Point Reflection - Properties

Properties

In even-dimensional Euclidean space, say 2N-dimensional space, the inversion in a point P is equivalent to N rotations over angles π in each plane of an arbitrary set of N mutually orthogonal planes intersecting at P. These rotations are mutually commutative. Therefore inversion in a point in even-dimensional space is an orientation-preserving isometry or direct isometry.

In odd-dimensional Euclidean space, say (2N + 1)-dimensional space, it is equivalent to N rotations over π in each plane of an arbitrary set of N mutually orthogonal planes intersecting at P, combined with the reflection in the 2N-dimensional subspace spanned by these rotation planes. Therefore it reverses rather than preserves orientation, it is an indirect isometry.

Geometrically in 3D it amounts to rotation about an axis through P by an angle of 180°, combined with reflection in the plane through P which is perpendicular to the axis; the result does not depend on the orientation (in the other sense) of the axis. Notations for the type of operation, or the type of group it generates, are, C_i, S₂, and 1×. The group type is one of the three symmetry group types in 3D without any pure rotational symmetry, see cyclic symmetries with n=1.

The following point groups in three dimensions contain inversion:

• C_nh and D_nh for even n
• S_2n and D_nd for odd n
• T_h, O_h, and I_h

Closely related to inverse in a point is reflection in respect to a plane, which can be thought of as a "inversion in a plane".