Editing Polyhedron (section)

=== 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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