Everything about Point Groups In Three Dimensions totally explained
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 which 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.
Group structure
SO(3) is a subgroup of
E+(3), which consists of
direct isometries, for example, isometries preserving
orientation; it contains those which leave the origin fixed.
O(3) is the
direct product of SO(3) and the group generated by
inversion (denoted by its matrix −
I):
» O(3) = SO(3) × ^2 by the action of a binary polyhedral group is a
Du Val singularity.
For point groups that reverse orientation, the situation is more complicated, as there are two
pin groups, so there are two possible binary groups corresponding to a given point group.
Further Information
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