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Point group - In mathematics, point group is a group of geometric symmetries (isometries) leaving a point fixed. Point groups can exist in a Euclidean space of any dimension.
Crystallographic point group - In crystallography, a crystallographic point group is a set of symmetry operations, like rotations or reflections, that leave a point fixed while moving each atom of the crystal to the position of an atom of the same kind. That is, an infinite crystal would look exactly the same before and after any of the operations in its point group.
Point groups in two dimensions - In geometry, a point group in two dimensions is an isometry group in two dimensions that leaves the origin fixed, or correspondingly, an isometry group of a circle. It is a subgroup of the orthogonal group O(2), the group of all isometries which leave the origin fixed, or correspondingly, the group of orthogonal matrices.
Point groups in three dimensions - In geometry a point group in 3D 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.
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