PIATRA . INSTITUTE

Crystallographic Groups

Symmetry patterns in 2D, 3D, and 4D

based on 1978, H. Brown et al., Crystallographic Groups of Four-Dimensional Space

17 Wallpaper Groups

Group 1 / 17: p1

Crystallographic groups

Crystallographic groups describe all possible discrete symmetries of periodic structures in n-dimensional Euclidean space. They are 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

Following 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 uses computational group theory for the complete classification.

claude opus 4.8July 2025·first cut. an interactive tour of the 17 wallpaper groups (plus 3D and 4D companions), each pattern generated as the orbit of a single motif under the cell point group on its Bravais lattice. adds an exact logic module reading off point-group order, rotation order, reflection and glide presence, and lattice type for every group, a calibration panel checking those exact integers against the textbook classification, eight assumptions separating the established theorem from the rendering simplifications, and a research companion on the plane-group classification.

v1.0July 2025

wallpaper layer: all 17 plane groups rendered by laying out one motif and its point-group orbit across a square, centred, or hexagonal lattice.
space-group companions: a 3D space-group viewer and a 4D crystallographic viewer with stereographic projection, hyperplane slicing, and 4D rotation controls.
logic module: exact per-group specs (point-group order, highest rotation order, reflection flag, glide flag, Bravais lattice) derived from the same construction the canvas draws.
calibration: computed group properties checked against the textbook classification (17 groups, p6 6-fold, p6m order-12 point group, 10 mirror groups), all exact.
assumptions: the Euclidean-plane, infinite-lattice, and crystallographic-restriction premises behind the closed count of 17 made explicit, alongside the discrete-glide and fixed-lattice simplifications.
research companion on the structure and proof history of the plane-group classification.

read the research companion →