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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