Editing Polyhedron (section)

== Symmetries ==
[[File:Revolução de poliedros 03.webm|thumb|280px|Some polyhedra rotating around a symmetrical axis (at [[Matemateca|Matemateca IME-USP]])]]
Many of the most studied polyhedra are highly [[Symmetry|symmetrical]]. Their appearance is unchanged by some reflection by plane or rotation around the [[axis of symmetry|axes]] passing through two opposite vertices, edges, or faces in space. Each symmetry may change the location of a given element, but the set of all vertices (likewise faces and edges) is unchanged. The collection of symmetries of a polyhedron is called its [[symmetry group]].<ref name=akl>{{citation
 | last = Mal'cev | first = A. V.
 | editor-last1 = Aleksandrov | editor-first1 = A. D.
 | editor-last2 = Kolmogorov | editor-first2 = A. N.
 | editor-last3 = Lavrent'ev | editor-first3 = M. A.
 | year = 2012
 | title = Mathematics: Its Content, Methods and Meaning
 | url = https://books.google.com/books?id=2cPDAgAAQBAJ&pg=RA2-PA278
 | page = Volume III, 278
 | publisher = Dover Publications
| isbn = 978-0-486-15787-0
 }}</ref>

=== By elements of polyhedron ===
All the elements (vertex, face, and edge) that can be superimposed on each other by symmetries are said to form a [[Symmetry orbit#Orbits and stabilizers|symmetry orbit]]. If these elements lie in the same orbit, the figure may be transitive on the orbit. Individually, they are [[isohedral]] (or face-transitive, meaning the symmetry transformations involve the polyhedra's faces in orbit),<ref name=mclean>{{citation
 | last = McLean | first = K. Robin
 | year = 1990
 | title = Dungeons, dragons, and dice
 | journal = [[The Mathematical Gazette]]
 | volume = 74 | issue = 469 | pages = 243–256
 | doi = 10.2307/3619822
 | jstor = 3619822
 | s2cid = 195047512
}} See p. 247.</ref>{{efn|1=The topological property of an isohedral polyhedra can be represented by a [[face configuration]]. All five [[Platonic solids]] and thirteen [[Catalan solid]]s are isohedra, as well as the infinite families of [[trapezohedra]] and [[bipyramid]]s. Some definitions of isohedra allow geometric variations including concave and self-intersecting forms.}} [[isotoxal]] (or edge-transitive, which involves the edge's polyhedra),<ref name=grunbaum-1997>{{citation
 | last = Grünbaum | first = Branko | authorlink = Branko Grünbaum 
 | year = 1997
 | title = Isogonal Prismatoids
 | journal = Discrete & Computational Geometry
 | volume = 18 | issue = 1 | pages = 13&ndash;52
 | doi = 10.1007/PL00009307
}}</ref> and [[isogonal figure|isogonal]] (or vertex-transitive, which involves the polyhedra's vertices). For example, the [[cube]] in which all the faces are in one orbit and involving the rotation and reflections in the orbit remains unchanged in its appearance; hence, the cube is face-transitive. The cube also has the other two such symmetries.<ref name=senechal>{{citation
 | last = Senechal | first = Marjorie
 | year = 1989
 | contribution = A Brief Introduction to Tilings
 | contribution-url = https://books.google.com/books?id=OToVjZW9CKMC&pg=PA12
 | editor-last = Jarić | editor-first = Marko
 | title = Introduction to the Mathematics of Quasicrystals
 | publisher = [[Academic Press]]
 | page = 12
}}</ref>

[[File:Hexahedron.svg|thumb|upright=0.6|The [[cube]] is a [[regular polyhedron]], because its faces, edges, and vertices are transitive to another, and the appearance is unchanged.]]
When three such symmetries belong to a polyhedron, it is known as a [[regular polyhedron]].<ref name=senechal /> There are nine regular polyhedra: five [[Platonic solid]]s (cube, [[regular octahedron|octahedron]], [[regular icosahedron|icosahedron]], [[regular tetrahedron|tetrahedron]], and [[regular dodecahedron|dodecahedron]]&mdash;all of which have regular polygonal faces) and four [[Kepler&ndash;Poinsot polyhedron]]s. Nevertheless, some polyhedrons may not possess one or two of those symmetries:
* A polyhedron with vertex-transitive and edge-transitive is said to be a [[quasiregular polyhedron|quasiregular]], although they have regular faces, and its dual is face-transitive and edge-transitive.
* A vertex- but not edge-transitive polyhedron with regular polygonal faces is said to be a [[Semiregular polyhedron|semiregular]].{{efn|1=This is one of several definitions of the term, depending on the author. Some definitions overlap with the quasi-regular class.}} and such polyhedrons are [[prism (geometry)|prisms]] and [[antiprism]]s. Its dual is face-transitive but not vertex-transitive, and every vertex is regular.
* A polyhedron with regular polygonal faces and vertex-transitive is said to be [[Uniform polyhedron|uniform]]. This class may be subdivided into a regular, quasi-regular, or semi-regular polyhedron, and may be convex or starry. The dual is face-transitive and has regular vertices but is not necessarily vertex-transitive. The uniform polyhedra and their duals are traditionally classified according to their degree of symmetry, and whether they are [[Convex polyhedron|convex]] or not.
* A face- and vertex-transitive (but not necessarily edge-transitive) polyhedra is said to be [[Noble polyhedron|noble]]. The regular polyhedra are also noble; they are the only noble uniform polyhedra. The duals of noble polyhedra are themselves noble.

Some polyhedra may have no [[reflection symmetry]] such that they have two enantiomorph forms, which are reflections of each other. Such symmetry is known for having [[Chirality (mathematics)|chirality]]. Examples include the [[snub cuboctahedron]] and [[snub icosidodecahedron]].

=== By point group in three dimensions ===
{{main|Point groups in three dimensions}}
The [[point groups in three dimensions|point group of polyhedra]] means a [[mathematical group]] endowed with its [[symmetry operation]]s so that the appearance of polyhedra remains preserved while transforming in three-dimensional space. The indicated transformation here includes the rotation around the axes, reflection through the plane, inversion through a center point, and a combination of these three.<ref name=powell>{{citation
 | last = Powell | first = R. C.
 | year = 2010
 | title = Symmetry, Group Theory, and the Physical Properties of Crystals
 | series = Lecture Notes in Physics
 | volume = 824
 | publisher = [[Springer Science+Business Media|Springer]]
 | url = https://books.google.com/books?id=ojq5BQAAQBAJ&pg=PA27
 | page = 27
 | isbn = 978-1-441-97598-0
 | doi = 10.1007/978-1-4419-7598-0
}}</ref>

[[Image:Symmetries of the tetrahedron.svg|thumb|upright=1.2|The regular tetrahedron has full tetrahedral symmetry: three-fold rotation around axis passing both vertex and triangular face, and two-fold rotation around axis through two edges, as well as the reflection plane through two faces and one edge]]
The [[polyhedral group]] is the symmetry group originally derived from the three Platonic solids: tetrahedron, octahedron, and icosahedron. These three have point groups respectively known as [[tetrahedral symmetry]], [[octahedral symmetry]], and [[icosahedral symmetry]]. Each of these focuses on the rotation group of polyhedra, known as the ''chiral polyhedral group'', whereas the additional reflection symmetry is known as the ''full polyhedral group''. One point group, [[pyritohedral symmetry]], includes the rotation of tetrahedral symmetry and additionally has three planes of reflection symmetry and some [[rotoreflection]]s. Overall, the mentioned polyhedral groups are summarized in the following bullets:<ref name=fsz>{{citation
 | last1 = Flusser | first1 = J.
 | last2 = Suk | first2 = T.
 | last3 = Zitofa | first3 = B.
 | year = 2017
 | title = 2D and 3D Image Analysis by Moments
 | publisher = [[John Wiley & Sons]]
 | isbn = 978-1-119-03935-8
 | page = 127&ndash;128
 | url = https://books.google.com/books?id=jwKLDQAAQBAJ&pg=PA128
}}</ref>
* chiral tetrahedral symmetry <math> \mathrm{T} </math>, the rotation group for a regular tetrahedron and has the order of twelve.
* full tetrahedral symmetry <math> \mathrm{T}_\mathrm{d} </math>, the symmetry group for a regular tetrahedron and has the order of twenty-four.
* pyritohedral symmetry <math> \mathrm{T}_\mathrm{h} </math>, the symmetry of a [[pyritohedron]] and has the order of twenty-four.
* chiral octahedral symmetry <math> \mathrm{O} </math>, the rotation group of both cube and regular octahedron and has the order twenty-four.
* full octahedral symmetry <math> \mathrm{O}_\mathrm{h} </math>, the symmetry group of both cube and regular octahedron and has order forty-eight.
* chiral icosahedral symmetry <math> \mathrm{I} </math>, the rotation group of both regular icosahedron and regular dodecahedron and has the order of sixty.
* full icosahedral symmetry <math> \mathrm{I}_\mathrm{h} </math>, the symmetry group of both regular icosahedron and regular dodecahedron and has the order of a hundred-twenty.

[[File:Square pyramid.png|thumb|upright=1|The [[square pyramid]] has pyramidal symmetry <math>C_{4\mathrm{v}}</math>. It shows the appearance is invariant by rotating every quarter of a full turn around its axis and possesses [[mirror symmetric]] relative to any perpendicular plane passing through its base's bisector]]
Point groups in three dimensions may also allow the preservation of polyhedra's appearance by the circulation around an axis. There are three various of these point groups:
* [[pyramidal symmetry]] <math> C_{n \mathrm{v}} </math>, allowing rotate the axis passing through the [[Apex (geometry)|apex]] and its [[Base (geometry)|base]], as well as reflection relative to perpendicular planes passing through the bisector of a base. This point group symmetry can be found in pyramids,{{sfnp|Flusser|Suk|Zitofa|2017|p=[https://books.google.com/books?id=jwKLDQAAQBAJ&pg=PA126 126]}} [[cupola (geometry)|cupola]]s, and [[rotunda (geometry)|rotunda]]s.
* [[prismatic symmetry]] <math> D_{n\mathrm{h}} </math>, similar to the pyramidal symmetry, but with additional transformation by reflecting it across a horizontal plane.{{sfnp|Flusser|Suk|Zitofa|2017|p=[https://books.google.com/books?id=jwKLDQAAQBAJ&pg=PA126 126]}} This may be achieved from the family of prisms and its dual [[bipyramid]]s.{{sfnp|Flusser|Suk|Zitofa|2017|p=[https://books.google.com/books?id=jwKLDQAAQBAJ&pg=PA126 126]}}
* [[antiprismatic symmetry]] <math> D_{n \mathrm{v}} </math>, which preserves the symmetry by rotating its half bottom and reflection across the horizontal plane.{{sfnp|Flusser|Suk|Zitofa|2017|p=[https://books.google.com/books?id=jwKLDQAAQBAJ&pg=PA126 126]}} Examples can be found in antiprisms.
A point group <math> C_{n \mathrm{h}} </math> consists of rotating around the axis of symmetry and reflection on the horizontal plane. In the case of <math> n = 1 </math>, the symmetry group only preserves the symmetry by a full rotation solely, ordinarily denoting <math> C_s </math>.<ref name=herbert>{{citation
 | last1 = Hergert | first1 = W.
 | last2 = Geilhufe | first2 = M.
 | year = 2018
 | title = Group Theory in Solid State Physics and Photonics: Problem Solving with Mathematica
 | url = https://books.google.com/books?id=6mvpCgAAQBAJ&pg=PA56
 | publisher = [[John Wiley & Sons]]
 | isbn = 978-3-527-41300-3
}}</ref> Polyhedra may have rotation only to preserve the symmetry, and the symmetry group may be considered as the [[cyclic group]] <math> C_n </math>.{{sfnp|Flusser|Suk|Zitofa|2017|p=[https://books.google.com/books?id=jwKLDQAAQBAJ&pg=PA125 125]}} Polyhedra with the rotoreflection and the rotation by the cyclic group is the point group <math> S_n </math>.{{sfnp|Hergert|Geilhufe|2018|p=[https://books.google.com/books?id=6mvpCgAAQBAJ&pg=PA57 57]}}
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