Every border ever stitched, if it repeats, is one of seven kinds

Every border below starts from your motif. The first only moves it along, step by step. The others also turn it over: they mirror it left to right or top to bottom, spin it half a turn, or flip it and move it at once, the way footprints fall: left, right, left.

Pick one to see it large. The coloured lines show the symmetries the page found in the finished border.

If your motif is already symmetrical, some ways merge. A diamond mirrored left to right looks exactly like a diamond moved along, because the diamond was already its own mirror image. The page looks at what you stitched, not what you asked for, and tells you when a border has moved into another family.

Here you decide how each copy of the motif is turned: up to four copies in a row and, if you like, a second row underneath. Press a copy to turn it. However you arrange them, the result falls into one of the seven families. Each new family you hit lights up below and stays lit the next time you open the page.

A result that looks new can be an old one seen from the other end: the same border, started one copy further along. That is why the page compares finished borders, not lists of copies.

A border is a band that repeats in one direction. Any move that leaves it unchanged must lay the band back on top of itself. That leaves very few possible moves besides simply sliding it one step.

A mirror can only lie along the band, on its midline, or across it. The only rotation that keeps the band in place is the half-turn: a quarter-turn would stand it across the page. A glide can only run lengthways; running any other way, doing it twice would shift the band off itself. That leaves four kinds of move: the mirror across, the mirror along, the glide and the half-turn.

Four moves, each present or not, give 16 combinations. But moves combine: two mirrors at right angles make a half-turn, so you cannot have the mirrors without it. Donald Crowe, a mathematician and co-author of “Symmetries of Culture” (1988), writes that nine of the 16 combinations are impossible. Seven remain: no move; the glide alone; the mirror along with the glide; the mirror across alone; the half-turn alone; the mirror across with the glide and the half-turn; all four. Switch on the moves below and see where you land.

The seven families, with the names John Conway gave them after the tracks left by someone moving that way: hopping on one foot, stepping, sidling, jumping with feet together.

Who first listed all seven depends on whom you ask. Gwen Fisher writes that Paul Niggli was the first to enumerate the seven border groups, in 1926. Slavik Jablan names three people together: George Pólya (1924), Niggli (1926) and Andreas Speiser (1927). Fisher also writes that Evgraf Fedorov appears to have been the first to count, in 1891, the 17 ways a pattern can repeat across a whole surface, like wallpaper.

Borders are far older than the proof. Jablan gives as an example glide borders from the Neolithic art of Butmir, in today's Bosnia, from around 3500 BC.

Because this is a limit of geometry rather than of a tradition, the list is the same on every continent. Donald Crowe shows all seven families on the pottery of San Ildefonso pueblo in New Mexico, and writes that all seven also occur in Kuba and Benin art in Africa. Lynn Bodner looked for them on the walls of the Alhambra.

What differs from place to place is how often each family turns up. Crowe gives two examples from the collections he studied: in Anasazi pottery from the south-western United States, some 70% of the identifiable borders have only half-turns; on smoking pipes from Begho in Ghana, nearly 70% have every symmetry possible. For Romanian embroidery we found no study that has counted the families.

Colour splits the families further: if swapping red for black also counts as a symmetry, Crowe counts 17 types of two-colour border. The seven families above concern shape alone: where there is a stitch and where there is not.

The link below keeps your exact motif and border. Whoever opens it sees what you stitched and can carry on from there. The stitch chart downloads as an image, one square per cross-stitch, ready to work on cloth.

For a geometry lesson, from about age ten up:

1. The border hunt. Pupils look for borders around the school and on the way home: fences, floor tiles, rugs, the edge of an exercise book. For each one they draw one step and name its family. Who can find all seven?
2. The footprints. In snow, in sand, or with paper footprints along a corridor: hop on one foot, walk normally, sidle, jump with feet together. Conway named the families after tracks like these.
3. The proof. In “Why there are only seven”, pupils switch the moves on two at a time and note which move appears by itself. Of the 16 combinations, how many survive?
4. The stitching. With the downloaded chart, each pupil stitches one step of their border on aida cloth. Laid end to end, the borders show whether the rule held.

The page needs no account, sends nothing anywhere, and remembers the families you found only on your own device.