Software for exploring polyhedra and printing nets for their physical construction. Includes uniform polyhedra, stellations, compounds, Johnson solids, etc.