Note 4

See The Essay on Crystals.
A structure (pattern) has a translative symmetry if it maps onto itself (i.e. if the structure remains unchanged in all respects) after a certain linear shift of this structure (i.e. after an operation of linear movement of the whole structure over a certain distance has been carried out).
A structure has a point symmetry if it maps onto itself after an operation (executed on that structure), such that at least one point of this structure has remained at its original location (for example a rotation, or a reflection).


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