Crystallographic groups mathematically describe all possible discrete symmetries of periodic structures in n-dimensional Euclidean space. They represent the complete set of isometries that preserve a crystal lattice.

2D Plane Groups (Wallpaper Groups)

The 17 plane symmetry groups, proven complete by Fedorov (1891) and Pólya (1924), classify all two-dimensional repetitive patterns. Each group is generated by combinations of translations, rotations (2-, 3-, 4-, or 6-fold), reflections, and glide reflections, subject to the crystallographic restriction theorem.

3D Space Groups

The 230 three-dimensional space groups, independently derived by Fedorov (1890), Schönflies (1891), and Barlow (1894), extend plane groups with additional symmetry operations including screw axes and glide planes. These groups are fundamental to X-ray crystallography and the International Tables for Crystallography.

4D Crystallographic Groups

Based on the systematic enumeration by Brown, Bülow, Neubüser, Wondratschek, and Zassenhaus (1978), there are 4,783 four-dimensional space groups (4,894 including enantiomorphic pairs). This visualization implements stereographic projection from 4D to 3D, with interactive controls for hyperplane sectioning and 4D rotations. The work extends Bieberbach's theorems and utilizes computational group theory for the complete classification.