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- Point_groups_in_three_dimensions abstract "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 isometries that leave the origin fixed, or correspondingly, the group of orthogonal matrices. O(3) itself is a subgroup of the Euclidean group E(3) of all isometries.Symmetry groups of objects are isometry groups. Accordingly, analysis of isometry groups is analysis of possible 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 chiral.The point groups in three dimensions are heavily used in chemistry, especially to describe the symmetries of a molecule and of molecular orbitals forming covalent bonds, and in this context they are also called molecular point groups.Finite Coxeter groups 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.".
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- Point_groups_in_three_dimensions wikiPageExternalLink CC_All.html.
- Point_groups_in_three_dimensions wikiPageExternalLink Solids.html.
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- Point_groups_in_three_dimensions wikiPageExternalLink crystal.pdf.
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- Point_groups_in_three_dimensions wikiPageRevisionID "604164035".
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- Point_groups_in_three_dimensions subject Category:Euclidean_symmetries.
- Point_groups_in_three_dimensions subject Category:Group_theory.
- Point_groups_in_three_dimensions type Abstraction100002137.
- Point_groups_in_three_dimensions type Attribute100024264.
- Point_groups_in_three_dimensions type EuclideanSymmetries.
- Point_groups_in_three_dimensions type Property104916342.
- Point_groups_in_three_dimensions type SpatialProperty105062748.
- Point_groups_in_three_dimensions type Symmetry105064827.
- Point_groups_in_three_dimensions comment "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 isometries that leave the origin fixed, or correspondingly, the group of orthogonal matrices. O(3) itself is a subgroup of the Euclidean group E(3) of all isometries.Symmetry groups of objects are isometry groups.".
- Point_groups_in_three_dimensions label "Point groups in three dimensions".
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