Wallpaper Group Designer
Every repeating pattern in the plane — every wallpaper, tiled floor, brick bond and Escher print — has one of exactly 17 symmetry groups. This board lets you draw freely with pen, line, circle and dot tools while the group you chose repeats each stroke through all of its operations: every reflection, glide reflection and rotation of the group, then every lattice translation. Switch groups and the same motif reorganises itself instantly, from p1 (pure translation) through the rectangular groups with their mirrors and glides to the square and hexagonal groups with 4- and 6-fold centres. The symmetry elements are not drawn by hand: the tool derives mirror lines, glide lines and rotation centres from the group’s operations and overlays them on the pattern, so you can see exactly where an F becomes its mirror image and why pg has no mirrors at all. Stretch the cell of the rectangular and oblique groups, zoom the cell size with the wheel, undo freely, and export the view as PNG or fully vector SVG, or a seamless rectangular tile (PNG or SVG) ready for CSS backgrounds and print. Your design autosaves in the browser and any state can be shared as a link.
Runs 100% in your browser — simulations are computed locally on your device.
Read the full guide to this tool
Notes
- The group operations follow the International Tables for Crystallography: (x, y) → (−x, y + ½) is the glide of pg, and the hexagonal groups use the 120° basis. Rotation centres are the fixed points of each operation; a reflection’s axis is found by splitting its translation part along the eigenvectors of its matrix.
- The cell drawn in dark outline is one conventional unit cell. For cm and cmm it is the centred rectangular cell, so the motif appears twice per cell offset by half a diagonal. Your strokes are stored in lattice coordinates, which is why changing the cell shape stretches the motif with it.
- Seamless tiles need a rectangular period, which every lattice except the oblique one (p1, p2) has: rectangular and square cells are their own tile, the hexagonal tile is a rectangle of width a and height a√3 holding two cells. Oblique designs still export as a full vector SVG of the view.
- Tip: the letter F is the classic probe for symmetry. Load the F preset and cycle through the groups: mirrors flip it, glides flip and shift it, and rotation centres spin it.
- Runs 100% in your browser — simulations are computed locally on your device.
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