Four moves, and only four
A repeating pattern is a motif plus the rigid motions that carry it onto itself. In the plane there are exactly four: slide it (translation), spin it (rotation), flip it (reflection), or flip-and-slide together (a glide reflection — the footprints of a walker).
That's the whole vocabulary. Every pattern below is built from these. Step through them.
One tile fills the wall
Pick a single asymmetric motif — here a blocky letter L, chosen so every flip and turn shows. Choose which moves are allowed, and the pattern writes itself: each move makes a copy, the copies make copies, and the plane fills.
Add a half-turn to plain repetition and you get p2; add mirrors and you climb toward p4 and p6. Warm tiles are the originals, cool ones are mirror-flipped.
Only 1, 2, 3, 4, 6
A repeating pattern can spin onto itself by a half-turn, a third, a quarter or a sixth — and by nothing else. Never a fifth. Never a seventh. This isn't a shortage of clever tilings; it is arithmetic, and the whole proof fits in one picture.
Take the shortest vector a between two points of the lattice. If turning the pattern by θ leaves it unchanged, then the turned copies R(+θ)a and R(−θ)a must land on lattice points too — and so must their sum. But that sum always lies along a, and it is exactly 2·cos θ times as long.
So 2·cos θ has to be a whole number. For a half-turn it is −2; a third, −1; a quarter, 0; a sixth, 1. For a fifth it is 0.618, and the sum lands in the gap between two lattice points, where nothing exists. The lattice cannot close. Decades later, Penrose got five-fold symmetry anyway — by giving up repetition altogether.
Reading a pattern's bones
Every group has a hidden scaffold of symmetry elements: rotation centres (a lens for 2-fold, triangle for 3, square for 4, hexagon for 6), solid mirror lines, and dashed glide lines. Find them and you've identified the group.
This is p4m — the richest square pattern. Reveal its scaffold, then strip back to the pattern.
The seventeen
This is all of them — proven complete by Fedorov in 1891. The same little L, run through every distinct symmetry of the plane. From p1, which only slides, to p6m, with its six-fold centres and mirrors in every direction. Tap any one to read its bones; recolour to taste.