|
|
| Line 1: |
Line 1: |
| In [[geometry]], a '''[[point group]] in three dimensions''' is an [[isometry group]] in three dimensions that leaves the origin fixed, or correspondingly, an isometry group of a [[sphere]]. It is a [[subgroup]] of the [[orthogonal group]] O(3), the group of all [[isometry|isometries]] that leave the origin fixed, or correspondingly, the group of [[orthogonal matrix|orthogonal matrices]]. O(3) itself is a subgroup of the [[Euclidean group]] ''E''(3) of all isometries.
| | [http://www.saferstdtesting.com/free-std-testing/free-std-testing-hialeah-fl Myrtle Benny] [http://premium.asslikethat.com/blog/9567 at home std testing] is how I'm called and I feel [http://www.seventeen.com/health/tips/std-symptom-chart comfortable] when individuals use the full name. He used to be unemployed but now he is a computer operator but his marketing never comes. California is exactly home std test kit where her home is but she requirements to at home std test transfer because of her family members. What I adore doing is doing ceramics [http://www.zs-imports.com/blogs/post/70017 www.zs-imports.com] but I haven't produced a at home std testing dime with it.<br><br>Look at my web site ... std testing at home ([http://Torontocartridge.com/uncategorized/the-ideal-way-to-battle-a-yeast-infection/ mouse click the next webpage]) |
| | |
| [[Symmetry group]]s of objects are isometry groups. Accordingly, analysis of isometry groups is analysis of possible [[symmetry|symmetries]]. All isometries of a bounded 3D object have one or more common fixed points. We choose the origin as one of them.
| |
| | |
| The symmetry group of an object is sometimes also called '''full symmetry group''', as opposed to its '''rotation group''' or '''proper symmetry group''', the intersection of its full symmetry group and the [[rotation group SO(3)]] of the 3D space itself. The rotation group of an object is equal to its full symmetry group [[if and only if]] the object is [[chirality (mathematics)|chiral]].
| |
| | |
| The point groups in three dimensions are heavily used in chemistry, especially to describe the symmetries of a [[molecule]] and of [[molecular orbital]]s forming [[covalent bond]]s, and in this context they are also called '''[[List of character tables for chemically important 3D point groups|molecular point groups]]'''.
| |
| | |
| [[Finite Coxeter group]]s are a special set of ''point groups'' generated purely by a set of reflectional mirrors passing through the same point. A rank ''n'' Coxeter group has ''n'' mirrors and is represented by a [[Coxeter-Dynkin diagram]]. [[Coxeter notation]] offers a bracketed notation equivalent to the Coxeter diagram, with markup symbols for rotational and other subsymmetry point groups.
| |
| | |
| ==Group structure==
| |
| SO(3) is a subgroup of [[Euclidean group#Direct and indirect isometries|''E''<sup>+</sup>(3)]], which consists of [[Euclidean group#Direct and indirect isometries|''direct isometries'']], i.e., isometries preserving [[orientation (mathematics)|orientation]]; it contains those that leave the origin fixed.
| |
| | |
| O(3) is the [[direct product of groups|direct product]] of SO(3) and the group generated by [[Inversion in a point|inversion]] (denoted by its matrix −''I''):
| |
| :O(3) = SO(3) × { ''I'' , −''I'' }
| |
| | |
| Thus there is a 1-to-1 correspondence between all direct isometries and all indirect isometries, through inversion. Also there is a 1-to-1 correspondence between all groups of direct isometries ''H'' in O(3) and all groups ''K'' of isometries in O(3) that contain inversion:
| |
| :''K'' = ''H'' × { ''I'' , −''I'' }
| |
| :''H'' = ''K'' ∩ SO(3)
| |
| For instance, if ''H'' is ''C''<sub>2</sub>, then ''K'' is ''C''<sub>2h</sub>, or if ''H'' is ''C''<sub>3</sub>, then ''K'' is ''S''<sub>6</sub>. (See lower down for the definitions of these groups.)
| |
| | |
| If a group of direct isometries ''H'' has a subgroup ''L'' of [[Index of a subgroup|index]] 2, then, apart from the corresponding group containing inversion there is also a corresponding group that contains indirect isometries but no inversion:
| |
| :''M'' = ''L'' ∪ ( (''H'' \ ''L'') × { − ''I'' } )
| |
| where isometry ( ''A'', ''I'' ) is identified with ''A''. An example would be ''C''<sub>4</sub> for ''H'' and ''S''<sub>4</sub> for ''M''.
| |
| | |
| Thus ''M'' is obtained from ''H'' by inverting the isometries in ''H'' \ ''L''. This group ''M'' is as abstract group isomorphic with ''H''. Conversely, for all isometry groups that contain indirect isometries but no inversion we can obtain a rotation group by inverting the indirect isometries. This is clarifying when categorizing isometry groups, see below.
| |
| | |
| In 2D the [[cyclic group]] of ''k''-fold [[rotation]]s ''C<sub>k</sub>'' is for every positive integer ''k'' a normal subgroup of O(2,'''R''') and SO(2,'''R'''). Accordingly, in 3D, for every axis the cyclic group of ''k''-fold rotations about that axis is a normal subgroup of the group of all rotations about that axis. Since any subgroup of index two is normal, the group of rotations (''C<sub>n</sub>'') is normal both in the group obtained by adding reflections in planes through the axis (''C<sub>nv</sub>'') and in the group obtained by adding a reflection plane perpendicular to the axis (''C<sub>nh</sub>'').
| |
| | |
| ==3D isometries that leave origin fixed==
| |
| The isometries of '''R'''<sup>''3''</sup> that leave the origin fixed, forming the group O(''3'','''R'''), can be categorized as follows:
| |
| *SO(''3'','''R'''):
| |
| **identity
| |
| **rotation about an axis through the origin by an angle not equal to 180°
| |
| **rotation about an axis through the origin by an angle of 180°
| |
| *the same with [[Inversion in a point|inversion]] ('''x''' is mapped to −'''x'''), i.e. respectively:
| |
| **inversion
| |
| **rotation about an axis by an angle not equal to 180°, combined with reflection in the plane through the origin perpendicular to the axis
| |
| **reflection in a plane through the origin
| |
| | |
| The 4th and 5th in particular, and in a wider sense the 6th also, are called [[improper rotation]]s.
| |
| | |
| See also the similar [[Euclidean group#Overview of isometries in up to three dimensions|overview including translations]].
| |
| | |
| ==Conjugacy==
| |
| When comparing the symmetry type of two objects, the origin is chosen for each separately, i.e. they need not have the same center. Moreover, two objects are considered to be of the same symmetry type if their symmetry groups are conjugate subgroups of O(3) (two subgroups ''H''<sub>1</sub>, ''H''<sub>2</sub> of a group ''G'' are [[Conjugacy class#Conjugacy of subgroups and general subsets|''conjugate'']], if there exists ''g'' ∈ ''G'' such that ''H''<sub>1</sub> = g<sup>−1</sup>''H''<sub>2</sub>''g'' ).
| |
| | |
| For example two 3D objects have the same symmetry type:
| |
| *if both have mirror symmetry, but with respect to a different mirror plane
| |
| *if both have 3-fold rotational symmetry, but with respect to a different axis.
| |
| In the case of multiple mirror planes and/or axes of rotation, two symmetry groups are of the same symmetry type [[if and only if]] there is a rotation mapping the whole structure of the first symmetry group to that of the second. (In fact there will be more than one such rotation, but not an infinite number as when there is only one mirror or axis.) The conjugacy definition would also allow a mirror image of the structure, but this is not needed, the structure itself is achiral. For example, if a symmetry group contains a 3-fold axis of rotation, it contains rotations in two opposite directions. (The structure '''''is''''' chiral for 11 pairs of '''''[[space group]]s''''' with a screw axis.)
| |
| | |
| ==Infinite isometry groups==
| |
| There are many infinite isometry groups; for example, the "[[cyclic group]]" (meaning that it is generated by one element – not to be confused with a [[torsion group]]) generated by a rotation by an irrational number of turns about an axis. We may create non-cyclical [[abelian group]]s by adding more rotations around the same axis. There are also non-abelian groups generated by rotations around different axes. These are usually (generically) [[free group]]s. They will be infinite unless the rotations are specially chosen.
| |
| | |
| All the infinite groups mentioned so far are not [[Closed (topology)|closed]] as [[topological group|topological subgroups]] of O(3). We now discuss topologically closed subgroups of O(3).
| |
| | |
| The whole O(3) is the symmetry group of spherical symmetry; [[SO(3)]] is the corresponding rotation group. The other infinite isometry groups consist of all [[rotation]]s about an axis through the origin, and those with additionally reflection in the planes through the axis, and/or reflection in the plane through the origin, perpendicular to the axis. Those with reflection in the planes through the axis, with or without reflection in the plane through the origin perpendicular to the axis, are the symmetry groups for the two types of cylindrical symmetry. Note that any physical object having infinite rotational symmetry will also have the symmetry of mirror planes through the axis.
| |
| | |
| See also [[Rotational symmetry#Rotational symmetry with respect to any angle|rotational symmetry with respect to any angle]].
| |
| | |
| ==Finite isometry groups==
| |
| Symmetries in 3D that leave the origin fixed are fully characterized by symmetries on a sphere centered at the origin. For finite 3D point groups, see also [[list of spherical symmetry groups|spherical symmetry groups]].
| |
| | |
| Up to conjugacy the set of finite 3D point groups consists of:
| |
| *7 infinite series with at most one more-than-2-fold rotation axis; they are the finite symmetry groups on an infinite [[Cylinder (geometry)|cylinder]], or equivalently, those on a finite cylinder. They are sometimes called the axial or prismatic point groups.
| |
| *7 point groups with multiple 3-or-more-fold rotation axes; they can also be characterized as point groups with multiple 3-fold rotation axes, because all 7 include these axes; with regard to 3-or-more-fold rotation axes the possible combinations are:
| |
| **4 3-fold axes
| |
| **4 3-fold axes and 3 4-fold axes
| |
| **10 3-fold axes and 6 5-fold axes
| |
| A selection of point groups is compatible with discrete [[translational symmetry]]: 27 from the 7 infinite series, and 5 of the 7 others, the 32 so-called crystallographic point groups. See also the [[crystallographic restriction theorem]].
| |
| | |
| ==The seven infinite series of axial groups==
| |
| | |
| The infinite series of axial or prismatic groups have an index ''n'', which can be any integer; in each series, the ''n''th symmetry group contains ''n''-fold [[rotational symmetry]] about an axis, i.e. symmetry with respect to a rotation by an angle 360°/''n''. ''n''=1 covers the cases of no rotational symmetry at all. There are four series with no other axes of rotational symmetry (see [[cyclic symmetries]]) and three with additional axes of 2-fold symmetry (see [[dihedral symmetry]]). They can be understood as [[point groups in two dimensions]] extended with an axial coordinate and reflections in it. They are related to the [[frieze group]]s;<ref>{{citation | first1=G.L. | last1=Fisher | first2=B. | last2=Mellor | title= Three-dimensional finite point groups and the symmetry of beaded beads | journal=[[Journal of Mathematics and the Arts]] | year=2007 | url=http://myweb.lmu.edu/bmellor/beadedbeads.pdf}}</ref> they can be interpreted as frieze-group patterns repeated ''n'' times around a cylinder.
| |
| | |
| The following table lists several notations for point groups: [[Hermann–Mauguin notation]], [[Arthur Moritz Schönflies|Schönflies]] notation, [[orbifold notation]], and [[Coxeter notation]]. The latter two are not only conveniently related to its properties, but also to the order of the group, see below. It is a unified notation, also applicable for [[wallpaper group]]s and [[frieze group]]s. The crystallographic groups have ''n'' restricted to 1, 2, 3, 4, and 6; removing crystallographic restriction allows any positive integer.
| |
| | |
| The series are:
| |
| {| class="wikitable"
| |
| |-
| |
| ! colspan=2 | Hermann–Mauguin
| |
| ! rowspan=2 | Schönflies
| |
| ! rowspan=2 | Orbifold
| |
| ! rowspan=2 | Coxeter
| |
| ! rowspan=2 | Frieze
| |
| ! rowspan=2 | Order
| |
| ! rowspan=2 | Abstract group
| |
| ! rowspan=2 | Comments
| |
| |-
| |
| ! Even ''n'' || Odd ''n''
| |
| |-
| |
| | colspan=2 | ''n''
| |
| | C<sub>''n''</sub>
| |
| | ''nn''
| |
| | [n]<sup>+</sup>
| |
| | p1
| |
| | ''n''
| |
| | Z<sub>''n''</sub>
| |
| | ''n''-fold rotational symmetry
| |
| |-
| |
| | {{overline|2''n''}}
| |
| | {{overline|''n''}}
| |
| | S<sub>2''n''</sub>
| |
| | ''n''x
| |
| | [2n<sup>+</sup>,2<sup>+</sup>]
| |
| | p11g
| |
| | 2''n''
| |
| | Z<sub>2''n''</sub>
| |
| | Not to be confused with the [[symmetric group]]s
| |
| |-
| |
| | ''n''/m
| |
| | {{overline|2''n''}}
| |
| | C<sub>''n''h</sub>
| |
| | ''n''*
| |
| | [n<sup>+</sup>,2]
| |
| | p11m
| |
| | 2''n''
| |
| | Z<sub>''n''</sub> × Z<sub>2</sub>
| |
| |
| |
| |-
| |
| | ''n''mm
| |
| | ''n''m
| |
| | C<sub>''n''v</sub>
| |
| | *''nn''
| |
| | [n]
| |
| | p1m1
| |
| | 2''n''
| |
| | Dih<sub>''n''</sub>
| |
| | Pyramidal symmetry; in biology, biradial symmetry
| |
| |-
| |
| | ''n''22
| |
| | ''n''2
| |
| | D<sub>''n''</sub>
| |
| | 22''n''
| |
| | [n,2]<sup>+</sup>
| |
| | p211
| |
| | 2''n''
| |
| | Dih<sub>''n''</sub>
| |
| | Dihedral symmetry
| |
| |-
| |
| | {{overline|2''n''}}2m
| |
| | {{overline|''n''}}m
| |
| | D<sub>''n''d</sub>, D<sub>''n''v</sub>
| |
| | 2*''n''
| |
| | [2n,2<sup>+</sup>]
| |
| | p2mg
| |
| | 4''n''
| |
| | Dih<sub>2''n''</sub>
| |
| | Antiprismatic symmetry
| |
| |-
| |
| | ''n''/mmm
| |
| | {{overline|2''n''}}2m
| |
| | D<sub>''n''h</sub>
| |
| | *22''n''
| |
| | [n,2]
| |
| | p2mm
| |
| | 4''n''
| |
| | Dih<sub>''n''</sub> × Z<sub>2</sub>
| |
| | Prismatic symmetry
| |
| |}
| |
| | |
| For odd ''n'' we have Z<sub>2''n''</sub> = Z<sub>''n''</sub> × Z<sub>2</sub> and Dih<sub>2''n''</sub> = Dih<sub>''n''</sub> × Z<sub>2</sub>.
| |
| | |
| The terms horizontal (h) and vertical (v), and the corresponding subscripts, refer to the additional mirror plane, that can be parallel to the rotation axis (vertical) or perpendicular to the rotation axis (horizontal).
| |
| | |
| The simplest nontrivial ones have [[Involution (mathematics)|Involution]]al symmetry (abstract group Z<sub>2</sub> ):
| |
| *''C''<sub>''i''</sub> – '''[[inverse (mathematics)|inversion]] symmetry'''
| |
| *''C''<sub>2</sub> – '''2-fold [[rotational symmetry]]'''
| |
| *''C''<sub>''s''</sub> – '''[[reflection symmetry]]''', also called '''[[symmetry (biology)#Bilateral symmetry|bilateral symmetry]]'''.
| |
| | |
| [[Image:Uniaxial.png|right|thumb|200px|Patterns on a cylindrical band illustrating the case ''n'' = 6 for each of the 7 infinite families of point groups. The symmetry group of each pattern is the indicated group.]]
| |
| The second of these is the first of the uniaxial groups ([[cyclic group]]s) ''C<sub>n</sub>'' of order ''n'' (also applicable in 2D), which are generated by a single rotation of angle 360°/''n''. In addition to this, one may add a mirror plane perpendicular to the axis, giving the group ''C<sub>nh</sub>'' of order 2''n'', or a set of ''n'' mirror planes containing the axis, giving the group ''C<sub>nv</sub>'', also of order 2''n''. The latter is the symmetry group for a regular ''n''-sided [[pyramid (geometry)|pyramid]]. A typical object with symmetry group ''C''<sub>''n''</sub> or ''D''<sub>''n''</sub> is a [[propeller]].
| |
| | |
| If both horizontal and vertical reflection planes are added, their intersections give ''n'' axes of rotation through 180°, so the group is no longer uniaxial. This new group of order 4''n'' is called ''D<sub>nh</sub>''. Its subgroup of rotations is the [[dihedral group]] ''D<sub>n</sub>'' of order 2''n'', which still has the 2-fold rotation axes perpendicular to the primary rotation axis, but no mirror planes. Note that in 2D ''D<sub>n</sub>'' includes reflections, which can also be viewed as flipping over flat objects without distinction of front- and backside, but in 3D the two operations are distinguished: the group contains "flipping over", not reflections.
| |
| | |
| There is one more group in this family, called ''D<sub>nd</sub>'' (or ''D<sub>nv</sub>''), which has vertical mirror planes containing the main rotation axis, but instead of having a horizontal mirror plane, it has an isometry that combines a reflection in the horizontal plane and a rotation by an angle 180°/''n''. ''D<sub>nh</sub>'' is the symmetry group for a regular ''(n+2)''-sided [[Prism (geometry)|prisms]] and also for a regular (2n)-sided [[bipyramid]]. ''D<sub>nd</sub>'' is the symmetry group for a regular ''(n+2)''-sided [[antiprism]], and also for a regular ''(2n)''-sided [[trapezohedron]]. ''D<sub>n</sub>'' is the symmetry group of a partially rotated prism.
| |
| | |
| The groups ''D''<sub>2</sub> and ''D''<sub>2''h''</sub> are noteworthy in that there is no special rotation axis. Rather, there are three perpendicular 2-fold axes. ''D''<sub>2</sub> is a subgroup of all the polyhedral symmetries (see below), and ''D''<sub>2''h''</sub> is a subgroup of the polyhedral groups T<sub>h</sub> and O<sub>h</sub>. ''D''<sub>2</sub> can occur in [[homotetramer]]s such as [[Concanavalin A]], in tetrahedral [[coordination compound]]s with four identical [[chiral ligand]]s, or in a molecule such as tetrakis(chlorofluoromethyl)methane if all the chlorofluoromethyl groups have the same chirality. The elements of ''D''<sub>2</sub> are in 1-to-2 correspondence with the rotations given by the [[Unit (ring theory)|unit]] [[Lipschitz quaternion]]s.
| |
| | |
| The group ''S<sub>n</sub>'' is generated by the combination of a reflection in the horizontal plane and a rotation by an angle 360°/n. For ''n'' odd this is equal to the group generated by the two separately, ''C<sub>nh</sub>'' of order 2''n'', and therefore the notation ''S<sub>n</sub>'' is not needed; however, for ''n'' even it is distinct, and of order ''n''. Like ''D<sub>nd</sub>'' it contains a number of [[improper rotation]]s without containing the corresponding rotations.
| |
| | |
| All symmetry groups in the 7 infinite series are different, except for the following four pairs of mutually equal ones:
| |
| *''C<sub>1h</sub>'' and ''C<sub>1v</sub>'': group of order 2 with a single reflection (''C<sub>s</sub>'' )
| |
| *''D''<sub>1</sub> and ''C''<sub>2</sub>: group of order 2 with a single 180° rotation
| |
| *''D''<sub>1''h''</sub> and ''C''<sub>2''v''</sub>: group of order 4 with a reflection in a plane and a 180° rotation through a line in that plane
| |
| *''D''<sub>1''d''</sub> and ''C''<sub>2''h''</sub>: group of order 4 with a reflection in a plane and a 180° rotation through a line perpendicular to that plane
| |
| | |
| ''S''<sub>2</sub> is the group of order 2 with a single inversion (''C<sub>i</sub>'' )
| |
| | |
| "Equal" is meant here as the same up to conjugacy in space. This is stronger than "up to algebraic isomorphism". For example, there are three different groups of order two in the first sense, but there is only one in the second sense. Similarly, e.g. ''S<sub>2n</sub>'' is algebraically isomorphic with Z''<sub>2n</sub>''.
| |
| | |
| The groups may be constructed as follows:
| |
| * C<sub>''n''</sub>. Generated by an element also called C<sub>''n''</sub>, which corresponds to a rotation by angle 2π/''n'' around the axis. Its elements are E (the identity), C<sub>''n''</sub>, C<sub>''n''</sub><sup>2</sup>, ..., C<sub>''n''</sub><sup>''n''−1</sup>, corresponding to rotation angles 0, 2π/''n'', 4π/''n'', ..., 2(''n'' − 1)π/''n''.
| |
| * S<sub>2''n''</sub>. Generated by element C<sub>2''n''</sub>σ<sub>h</sub>, where </sub>σ<sub>h</sub> is a reflection in the direction of the axis. Its elements are the elements of C<sub>''n''</sub> with C<sub>''2n''</sub>σ<sub>h</sub>, C<sub>2''n''</sub><sup>3</sup>σ<sub>h</sub>, ..., C<sub>2''n''</sub><sup>2''n''−1</sup>σ<sub>h</sub> added.
| |
| * C<sub>''n''h</sub>. Generated by element C<sub>''n''</sub> and reflection σ<sub>h</sub>. Its elements are the elements of group C<sub>''n''</sub>, with elements σ<sub>h</sub>, C<sub>''n''</sub>σ<sub>h</sub>, C<sub>''n''</sub><sup>2</sup>σ<sub>h</sub>, ..., C<sub>''n''</sub><sup>''n''−1</sup>σ<sub>h</sub> added.
| |
| * C<sub>''n''v</sub>. Generated by element C<sub>''n''</sub> and reflection σ<sub>v</sub> in a direction in the plane perpendicular to the axis. Its elements are the elements of group C<sub>''n''</sub>, with elements σ<sub>v</sub>, C<sub>''n''</sub>σ<sub>v</sub>, C<sub>''n''</sub><sup>2</sup>σ<sub>v</sub>, ..., C<sub>''n''</sub><sup>''n''−1</sup>σ<sub>v</sub> added.
| |
| * D<sub>''n''</sub>. Generated by element C<sub>''n''</sub> and 180° rotation U = σ<sub>h</sub>σ<sub>v</sub> around a direction in the plane perpendicular to the axis. Its elements are the elements of group C<sub>''n''</sub>, with elements U, C<sub>''n''</sub>U, C<sub>''n''</sub><sup>2</sup>U, ..., C<sub>''n''</sub><sup>''n'' − 1</sup>U added.
| |
| * D<sub>''n''d</sub>. Generated by elements C<sub>2''n''</sub>σ<sub>h</sub> and σ<sub>v</sub>. Its elements are the elements of group C<sub>''n''</sub> and the additional elements of S<sub>2''n''</sub> and C<sub>''n''v</sub>, with elements C<sub>2''n''</sub>σ<sub>h</sub>σ<sub>v</sub>, C<sub>2''n''</sub><sup>3</sup>σ<sub>h</sub>σ<sub>v</sub>, ..., C<sub>2''n''</sub><sup>2''n'' − 1</sup>σ<sub>h</sub>σ<sub>v</sub> added.
| |
| * D<sub>''n''h</sub>. Generated by elements C<sub>''n''</sub>, σ<sub>h</sub>, and σ<sub>v</sub>. Its elements are the elements of group C<sub>''n''</sub> and the additional elements of C<sub>''n''h</sub>, C<sub>''n''v</sub>, and D<sub>''n''</sub>.
| |
| | |
| Taking ''n'' to ∞ yields groups with continuous axial rotations:
| |
| | |
| {| class="wikitable"
| |
| |-
| |
| ! H–M
| |
| ! Schönflies
| |
| ! Orbifold
| |
| ! Coxeter
| |
| ! Limit of
| |
| ! Abstract group
| |
| |-
| |
| | ∞
| |
| | C<sub>∞</sub>
| |
| | ∞∞
| |
| | [∞]<sup>+</sup>
| |
| | C<sub>''n''</sub>
| |
| | SO(2)
| |
| |-
| |
| | {{overbar|∞}}, ∞/m
| |
| | C<sub>∞h</sub>
| |
| | ∞*
| |
| | [2,∞<sup>+</sup>]
| |
| | C<sub>''n''h</sub>, S<sub>2''n''
| |
| | SO(2) × Z<sub>2</sub>
| |
| |-
| |
| | ∞m
| |
| | C<sub>∞v</sub>
| |
| | *∞∞
| |
| | [∞]
| |
| | C<sub>''n''v</sub>
| |
| | O(2)
| |
| |-
| |
| | ∞2
| |
| | D<sub>∞</sub>
| |
| | 22∞
| |
| | [2,∞]<sup>+</sup>
| |
| | D<sub>''n''</sub>
| |
| | O(2)
| |
| |-
| |
| | {{overbar|∞}}m, ∞/mm
| |
| | D<sub>∞h</sub>
| |
| | *22∞
| |
| | [2,∞]
| |
| | D<sub>''n''h</sub>, D<sub>''n''d</sub>
| |
| | O(2) × Z<sub>2</sub>
| |
| |}
| |
| | |
| ==The seven remaining point groups==<!-- This section is linked from [[Polyhedron]] -->
| |
| The remaining point groups are said to be of very high or [[polyhedron|polyhedral]] symmetry because they have more than one rotation axis of order greater than 2. Here, C<sub>n</sub> denotes an axis of rotation through 360°/n and S<sub>n</sub> denotes an axis of improper rotation through the same. In parentheses are the [[orbifold notation]], [[Coxeter notation]], the full [[Hermann–Mauguin notation]], and the abbreviated one if different. The groups are:
| |
| | |
| *'''T''' (332, [3,3]<sup>+</sup>, 23) of order 12 – '''chiral [[tetrahedral symmetry]]'''. There are four C<sub>3</sub> axes, each through two vertices of a [[cube]] (body diagonals) or one of a regular [[tetrahedron]], and three C<sub>2</sub> axes, through the centers of the cube's faces, or the midpoints of the tetrahedron's edges. This group is [[isomorphic]] to ''A''<sub>4</sub>, the [[alternating group]] on 4 elements, and is the rotation group for a regular tetrahedron. It is a [[normal subgroup]] of T<sub>d</sub>, T<sub>h</sub>, and the octahedral symmetries. The elements of the group correspond 1-to-2 to the rotations given by the 24 [[Unit (ring theory)|unit]] [[Hurwitz quaternion]]s (the "[[binary tetrahedral group]]").
| |
| | |
| *'''T<sub>d</sub>''' (*332, [3,3] {{overline|4}}3m) of order 24 – '''full [[tetrahedral symmetry]]'''. This group has the same rotation axes as T, but with six mirror planes, each containing two edges of the cube or one edge of the tetrahedron, a single C<sub>2</sub> axis and two C<sub>3</sub> axes. The C<sub>2</sub> axes are now actually S<sub>4</sub> axes. This group is the symmetry group for a regular [[tetrahedron]]. T<sub>d</sub> is isomorphic to ''S''<sub>4</sub>, the [[symmetric group]] on 4 letters, because there is a 1-to-1 correspondence between the elements of T<sub>d</sub> and the 24 permutations of the four 3-fold axes. An object of ''C''<sub>3v</sub> symmetry under one of the 3-fold axes gives rise under the action of T<sub>d</sub> to an [[Group action#orbstab|orbit]] consisting of four such objects, and T<sub>d</sub> corresponds to the set of permutations of these four objects. T<sub>d</sub> is a normal subgroup of O<sub>h</sub>. See also [[Tetrahedron#The isometries of the regular tetrahedron|the isometries of the regular tetrahedron]].
| |
| | |
| *'''T<sub>h</sub>''' (3*2, [3<sup>+</sup>,4], 2/m{{overline|3}}, m{{overline|3}}) of order 24 – '''[[tetrahedral symmetry|pyritohedral symmetry]]'''.[[Image:Volleyball seams diagram.png|thumb|The seams of a [[volleyball (ball)|volleyball]] have T<sub>h</sub> symmetry.]] This group has the same rotation axes as ''T'', with mirror planes parallel to the cube faces. The C<sub>3</sub> axes become S<sub>6</sub> axes, and there is inversion symmetry. T<sub>h</sub> is isomorphic to ''A''<sub>4</sub> × ''C''<sub>2</sub> (since T and C<sub>i</sub> are both normal subgroups), and not to the [[symmetric group]] S<sub>4</sub>. It is the symmetry of a cube with on each face a line segment dividing the face into two equal rectangles, such that the line segments of adjacent faces do not meet at the edge. The symmetries correspond to the even permutations of the body diagonals and the same combined with inversion. It is also the symmetry of a [[pyritohedron]], which is similar to the cube described, with each rectangle replaced by a pentagon with one symmetry axis and 4 equal sides and 1 different side (the one corresponding to the line segment dividing the cube's face); i.e., the cube's faces bulge out at the dividing line and become narrower there. It is a subgroup (but not a normal subgroup) of the full icosahedral symmetry group (as isometry group, not just as abstract group), with 4 of the 10 3-fold axes. It is a normal subgroup of O<sub>h</sub>.
| |
| | |
| *'''O''' (432, [4,3]<sup>+</sup>, 432) of order 24 – '''chiral [[octahedral symmetry]]'''. This group is like T, but the C<sub>2</sub> axes are now C<sub>4</sub> axes, and additionally there are 6 C<sub>2</sub> axes, through the midpoints of the edges of the cube. This group is also isomorphic to ''S''<sub>4</sub> because its elements are in 1-to-1 correspondence to the 24 permutations of the 3-fold axes, as with T. An object of ''D''<sub>3</sub> symmetry under one of the 3-fold axes gives rise under the action of O to an [[Group action#orbstab|orbit]] consisting of four such objects, and O corresponds to the set of permutations of these four objects. It is the rotation group of the [[Cube (geometry)|cube]] and [[octahedron]]. Representing rotations with [[quaternion]]s, O is made up of the 24 [[Unit (ring theory)|unit]] [[Hurwitz quaternion]]s and the 24 [[Lipschitz quaternion]]s of squared norm 2 normalized by dividing by <math>\sqrt 2</math>. As before, this is a 1-to-2 correspondence.
| |
| | |
| *'''O<sub>h</sub>''' (*432, [4,3], 4/m{{overline|3}}2/m, m{{overline|3}}m) of order 48 – '''full octahedral symmetry'''. This group has the same rotation axes as ''O'', but with mirror planes, comprising both the mirror planes of ''T<sub>d</sub>'' and ''T<sub>h</sub>''. This group is isomorphic to ''S''<sub>4</sub> × ''C''<sub>2</sub> (because both O and ''C<sub>i</sub>'' are normal subgroups), and is the symmetry group of the [[Cube (geometry)|cube]] and [[octahedron]]. See also [[octahedral symmetry#The isometries of the cube|the isometries of the cube]].
| |
| | |
| *'''I''' (532, [5,3]<sup>+</sup>, 532) of order 60 – '''chiral [[icosahedral symmetry]]'''; the rotation group of the [[icosahedron]] and the [[dodecahedron]]. It is a [[normal subgroup]] of [[index of a subgroup|index]] 2 in the full group of symmetries '''I<sub>h</sub>'''. The group contains 10 versions of ''D<sub>3</sub>'' and 6 versions of ''D<sub>5</sub>'' (rotational symmetries like prisms and antiprisms). It also contains five versions of T<sub>h</sub> (see [[Compound of five tetrahedra]]). The group '''I''' is [[isomorphic]] to ''A''<sub>5</sub>, the [[alternating group]] on 5 letters, since its elements correspond 1-to-1 with even permutations of the five T<sub>h</sub> symmetries (or the five tetrahedra just mentioned).
| |
| | |
| *'''I<sub>h</sub>''' (*532, [5,3], {{overline|5}}{{overline|3}}2/m, {{overline|5}}{{overline|3}}m) of order 120 – '''full icosahedral symmetry'''; the symmetry group of the icosahedron and the dodecahedron. The group '''I<sub>h</sub>''' is isomorphic to ''A''<sub>5</sub> × ''C''<sub>2</sub> because I and ''C''<sub>i</sub> are both normal subgroups. The group contains 10 versions of ''D<sub>3d</sub>'', 6 versions of ''D<sub>5d</sub>'' (symmetries like antiprisms), and 5 versions of T<sub>h</sub>.
| |
| | |
| The continuous groups related to these groups are:
| |
| * '''K''' or SO(3), all possible rotations.
| |
| * '''K<sub>h</sub>''' or O(3), all possible rotations and reflections.
| |
| As noted above for infinite rotation groups, any physical object having K symmetry will also have K<sub>h</sub> symmetry.
| |
| | |
| ==Relation between orbifold notation and order==
| |
| The order of each group is 2 divided by the [[orbifold]] [[Euler characteristic]]; the latter is 2 minus the sum of the feature values, assigned as follows:
| |
| *''n'' without or before * counts as (''n''−1)/''n''
| |
| *''n'' after * counts as (''n''−1)/(2''n'')
| |
| * * and x count as 1
| |
| This can also be applied for [[wallpaper group]]s and [[frieze group]]s: for them, the sum of the feature values is 2, giving an infinite order; see [[2D crystallographic group#Why there are exactly seventeen groups|orbifold Euler characteristic for wallpaper groups]]
| |
| | |
| ==Rotation groups==
| |
| The rotation groups, i.e. the finite subgroups of SO(3), are: the cyclic groups ''C''<sub>n</sub> (the rotation group of a regular [[pyramid (geometry)|pyramid]]), the dihedral groups ''D''<sub>n</sub> (the rotation group of a regular [[prism (geometry)|prism]], or regular [[bipyramid]]), and the rotation groups ''T'', ''O'' and ''I'' of a regular [[tetrahedron]], [[octahedron]]/[[cube]] and [[icosahedron]]/[[dodecahedron]].
| |
| | |
| In particular, the dihedral groups ''D''<sub>3</sub>, ''D''<sub>4</sub> etc. are the rotation groups of plane regular polygons embedded in three-dimensional space, and such a figure may be considered as a degenerate regular prism. Therefore it is also called a ''[[dihedron]]'' (Greek: solid with two faces), which explains the name ''dihedral group''.
| |
| | |
| *An object with symmetry group ''C<sub>n</sub>'', ''C<sub>nh</sub>'', ''C<sub>nv</sub>'' or ''S<sub>2n</sub>'' has rotation group ''C<sub>n</sub>.
| |
| *An object with symmetry group ''D<sub>n</sub>'', ''D<sub>nh</sub>'', or ''D<sub>nd</sub>'' has rotation group ''D<sub>n</sub>.
| |
| *An object with one of the other seven symmetry groups has as rotation group the corresponding one without subscript: ''T'', ''O'' or ''I''.
| |
| | |
| The rotation group of an object is equal to its full symmetry group [[if and only if]] the object is [[chirality (mathematics)|chiral]]. In other words, the chiral objects are those with their symmetry group in the list of rotation groups.
| |
| | |
| Given in [[Schönflies notation]], [[Coxeter notation]], ([[orbifold notation]]), the rotation subgroups are:
| |
| {| class=wikitable
| |
| !Reflectional
| |
| !Reflection/rotational
| |
| !Improper rotation
| |
| !Rotation
| |
| |- align=center
| |
| | ''C<sub>nv</sub>'', [n], (*nn)
| |
| | ''C<sub>nh</sub>'', [n<sup>+</sup>,2], (n*)
| |
| | ''S<sub>2n</sub>'', [2n<sup>+</sup>,2<sup>+</sup>], (nx)
| |
| | ''C<sub>n</sub>'', [n]<sup>+</sup>, (nn)
| |
| |- align=center
| |
| | ''D<sub>nh</sub>'', [2,n], (*n22)
| |
| | ''D<sub>nd</sub>'', [2<sup>+</sup>,2n], (2*n)
| |
| |
| |
| | ''D<sub>n</sub>'', [2,n]<sup>+</sup>, (n22)
| |
| |- align=center
| |
| | ''T<sub>d</sub>'', [3,3], (*332)
| |
| | ''T<sub>h</sub>'', [3<sup>+</sup>,4], (3*2)
| |
| |
| |
| | ''T'', [3,3]<sup>+</sup>, (332)
| |
| |- align=center
| |
| | ''O<sub>h</sub>'', [4,3], (*432)
| |
| |
| |
| |
| |
| | ''O'', [4,3]<sup>+</sup>, (432)
| |
| |- align=center
| |
| | ''I<sub>h</sub>'', [5,3], (*532)
| |
| |
| |
| |
| |
| | ''I'', [5,3]<sup>+</sup>, (532)
| |
| |-
| |
| |}
| |
| | |
| ==Correspondence between rotation groups and other groups==
| |
| The following groups contain [[Inversion in a point|inversion]]:
| |
| *''C''<sub>''nh''</sub> and ''D''<sub>''nh''</sub> for even ''n''
| |
| *''S''<sub>2''n''</sub> and ''D''<sub>''nd''</sub> for odd ''n'' (''S<sub>2</sub>'' = ''C<sub>i</sub>'' is the group generated by inversion; ''D''<sub>''1d''</sub> = ''C''<sub>''2h''</sub>)
| |
| *''T<sub>h</sub>'', ''O<sub>h</sub>'', and ''I<sub>h</sub>''
| |
| | |
| As explained above, there is a 1-to-1 correspondence between these groups and all rotation groups:
| |
| *''C''<sub>''nh''</sub> for even ''n'' and ''S''<sub>2''n''</sub> for odd ''n'' correspond to ''C''<sub>''n''</sub>
| |
| *''D''<sub>''nh''</sub> for even ''n'' and ''D''<sub>''nd''</sub> for odd ''n'' correspond to ''D''<sub>''n''</sub>
| |
| *''T<sub>h</sub>'', ''O<sub>h</sub>'', and ''I<sub>h</sub>'' correspond to ''T'', ''O'', and ''I'', respectively.
| |
| | |
| The other groups contain indirect isometries, but not inversion:
| |
| *''C''<sub>''nv''</sub>
| |
| *''C''<sub>''nh''</sub> and ''D''<sub>''nh''</sub> for odd ''n''
| |
| *''S''<sub>2''n''</sub> and ''D''<sub>''nd''</sub> for even ''n''
| |
| *''T<sub>d</sub>''
| |
| | |
| They all correspond to a rotation group ''H'' and a subgroup ''L'' of index 2 in the sense that they are obtained from ''H'' by inverting the isometries in ''H'' \ ''L'', as explained above:
| |
| *''C''<sub>''n''</sub> is subgroup of ''D''<sub>''n''</sub> of index 2, giving ''C''<sub>''nv''</sub>
| |
| *''C''<sub>''n''</sub> is subgroup of ''C''<sub>''2n''</sub> of index 2, giving ''C''<sub>''nh''</sub> for odd ''n'' and ''S''<sub>2''n''</sub> for even ''n''
| |
| *''D''<sub>''n''</sub> is subgroup of ''D''<sub>''2n''</sub> of index 2, giving ''D''<sub>''nh''</sub> for odd ''n'' and ''D''<sub>''nd''</sub> for even ''n''
| |
| *T is subgroup of ''O'' of index 2, giving ''T<sub>d</sub>''
| |
| | |
| ==Maximal symmetries==
| |
| There are two discrete point groups with the property that no discrete point group has it as proper subgroup: ''O<sub>h</sub>'' and ''I<sub>h</sub>''. Their largest common subgroup is ''T<sub>h</sub>''. The two groups are obtained from it by changing 2-fold rotational symmetry to 4-fold, and adding 5-fold symmetry, respectively. Alternatively the two groups are generated by adding for each a reflection plane to ''T<sub>h</sub>''.
| |
| | |
| There are two crystallographic point groups with the property that no crystallographic point group has it as proper subgroup: ''O<sub>h</sub>'' and ''D<sub>6h</sub>''. Their maximal common subgroups, depending on orientation, are ''D<sub>3d</sub>'' and ''D<sub>2h</sub>''.
| |
| | |
| ==The groups arranged by abstract group type==
| |
| Below the groups explained above are arranged by abstract group type.
| |
| | |
| The smallest abstract groups that are ''not'' any symmetry group in 3D, are the [[quaternion group]] (of order 8), Z<sub>3</sub> × Z<sub>3</sub> (of order 9), the [[dicyclic group]] Dic<sub>3</sub> (of order 12), and 10 of the 14 groups of order 16.
| |
| | |
| The column "# of order 2 elements" in the following tables shows the total number of isometry subgroups of types ''C<sub>2</sub>'', ''C<sub>i</sub>'', ''C<sub>s</sub>''. This total number is one of the characteristics helping to distinguish the various abstract group types, while their isometry type helps to distinguish the various isometry groups of the same abstract group.
| |
| | |
| Within the possibilities of isometry groups in 3D, there are infinitely many abstract group types with 0, 1 and 3 elements of order 2, there are two with 2''n'' + 1 elements of order 2, and there are three with 2''n'' + 3 elements of order 2 (for each ''n'' ≥ 2 ). There is never a positive even number of elements of order 2.
| |
| | |
| {{anchor|Cyclic 3D symmetry groups}}
| |
| | |
| ===Symmetry groups in 3D that are cyclic as abstract group===
| |
| The [[symmetry group]] for ''n''-fold rotational [[symmetry]] is ''C<sub>n</sub>''; its abstract group type is [[cyclic group]] Z<sub>n</sub>, which is also denoted by ''C<sub>n</sub>''. However, there are two more infinite series of symmetry groups with this abstract group type:
| |
| *For even order 2''n'' there is the group [[Improper rotation|''S<sub>2n</sub>'']] (Schoenflies notation) generated by a rotation by an angle 180°/n about an axis, combined with a reflection in the plane perpendicular to the axis. For ''S<sub>2</sub>'' the notation ''C<sub>i</sub>'' is used; it is generated by inversion.
| |
| *For any order 2''n'' where ''n'' is odd, we have ''C<sub>nh</sub>''; it has an ''n''-fold rotation axis, and a perpendicular plane of reflection. It is generated by a rotation by an angle 360°/''n'' about the axis, combined with the reflection. For ''C''<sub>1''h''</sub> the notation ''C<sub>s</sub>'' is used; it is generated by reflection in a plane.
| |
| | |
| Thus we have, with bolding of the 10 cyclic crystallographic point groups, for which the [[crystallographic restriction theorem|crystallographic restriction]] applies:
| |
| {| class=wikitable
| |
| |-
| |
| !Order !!Isometry groups !! Abstract group !! # of order 2 elements
| |
| |- align=center
| |
| |1 || '''''C<sub>1</sub>''''' || Z<sub>1</sub> || 0
| |
| |- align=center
| |
| |2 || '''''C<sub>2</sub>''''', '''''C<sub>i</sub>''''', '''''C<sub>s</sub>'''''|| Z<sub>2</sub> || 1
| |
| |- align=center
| |
| |3 || '''''C<sub>3</sub>''''' || Z<sub>3</sub> || 0
| |
| |- align=center
| |
| |4 || '''''C<sub>4</sub>''''', '''''S<sub>4</sub>''''' || Z<sub>4</sub> || 1
| |
| |- align=center
| |
| |5 || ''C<sub>5</sub>'' || Z<sub>5</sub> || 0
| |
| |- align=center
| |
| |6 || '''''C<sub>6</sub>''''', '''''S<sub>6</sub>''''', '''''C<sub>3h</sub>''''' || Z<sub>6</sub> = Z<sub>3</sub> × Z<sub>2</sub> || 1
| |
| |- align=center
| |
| |7 || ''C<sub>7</sub>'' || Z<sub>7</sub> || 0
| |
| |- align=center
| |
| |8 || ''C<sub>8</sub>'', ''S<sub>8</sub>'' || Z<sub>8</sub> || 1
| |
| |- align=center
| |
| |9 || ''C<sub>9</sub>'' || Z<sub>9</sub> || 0
| |
| |- align=center
| |
| |10 || ''C<sub>10</sub>'', ''S<sub>10</sub>'', ''C<sub>5h</sub>'' || Z<sub>10</sub> = Z<sub>5</sub> × Z<sub>2</sub> || 1
| |
| |}
| |
| etc.
| |
| | |
| ===Symmetry groups in 3D that are dihedral as abstract group===
| |
| In 2D [[dihedral group]] ''D<sub>n</sub>'' includes reflections, which can also be viewed as flipping over flat objects without distinction of front- and backside.
| |
| | |
| However, in 3D the two operations are distinguished: the symmetry group denoted by ''D<sub>n</sub>'' contains ''n'' 2-fold axes perpendicular to the ''n''-fold axis, not reflections. ''D<sub>n</sub>'' is the [[rotation group SO(3)|rotation group]] of the ''n''-sided [[prism (geometry)|prism]] with regular base, and ''n''-sided [[bipyramid]] with regular base, and also of a regular, ''n''-sided [[antiprism]] and of a regular, ''n''-sided [[trapezohedron]]. The group is also the full symmetry group of such objects after making them [[chirality (mathematics)|chiral]] by e.g. an identical chiral marking on every face, or some modification in the shape.
| |
| | |
| The abstract group type is [[dihedral group]] Dih<sub>''n''</sub>, which is also denoted by ''D<sub>n</sub>''. However, there are three more infinite series of symmetry groups with this abstract group type:
| |
| | |
| *''C<sub>nv</sub>'' of order 2''n'', the symmetry group of a regular ''n''-sided [[Pyramid (geometry)|pyramid]]
| |
| *''D<sub>nd</sub>'' of order 4''n'', the symmetry group of a regular ''n''-sided [[antiprism]]
| |
| *''D<sub>nh</sub>'' of order 4''n'' for odd ''n''. For ''n'' = 1 we get ''D''<sub>2</sub>, already covered above, so ''n'' ≥ 3.
| |
| | |
| Note the following property:
| |
| :Dih<sub>''4n+2''</sub> <math>\cong</math> Dih<sub>''2n+1''</sub> × Z<sub>2</sub>
| |
| | |
| Thus we have, with bolding of the 12 crystallographic point groups, and writing ''D<sub>1d</sub>'' as the equivalent ''C<sub>2h</sub>'':
| |
| {| class=wikitable
| |
| |-
| |
| !Order !!Isometry groups !! Abstract group !! # of order 2 elements
| |
| |- align=center
| |
| |4 || '''''D<sub>2</sub>''''', '''''C<sub>2v</sub>''''', '''''C<sub>2h</sub>''''' || Dih<sub>2</sub> = Z<sub>2</sub> × Z<sub>2</sub> || 3
| |
| |- align=center
| |
| |6 || '''''D<sub>3</sub>''''', '''''C<sub>3v</sub>''''' || Dih<sub>3</sub> || 3
| |
| |- align=center
| |
| |8 || '''''D<sub>4</sub>''''', '''''C<sub>4v</sub>''''', '''''D<sub>2d</sub>''''' || Dih<sub>4</sub> || 5
| |
| |- align=center
| |
| |10 || ''D''<sub>5</sub>, ''C''<sub>5''v''</sub> || Dih<sub>5</sub> || 5
| |
| |- align=center
| |
| |12 || '''''D<sub>6</sub>''''', '''''C<sub>6v</sub>''''', '''''D<sub>3d</sub>''''', '''''D<sub>3h</sub>''''' || Dih<sub>6</sub> = Dih<sub>3</sub> × Z<sub>2</sub> || 7
| |
| |- align=center
| |
| |14 || ''D''<sub>7</sub>, ''C''<sub>7''v''</sub> || Dih<sub>7</sub> || 7
| |
| |- align=center
| |
| |16 || ''D''<sub>8</sub>, ''C''<sub>8''v''</sub>, ''D''<sub>4''d''</sub> || Dih<sub>8</sub> || 9
| |
| |- align=center
| |
| |18 || ''D''<sub>9</sub>, ''C''<sub>9''v''</sub> || Dih<sub>9</sub> || 9
| |
| |- align=center
| |
| |20 || ''D''<sub>10</sub>, ''C''<sub>10''v''</sub>, ''D''<sub>5''h''</sub>, ''D''<sub>5''d''</sub> || Dih<sub>10</sub> = ''D''<sub>5</sub> × Z<sub>2</sub> || 11
| |
| |}
| |
| etc.
| |
| | |
| ===Other===
| |
| ''C<sub>2n,h</sub>'' of order 4''n'' is of abstract group type Z<sub>2''n''</sub> × Z<sub>2</sub>. For ''n'' = 1 we get Dih<sub>2</sub>, already covered above, so ''n'' ≥ 2.
| |
| | |
| Thus we have, with bolding of the 2 cyclic crystallographic point groups:
| |
| {| class=wikitable
| |
| |-
| |
| !Order !!Isometry group !! Abstract group !! # of order 2 elements !! [[Cycle diagram]]
| |
| |- align=center
| |
| |8 || '''''C<sub>4h</sub>''''' || Z<sub>4</sub> × Z<sub>2</sub> || 3 || [[Image:GroupDiagramMiniC2C4.png]]
| |
| |- align=center
| |
| |12 || '''''C<sub>6h</sub>''''' ||Z<sub>6</sub> × Z<sub>2</sub> = Z<sub>3</sub> × Z<sub>2</sub> × Z<sub>2</sub> = Z<sub>3</sub> × Dih<sub>2</sub>|| 3 || [[Image:GroupDiagramMiniC2C6.png]]
| |
| |- align=center
| |
| |16 || ''C<sub>8h</sub>'' || Z<sub>8</sub> × Z<sub>2</sub> || 3 || [[Image:GroupDiagramMiniC2C8.png]]
| |
| |- align=center
| |
| |20 || ''C<sub>10h</sub>'' || Z<sub>10</sub> × Z<sub>2</sub> = Z<sub>5</sub> × Z<sub>2</sub> × Z<sub>2</sub> || 3 ||
| |
| |}
| |
| etc.
| |
| | |
| ''D<sub>nh</sub>'' of order 4''n'' is of abstract group type Dih<sub>n</sub> × Z<sub>2</sub>. For odd ''n'' this is already covered above, so we have here ''D''<sub>2''n''h</sub> of order 8''n'', which is of abstract group type Dih<sub>2''n''</sub> × Z<sub>2</sub> (''n''≥1).
| |
| | |
| Thus we have, with bolding of the 3 dihedral crystallographic point groups:
| |
| {| class=wikitable
| |
| |-
| |
| !Order !!Isometry group !! Abstract group !! # of order 2 elements !! [[Cycle diagram]]
| |
| |- align=center
| |
| |8 || '''''D<sub>2h</sub>''''' || Dih<sub>2</sub> × Z<sub>2</sub> || 7 ||[[Image:GroupDiagramMiniC2x3.png]]
| |
| |- align=center
| |
| |16 || '''''D<sub>4h</sub>''''' || Dih<sub>4</sub> × Z<sub>2</sub> || 11 ||[[Image:GroupDiagramMiniC2D8.png]]
| |
| |- align=center
| |
| |24 || '''''D<sub>6h</sub>''''' || Dih<sub>6</sub> × Z<sub>2</sub>|| 15 ||
| |
| |- align=center
| |
| |32 || ''D<sub>8h</sub>'' || Dih<sub>8</sub> × Z<sub>2</sub> || 19 ||
| |
| |}
| |
| etc.
| |
| | |
| The remaining seven are, with bolding of the 5 crystallographic point groups (see also above):
| |
| | |
| *order 12: of type ''A''<sub>4</sub> ([[alternating group]]): '''T'''
| |
| *order 24:
| |
| **of type ''S''<sub>4</sub> ([[symmetric group]], not to be confused with the symmetry group with this notation): '''T<sub>d</sub>''', '''O'''
| |
| **of type ''A''<sub>4</sub> × Z<sub>2</sub>: '''T<sub>h</sub>''' .
| |
| *order 48, of type ''S''<sub>4</sub> × Z<sub>2</sub>: '''O<sub>h</sub>'''
| |
| *order 60, of type ''A''<sub>5</sub>: ''I''
| |
| *order 120, of type ''A''<sub>5</sub> × Z<sub>2</sub>: ''I<sub>h</sub>''
| |
| See also [[icosahedral symmetry]].
| |
| | |
| ==Impossible discrete symmetries==
| |
| Since the overview is exhaustive, it also shows implicitly what is ''not'' possible as discrete symmetry group. For example:
| |
| *a ''C''<sub>6</sub> axis in one direction and a ''C''<sub>3</sub> in another
| |
| *a ''C''<sub>5</sub> axis in one direction and a ''C''<sub>4</sub> in another
| |
| *a ''C''<sub>3</sub> axis in one direction and another ''C''<sub>3</sub> axis in a perpendicular direction
| |
| etc.
| |
| | |
| ==Fundamental domain==
| |
| The [[fundamental domain]] of a point group is a [[conic solid]]. An object with a given symmetry in a given orientation is characterized by the fundamental domain. If the object is a surface it is characterized by a surface in the fundamental domain continuing to its radial bordal faces or surface. If the copies of the surface do not fit, radial faces or surfaces can be added. They fit anyway if the fundamental domain is bounded by reflection planes.
| |
| | |
| For a polyhedron this surface in the fundamental domain can be part of an arbitrary plane. For example, in the [[disdyakis triacontahedron]] one full face is a fundamental domain. Adjusting the orientation of the plane gives various possibilities of combining two or more adjacent faces to one, giving various other polyhedra with the same symmetry. The polyhedron is convex if the surface fits to its copies and the radial line perpendicular to the plane is in the fundamental domain.
| |
| | |
| Also the surface in the fundamental domain may be composed of multiple faces.
| |
| | |
| ==Binary polyhedral groups==
| |
| The map Spin(3) → SO(3) is the double cover of the rotation group by the [[spin group]] in 3 dimensions. (This is the only connected cover of SO(3), since Spin(3) is simply connected.)
| |
| By the [[lattice theorem]], there is a [[Galois connection]] between subgroups of Spin(3) and subgroups of SO(3) (rotational point groups): the image of a subgroup of Spin(3) is a rotational point group, and the preimage of a point group is a subgroup of Spin(3).
| |
| | |
| The preimage of a finite point group is called a '''binary polyhedral group''', represented as <l,n,m>, and is called by the same name as its point group, with the prefix '''binary''', with double the order of the related [[polyhedral group]] (l,m,n). For instance, the preimage of the [[icosahedral group]] (2,3,5) is the [[binary icosahedral group]], <2,3,5>.
| |
| | |
| The binary polyhedral groups are:
| |
| * <math>A_n</math>: [[binary cyclic group]] of an (''n'' + 1)-gon
| |
| * <math>D_n</math>: [[binary dihedral group]] of an ''n''-gon, <2,2,n>
| |
| * <math>E_6</math>: [[binary tetrahedral group]], <2,3,3>
| |
| * <math>E_7</math>: [[binary octahedral group]], <2,3,4>
| |
| * <math>E_8</math>: [[binary icosahedral group]], <2,3,5>
| |
| These are classified by the [[ADE classification]], and the quotient of '''C'''<sup>2</sup> by the action of a binary polyhedral group is a [[Du Val singularity]].<ref>[http://enriques.mathematik.uni-mainz.de/burban/singul.pdf Du Val Singularities, by Igor Burban]</ref>
| |
| | |
| For point groups that reverse orientation, the situation is more complicated, as there are two [[pin group]]s, so there are two possible binary groups corresponding to a given point group.
| |
| | |
| Note that this is a covering of ''groups,'' not a covering of ''spaces'' – the sphere is [[simply connected]], and thus has no [[covering space]]s. There is thus no notion of a "binary polyhedron" that covers a 3-dimensional polyhedron. Binary polyhedral groups are discrete subgroups of a Spin group, and under a representation of the spin group act on a vector space, and may stabilize a polyhedron in this representation – under the map Spin(3) → SO(3) they act on the same polyhedron that the underlying (non-binary) group acts on, while under [[spin representation]]s or other representations they may stabilize other polyhedra.
| |
| | |
| This is in contrast to [[projective polyhedra]] – the sphere does cover [[projective space]] (and also [[lens space]]s), and thus a tessellation of projective space or lens space yields a distinct notion of polyhedron.
| |
| | |
| ==See also==
| |
| <div style="-moz-column-count:2; column-count:2;">
| |
| *[[List of spherical symmetry groups]]
| |
| *[[List of character tables for chemically important 3D point groups]]
| |
| *[[Point groups in two dimensions]]
| |
| *[[Symmetry]]
| |
| *[[Euclidean plane isometry]]
| |
| *[[Group action]]
| |
| *[[Point group]]
| |
| *[[Crystal system]]
| |
| *[[Space group]]
| |
| *[[List of small groups]]
| |
| *[[Molecular symmetry]]
| |
| </div>
| |
| | |
| ==Footnotes==
| |
| {{reflist}}
| |
| | |
| ==References==
| |
| {{Refimprove|date=May 2010}}
| |
| {{refbegin}}
| |
| * {{Citation | authorlink = Harold Scott MacDonald Coxeter | last = Coxeter | first = H. S. M. | title = Regular Complex Polytopes | publisher = Cambridge University Press | year = 1974 | chapter = 7 The Binary Polyhedral Groups | pages = [http://books.google.com/books?id=9BY9AAAAIAAJ&pg=PA73 73–82] }}.
| |
| *{{cite book | author=Coxeter, H. S. M. and Moser, W. O. J. | title=Generators and Relations for Discrete Groups, 4th edition | location=New York | publisher=Springer-Verlag | year=1980 | isbn=0-387-09212-9}} 6.5 The binary polyhedral groups, p. 68
| |
| *{{Citation | last1=Conway | first1=John Horton | author1-link=John Horton Conway | last2=Huson | first2=Daniel H. | title= The Orbifold Notation for Two-Dimensional Groups | publisher=Springer Netherlands | doi=10.1023/A:1015851621002 | year=2002 | journal=Structural Chemistry | volume=13 | issue=3 | pages=247–257}}
| |
| {{refend}}
| |
| | |
| ==External links==
| |
| *[http://newton.ex.ac.uk/research/qsystems/people/goss/symmetry/Solids.html Graphic overview of the 32 crystallographic point groups] – form the first parts (apart from skipping ''n''=5) of the 7 infinite series and 5 of the 7 separate 3D point groups
| |
| *[http://newton.ex.ac.uk/research/qsystems/people/goss/symmetry/CC_All.html Overview of properties of point groups]
| |
| *[http://homepage.mac.com/dmccooey/polyhedra/Simplest.html Simplest Canonical Polyhedra of Each Symmetry Type] (uses Java)
| |
| * [http://www.stanford.edu/~yishuwei/crystal.pdf] Point Groups and Crystal Systems, by Yi-Shu Wei, pp. 4–6
| |
| * [http://www.geom.uiuc.edu/docs/reference/CRC-formulas/node45.html The Geometry Center: 10.1 Formulas for Symmetries in Cartesian Coordinates (three dimensions)]
| |
| | |
| {{DEFAULTSORT:Point Groups In Three Dimensions}}
| |
| [[Category:Euclidean symmetries]]
| |
| [[Category:Group theory]]
| |