![]() All isometries of a bounded 3D object have one or more common fixed points. Accordingly, analysis of isometry groups is analysis of possible symmetries. ![]() Symmetry groups of objects are isometry groups. O(3) itself is a subgroup of the Euclidean group E(3) of all isometries. 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. ![]() 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. Groups of point isometries in 3 dimensions
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