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Crystal structure is determined by geometrical operations that occur naturally and can be explained mathematically. These basic mathematical operations produce recognizable and repeatable patterns in two dimensions and in three dimensions. First the 2-D operations are examined: translations, rotations, reflections and glides are explained and demonstrated using animations. The 17 unique 2-D plane groups may be further explored with an interactive space group activity. These same mathematical operations are then extended to three dimensions to produce the 32 unique crystal classes and 230 3-D space groups.
Topics covered in Chapter 11 include: Operations in Two Dimensions Combinations of Operations: Planar Point Groups Plane Lattices: Combinations of Rotations and Translations Plane Groups = Translation + Rotation + Reflection + Glide Operations in Three Dimensions Combinations of Operations in 3-D: Crystal Classes Space Lattices Space Groups