THE 17 WALLPAPER GROUPS, EXPLAINED

Every flat pattern that repeats in two directions — a tiled floor, a brick wall, a printed fabric, an Islamic mosaic — has exactly one of 17 possible symmetry structures. Here is the theorem behind that number, what the 17 groups are, how to tell them apart by eye, and where each one hides in the real world.

Jump to: the four symmetries · the theorem · the 17 groups · how to identify them · in the wild · friezes · rosettes · the orbifold shortcut

// the four symmetries

A symmetry of a pattern is a movement of the plane — an isometry — that leaves the pattern looking exactly as it did. For a repeating flat pattern there are only four kinds:

Translation — slide the pattern along and it lines up with itself. Every wallpaper pattern has translations in two independent directions; that is what "repeating" means. The two shortest ones span the pattern's lattice, and the parallelogram they form is its unit cell.
Rotation — turn the pattern about a point and it lines up. A point you can turn about by 180° is a 2-fold center; 120° is 3-fold; 90° is 4-fold; 60° is 6-fold.
Reflection — flip the pattern across a line (a mirror) and it lines up.
Glide reflection — flip across a line and slide along it by half a repeat. Footprints in snow are the classic example: the left print reflected and shifted gives the right print, but no plain mirror maps one onto the other.

A pattern's symmetry group is the complete set of movements that leave it unchanged. Two patterns that look nothing alike — a brick wall and a woven basket — can have the same group, because what is being compared is the structure of the movements, not the picture.

A tiled floor photographed at an angle, classified as p4g, with mirror lines, glide lines, and rotation centers drawn over the photo — Symmetry Hunter drawing the measured symmetries of a real floor: mirrors solid, glides dashed, rotation centers as markers. The reading here is p4g.

// the theorem

Theorem (Fedorov 1891; rediscovered independently by Pólya and Niggli in 1924). Let G be a group of isometries of the Euclidean plane that is discrete and contains translations in two independent directions. Then G is one of exactly 17 groups, up to a change of coordinates.

"Discrete" rules out patterns with arbitrarily small repeats; "two independent directions" is what makes it wallpaper rather than a strip. In three dimensions the same question — the symmetry groups of crystals — has 230 answers, the space groups, which is what Fedorov was actually after.

The proof is a classification in four steps, and each step is something you can see in a pattern:

The translations form a lattice. Take the two shortest independent translations; every translation in the group is a whole-number combination of them. Up to shape there are only five kinds of plane lattice — oblique, rectangular, centered rectangular (rhombic), square, and hexagonal — the five two-dimensional Bravais lattices.
The crystallographic restriction. Any rotation in the group must map the lattice onto itself. Write the rotation by angle θ in lattice coordinates: it becomes a matrix of whole numbers, so its trace, 2 cos θ, must be a whole number. That leaves cos θ ∈ {−1, −½, 0, ½, 1}, i.e. θ ∈ {180°, 120°, 90°, 60°, 360°}: only 2-, 3-, 4-, and 6-fold rotations can exist. Five-fold and eight-fold symmetry are impossible for anything that repeats — the crystallographic restriction theorem.
The point group. Forget the translations and look only at which rotations and reflections occur. The result is a finite group of the allowed kinds: cyclic C_1,2,3,4,6 (rotations only) or dihedral D_1,2,3,4,6 (rotations plus mirrors) — ten possibilities, each compatible with only some of the five lattices.
Mirrors or glides? Finally, for each reflection ask whether it can be realized as a true mirror or is forced to be a glide. This is the step that splits pm from pg, pmm from pmg and pgg, p4m from p4g — and, on the hexagonal lattice, the two ways of orienting three mirrors give p3m1 and p31m. Thirteen groups have every reflection as a true mirror; four (pg, pmg, pgg, p4g) need at least one glide with no mirror parallel to it. Thirteen plus four is seventeen.

This is also why a Penrose tiling is refused as a wallpaper pattern: its five-fold order is real, but by step 2 it can never repeat, so it belongs to no wallpaper group at all. The Alhambra in Granada is the traditional place to go hunting for all 17; whether every one of them is really there has been argued in the literature for fifty years.

// the 17 groups

Each diagram takes one asymmetric glyph and repeats it by every symmetry of the group: cyan copies are rotations and translations of the original, amber copies are its mirror images — so amber appearing at all means the group has reflections or glides. The white outline is one unit cell. Symbols are the standard crystallographic (Hermann–Mauguin) names that Symmetry Hunter uses; the small symbol beneath is Conway's orbifold notation, explained below. Each name links to Wikipedia's entry for that group. Rarity is how often the group turns up in the built world, measured from field data — it is what the app's collection scores.

Group	Lattice	Rotations	Mirrors	Glides	Where you see it	Rarity
p1 o	oblique	none	none	none	A single asymmetric motif simply repeated — many printed fabrics and wrapping papers.	Uncommon
p2 2222	oblique	2-fold	none	none	Asymmetric motifs paired head-to-tail; S- and Z-shaped repeats; some parquet.	Uncommon
pm **	rectangular	none	one direction	none	Stripes of mirror-symmetric motifs; vertical bands of leaves or arches.	Uncommon
pg ××	rectangular	none	none	one direction	The glide-only group: alternating rows of a motif and its reflection, shifted by half a repeat — rare in the wild, the collector's trophy.	Legendary
cm *×	rhombic	none	one direction	between the mirrors	Fish-scale and scallop patterns; chevrons and zigzags with a vertical mirror.	Uncommon
pmm *2222	rectangular	2-fold	two directions	none	Rectangular grids: window panes, stacked-bond brick, plain rectangular tiles.	Common
pmg 22*	rectangular	2-fold	one direction	perpendicular to the mirrors	Rows of symmetric arches alternating up and down; some ironwork and lattice fences.	Rare
pgg 22×	rectangular	2-fold	none	two directions	Herringbone parquet and herringbone brick — glides both ways, not a single mirror.	Epic
cmm 2*22	rhombic	2-fold	two directions	yes	Running-bond brick walls, offset checkerboards, diamond lattices, most knit and weave diagrams.	Common
p4 442	square	4-fold, 2-fold	none	none	Pinwheel tilings and swirling square motifs with no mirror line anywhere.	Uncommon
p4m *442	square	4-fold, 2-fold	four directions	yes	The checkerboard, the square grid, most bathroom and kitchen tile — the most common pattern on Earth.	Common
p4g 4*2	square	4-fold, 2-fold	two directions	diagonal	Basket weave; squares rotated 90° in alternation; the 4-fold centers sit between the mirrors, not on them.	Rare
p3 333	hexagonal	3-fold	none	none	Escher's interlocking lizards; three-armed motifs with no reflection.	Rare
p3m1 *333	hexagonal	3-fold	three directions	yes	The two-colored triangular tiling (when the colors count); every 3-fold center lies on a mirror.	Epic
p31m 3*3	hexagonal	3-fold	three directions	yes	Trefoil and knot patterns in Islamic and Celtic work; some 3-fold centers lie off the mirrors — the tell that separates it from p3m1.	Epic
p6 632	hexagonal	6-fold, 3-fold, 2-fold	none	none	Six-armed whirls and chiral florets — hexagonal, spinning, mirror-free.	Rare
p6m *632	hexagonal	6-fold, 3-fold, 2-fold	six directions	yes	Honeycomb hex tile, the plain triangular tiling, the marble floor of squares, triangles, and hexagons — the second most common pattern anywhere.	Common

// how to tell them apart

Identifying a wallpaper group by eye is a short decision tree. Find the highest-order rotation center first, then ask about mirrors, then about glides.

No rotation at all
- Mirrors present → are there glide lines between the mirrors? yes → cm · no → pm
- No mirrors → glides present? yes → pg · no → p1
Highest rotation is 2-fold (180°)
- Mirrors in two directions → are all 2-fold centers on mirrors? yes → pmm · no → cmm
- Mirrors in one direction only → pmg
- No mirrors → glides present? yes → pgg · no → p2
Highest rotation is 3-fold (120°)
- Mirrors → are all 3-fold centers on mirrors? yes → p3m1 · no → p31m
- No mirrors → p3
Highest rotation is 4-fold (90°)
- Mirrors → do mirrors pass through the 4-fold centers? yes → p4m · no → p4g
- No mirrors → p4
Highest rotation is 6-fold (60°)
- Mirrors → p6m · no mirrors → p6

Colors change the answer

A checkerboard is p4m. But if you decide the two colors are "the same" and look only at the grid of lines, the pattern becomes a plain square grid — also p4m, with a smaller unit cell. Do the same with a two-colored triangular tiling and it jumps from p3m1 to p6m: the coloring was breaking the 6-fold centers. Whether colors count is a choice, and the honest thing is to say which choice you made. Symmetry Hunter classifies colors-as-distinct by default and has a Structure toggle for the other reading.

// in the wild

Real patterns, not diagrams. These plates are public domain — most are from Owen Jones's The Grammar of Ornament (1856), via Wikimedia Commons — and every one was classified by the app's own engine to confirm the label.

p6m Owen Jones, The Grammar of Ornament (1856) — a star-and-triangle lattice with six-fold centers and mirrors in six directions.

p4m Square lattice with mirrors through every 4-fold center.

p4g Four-fold centers between the mirrors, not on them.

cmm Rhombic lattice, mirrors both ways, 2-fold centers between them.

cm One mirror direction, with glides running between the mirrors.

pm Parallel mirrors and nothing else — no rotation at all.

p31m Three-fold centers with some of them off the mirror lines.

p31m Another p31m — the same group, an entirely different picture.

refused Penrose rhombus tiling: five-fold order that never repeats — no wallpaper group. Its centre reports as a d5 rosette.

// the 7 frieze groups

Patterns that repeat along a single strip — borders, railings, cornices, a line of footprints — have only one translation direction, and the same reasoning yields exactly seven frieze groups. John Conway gave them names you can act out.

Group	Conway name	Symmetries	Where you see it	Rarity
p1 ∞∞	hop	translation only	A motif repeated along a border with no other symmetry.	Uncommon
p11g ∞×	step	glide reflection	Footprints in snow; alternating leaves on a stem. The frieze trophy.	Epic
p1m1 *∞∞	sidle	vertical mirrors	Rows of symmetric arches, balusters, or shields.	Uncommon
p2 22∞	spinning hop	2-fold rotations	S-scroll borders; a motif and its 180° turn alternating.	Uncommon
p2mg 2*∞	spinning sidle	vertical mirrors + glide + 2-fold	Alternating up/down symmetric motifs — many Greek key and wave borders.	Rare
p11m ∞*	jump	horizontal mirror	A border symmetric about its centerline, like a reflected wave.	Uncommon
p2mm *22∞	spinning jump	both mirrors + 2-fold	Rectangular railings, dentil cornices, beaded borders — the commonest frieze.	Common

// rosettes

A cyclic rosette: five rotated copies of a glyph, no mirrors — c5 — cyclic: rotations only

A finite design with a center — a medallion, a mandala, a hubcap, a snowflake — does not repeat at all, so the crystallographic restriction does not apply and any rotation order is allowed. There are just two families, the point groups of the plane: the cyclic groups cn (n-fold rotation, no mirrors — a pinwheel) and the dihedral groups dn (n-fold rotation plus n mirrors — a snowflake is d6, a regular pentagon d5). This is where a Penrose tiling lands: refused as wallpaper, it reports its five-fold sun as a d5 rosette.

// the orbifold shortcut

There is a second, more modern way to see why the answer is 17, due to John Conway. Fold a pattern up by all of its symmetries — glue every point to every point it is symmetric with — and you get a small surface with some special points, an orbifold. Conway's notation just lists the features: a digit for each kind of rotation center (a gyration), * where a mirror line appears (with the rotation centers on mirrors written after the star), × for a glide with no mirror parallel to it (a miracle), and o for a plain wrap-around with nothing else.

Conway's "magic theorem"

Every feature has a price: a gyration of order n costs (n−1)/n; a mirror * costs 1, and a rotation center of order n sitting on a mirror costs (n−1)/2n; a miracle × costs 1; a wonder o costs 2. A pattern repeats in two directions exactly when the total comes to 2. Check it: p6 = 632 → 5⁄6 + 2⁄3 + 1⁄2 = 2. p4m = *442 → 1 + 3⁄8 + 3⁄8 + 1⁄4 = 2. pg = ×× → 1 + 1 = 2. List every way to spend exactly 2 euros in this shop and you get seventeen receipts — the 17 wallpaper groups. (Spend less than 2 and you get the frieze and rosette groups; the seven friezes are the ways to reach 2 with the infinite gyration ∞ allowed.)

// further reading

Wallpaper group on Wikipedia — the reference article, with a diagram and example images for each of the 17.
Conway, Burgiel & Goodman-Strauss, The Symmetries of Things (2008) — the definitive modern treatment and the source of the orbifold notation and the magic theorem.
Grünbaum & Shephard, Tilings and Patterns (1987) — the encyclopedic reference.
Owen Jones, The Grammar of Ornament (1856) — public-domain plates covering essentially all 17 groups; several are Symmetry Hunter's practice patterns.
The Alhambra question: Grünbaum, "What symmetry groups are present in the Alhambra?", Notices of the AMS 53 (2006).

Put it to the test.
Symmetry Hunter identifies the group from a photo, draws every mirror, glide, and rotation center on it, and keeps score as you collect all 17 in the wild.

Download Symmetry Hunter on the App Store