Paint a motif and watch it tile the plane in all 17 wallpaper symmetries
Every pattern that repeats in two directions, from bathroom tiles to crystal structures, has one of exactly 17 kinds of symmetry.
Click any pattern to enlarge it with its fundamental domain highlighted, then press Back to Gallery. All 17 start from the same lopsided motif.
Click any tile to view the wallpaper group in detail. Each of the 17 groups represents a distinct way to tile the plane using symmetry operations.
A wallpaper group is the symmetry group of a pattern that repeats in two independent directions, tiling the entire plane. One of the great theorems of crystallography states that there are exactly 17 distinct wallpaper groups. Every repeating 2D pattern -- from bathroom tiles to medieval mosaics to atomic crystal structures -- falls into one of these 17 categories.
This classification was completed by Fedorov in 1891 and independently by Polya in 1924. Remarkably, all 17 groups appear in the decorative arts of the Alhambra palace in Granada, created centuries before the mathematical classification existed.
A gallery of all 17 wallpaper groups, each generated from the same asymmetric motif. Click any pattern to see it enlarged with its fundamental domain highlighted. Notice how the same base shape produces 17 entirely different tilings through different symmetry operations.
Key insight: The 17 groups are constrained by the crystallographic restriction: only 2-fold, 3-fold, 4-fold, and 6-fold rotational symmetries are compatible with translational periodicity. This is why 5-fold and 7-fold symmetric wallpaper patterns are impossible (but quasicrystals can achieve 5-fold -- see the Penrose tilings lesson).
Paint inside the dashed fundamental domain and watch your strokes copy across the plane. Switch groups to see the same drawing turn into very different patterns, from p1 to p6m.
Draw inside the fundamental domain (dashed outline) to see your strokes replicated by the symmetry group across the canvas.
Paint inside a fundamental domain and watch your design replicate across the plane according to the selected wallpaper group. Try different groups to see how the same drawing can produce dramatically different overall patterns -- from simple translations (p1) to the full hexagonal symmetry of p6m.
Key insight: The fundamental domain is the smallest region that generates the entire pattern under the group action. For p1, it is a full parallelogram. For p6m (the most symmetric group), it is a tiny triangle -- 1/12 of the hexagonal unit cell. More symmetry means a smaller fundamental domain and more copies per unit cell.
Choose a lattice type and see which mirror lines, glide axes, and rotation centers it allows. Notice that more lattice symmetry permits more wallpaper groups.
Every wallpaper group is built on one of five Bravais lattice types. The lattice determines which rotation orders and reflection symmetries are possible. The cyan outline shows the unit cell; colored dots mark rotation centers of different orders.
Every wallpaper pattern is built on one of 5 Bravais lattice types: oblique, rectangular, centered rectangular, square, and hexagonal. Each lattice constrains which symmetries are possible. Explore how mirror lines, glide reflections, and rotation centers are arranged on each lattice type.
Key insight: The oblique lattice (most general) supports only p1 and p2. The rectangular lattice adds mirror lines, enabling pm, pg, pmm, pmg, pgg. The square lattice allows 4-fold rotations (p4, p4m, p4g). The hexagonal lattice unlocks 3-fold and 6-fold symmetries (p3, p3m1, p31m, p6, p6m). More lattice symmetry = more wallpaper groups.
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