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Point groups in three dimensions
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Selected point groups in three dimensions<table><tbody><tr><td><a href="/facts/List_of_spherical_symmetry_groups/3X3YcHoQ">Involutional symmetry</a>Cs, (*)[ ] = </td><td><a href="/facts/Cyclic_symmetry_in_three_dimensions/wQUCezbQ">Cyclic symmetry</a>Cnv, (*nn)[n] = </td><td><a href="/facts/Dihedral_symmetry_in_three_dimensions/xNRpsIAF">Dihedral symmetry</a>Dnh, (*n22)[n,2] = </td></tr><tr><th colspan="4"><a href="/facts/Polyhedral_group/ynwsQLXE">Polyhedral group</a>, [n,3], (*n32)</th></tr><tr><td><a href="/facts/Tetrahedral_symmetry/hIVlf9pu">Tetrahedral symmetry</a>Td, (*332)[3,3] = </td><td><a href="/facts/Octahedral_symmetry/qLqhYxgm">Octahedral symmetry</a>Oh, (*432)[4,3] = </td><td><a href="/facts/Icosahedral_symmetry/pDlwAy23">Icosahedral symmetry</a>Ih, (*532)[5,3] = </td></tr></tbody></table> <p>In <a href="/facts/Geometry/wTQf5ffQ">geometry</a>, a <a href="/facts/Point_group/41NSqUN6">point group</a> in three dimensions is an <a href="/facts/Isometry_group/X5uyVIdJ">isometry group</a> in three dimensions that leaves the origin fixed, or correspondingly, an isometry group of a <a href="/facts/Sphere/jywYLNM2">sphere</a>. It is a <a href="/facts/Subgroup/r91mBgpa">subgroup</a> of the <a href="/facts/Orthogonal_group/jKWM1KDM">orthogonal group</a> O(3), the <a href="/facts/Group_(mathematics)/IQTYqINt">group</a> of all <a href="/facts/Isometry/K5wX8ah6">isometries</a> that leave the origin fixed, or correspondingly, the group of <a href="/facts/Orthogonal_matrix/WDXBXYB5">orthogonal matrices</a>. O(3) itself is a subgroup of the <a href="/facts/Euclidean_group/VdTSdwzl">Euclidean group</a> E(3) of all isometries. </p><p><a href="/facts/Symmetry_group/CNqeseMp">Symmetry groups</a> of geometric objects are isometry groups. Accordingly, analysis of isometry groups is analysis of possible <a href="/facts/Symmetry/hpu7DK8p">symmetries</a>. All isometries of a <a href="/facts/Bounded_set/yCldjtoy">bounded</a> (finite) 3D object have one or more common fixed points. We follow the usual convention by choosing the <a href="/facts/Origin_(mathematics)/Lm26tRCB">origin</a> as one of them. </p><p>The symmetry group of an object is sometimes also called its full symmetry group, as opposed to its proper symmetry group, the intersection of its full symmetry group with <a href="/facts/Euclidean_group/VdTSdwzl">E+(3)</a>, which consists of all <a href="/facts/Euclidean_group/VdTSdwzl"><i>direct isometries</i></a>, i.e., isometries preserving <a href="/facts/Orientation_(mathematics)/lGXs42Nw">orientation</a>. For a bounded object, the proper symmetry group is called its rotation group. It is the intersection of its full symmetry group with <a href="/facts/Rotation_group_SO(3)/yXbKPzhT">SO(3)</a>, the full rotation group of the 3D space. The rotation group of a bounded object is equal to its full symmetry group <a href="/facts/If_and_only_if/bYSxGJ66">if and only if</a> the object is <a href="/facts/Chirality_(mathematics)/FtKNlHTb">chiral</a>. </p><p>The point groups that are generated purely by a finite set of reflection mirror planes passing through the same point are the <a href="/facts/Finite_Coxeter_group/bgzHFm1Q">finite Coxeter groups</a>, represented by <a href="/facts/Coxeter_notation/ArUw6goe">Coxeter notation</a>. </p><p>The point groups in three dimensions are widely used in <a href="/facts/Chemistry/SrNqkGVm">chemistry</a>, especially to describe the symmetries of a <a href="/facts/Molecule/N9ljSfFq">molecule</a> and of <a href="/facts/Molecular_orbital/0dP30jaa">molecular orbitals</a> forming <a href="/facts/Covalent_bond/EzMLf3Pz">covalent bonds</a>, and in this context they are also called <a href="/facts/List_of_character_tables_for_chemically_important_3D_point_groups/yhvVb4Y2">molecular point groups</a>. </p>