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Symmetry, in counted-thread stitches

A border can repeat in only seven ways. Try to find an eighth.

Every pattern that repeats along a band, whether the hem of a blouse, the frieze of a temple or the rim of a plate, obeys, if you look only at its shape, one of the seven rules below. Not eight. Tap any band to stitch into it: your motif appears on all seven, carried along by each rule.

The motif, one repeat: 16 stitches wide, 10 high. Tap or drag across the squares to stitch or unpick. The dotted line is the middle of the band.

1. Hop p1The motif just moves along, always the same way up.
2. Step p11gEach copy is flipped top to bottom and moved on by half a period, like footprints in snow.
3. Sidle p1m1Upright mirrors: the motif and its reflection stand face to face.
4. Spinning hop p2Every other copy stands on its head, as if given half a turn.
5. Spinning sidle p2mgUpright mirrors with a half-turn between each pair, which brings in the step as well.
6. Jump p11mOne mirror along the middle: the bottom half repeats the top.
7. Spinning jump p2mmMirrors both ways, and so half-turns too: the most symmetric of the seven.

You can stitch straight onto the bands too: whatever you add to one appears on all seven, each by its own rule. The names are the mathematician John Conway’s, who pictured them as the tracks left by someone walking, hopping or jumping; in brackets is the crystallographers’ notation.

Why is there no eighth kind of border?

Because a band doesn’t let you do much with it. Any move that lays it exactly back on itself must carry the middle line back onto the middle line. That leaves four possibilities: leave the band as it is, flip it top to bottom, turn it left to right, or do both at once, which is half a turn. Each can be combined with a slide along the band.

That makes four switches, and so 16 possible combinations. Turn them on in any order and watch what happens to the band.

Bands found: 0 of 7

  • Hop
  • Step
  • Sidle
  • Spinning hop
  • Spinning sidle
  • Jump
  • Spinning jump

Some switches turn others on. Two mirrors that cross make a half-turn at the point where they meet. The mirror across, added to the step, also makes a half-turn, this time halfway between the mirrors. The mirror along, added to the step, shifts the pattern by half a period, so you really get the same jump, only denser. And the mirror across next to a half-turn forces a choice: if the centre of the turn sits on the mirror, the mirror along appears; if it sits halfway between two mirrors, the step appears.

So the 16 combinations leave seven different bands. This is a mathematical proof, true before any museum is visited: no repeating border, however old or far away, can fall outside the seven, as long as you treat it as a band without end and leave colour aside. Real borders end somewhere and have their slips of execution, and when you classify it you name the rule the pattern follows, not the wanderings of a thread.

How many ways are there to repeat a pattern on a wall? Seventeen

On a wall, a floor or a tablecloth, the pattern repeats in two directions at once. New moves appear: turns by a third, a quarter or a sixth, mirrors on the diagonal. Yet the list closes quickly: there are exactly 17 ways. A fifth of a turn never appears in any repeating pattern: a five-pointed star can sit on a tile, but the whole wall will never land on itself after a fifth of a turn.

no turns
½ turns
⅓ turns
¼ turns
⅙ turns

Tap the wall to add stitches. The kinds with turns of a third or a sixth are laid on hexagons: squares cannot hold them.

The full list of 17 was drawn up in 1891 by the Russian crystallographer Evgraf Fedorov, who was studying the geometry of crystals; the German Leonhard Sohncke had found the first 13 in 1874. In 1924 the mathematician George Pólya illustrated all 17, each with a tiling. The seven borders were sorted out at about the same time, in the 1920s, with no single discoverer to claim them.

Are all 17 in the Alhambra?

It is often said that the Alhambra palace in Granada, decorated in the 13th and 14th centuries, contains all 17 ways of repeating a pattern on a wall. The monument’s own trust says so on its website. The people who have counted do not agree.

How many kinds each found in the Alhambra
WhoWhenHow manyWhat they say
Edith Müller1944, doctoral thesis, Zürich12Documented 12 kinds. From her remark that small changes would have given two more, others wrongly understood that she had found 14.
José María Montesinos1987, book (Springer)17Offers photographs for all 17. A specialist review said the question was settled “once and for all”.
Rafael Pérez-Gómez1987, article17The article’s title: “The four regular mosaics missing in the Alhambra”; he searched for the ones not yet found.
Branko Grünbaumseveral days in 1983; published 200613Found Müller’s 12 and one more. Montesinos, he writes, counts colours when it suits him, uses a single tile or the back of a chair, and shows patterns so worn their design can no longer be seen.
Patronato de la Alhambraofficial website17Says the Alhambra is the only ancient monument where all 17 are present, without saying by which rules it counted.

The quarrel is about rules: which part of the decoration you count, how large and how well preserved it must be. The knottiest is the question of colour: do you count it or not? Look at the pattern below. With the colours it is one kind; without them, another.

Branko Grünbaum makes a remark worth keeping: symmetry groups meant nothing to the craftsmen who decorated the Alhambra. They were making beautiful patterns, and sometimes, he writes, they seem to have broken the symmetry on purpose, so it would not be monotonous.

M. C. Escher visited the Alhambra in 1922 and again in 1936 and copied patterns from its walls. Late in 1937 his brother, a geologist, sent him a reading list; on it was Pólya’s paper, which Escher copied out by hand, all 17 tilings included. Many of the tilings of birds, fish and lizards he drew afterwards are walls of the same 17 kinds.

What kind of border is this?

You can classify any border with three questions, asked in this order:

  1. Does it have upright mirrors, across the band?
    • yes → Does it have a mirror along the middle of the band?
      • yes → Spinning jump p2mm
      • no → Does it have half-turns (does it look the same upside down)?
        • yes → Spinning sidle p2mg
        • no → Sidle p1m1
    • no → Does it have a mirror along the middle of the band?
      • yes → Jump p11m
      • no → Does it have a step: a copy flipped top to bottom and moved half a period on?
        • yes → Step p11g
        • no → Does it have half-turns (does it look the same upside down)?
          • yes → Spinning hop p2
          • no → Hop p1

Below are real borders, from museums and buildings. Colour doesn’t count: look only at the shapes. Choose the kind, then open the explanation.

A red woven band with X-shaped stars in several colours and sawtooth edges.

The border of a rug from Bezdead, Romania

The explanation

Spinning jump p2mm. Each X-star mirrors both left to right and top to bottom, and the top teeth mirror the bottom ones. Two crossing mirrors bring half-turns too. The stars change colour one to the next and the weaver did not work to the millimetre, but the shape follows this rule.

Rug from Bezdead, Dâmbovița. Photo: Felix Pirvan, CC BY-SA 4.0, cropped.

A twisted rope carved in wood, its strands all slanting the same way.

The rope carved on the church gate at Breb, Maramureș

The explanation

Spinning hop p2. All the strands slant the same way, so there is no mirror: in a mirror the slant would reverse. Turned upside down, though, the rope looks the same. Light from above makes it look less symmetric than it is; look at the carved shape.

Church gate at Breb, Maramureș. Photo: DiaRo, CC BY-SA 4.0, cropped and rotated.

A grey zigzag between two straight lines, with triangles hatched in red and green.

The border of a painted tomb ceiling at Thebes, Egypt

The explanation

Spinning sidle p2mg. The zigzag has upright mirrors through every point and half-turns at the middle of every side. Flipped top to bottom and moved on half a period, it lands on itself: that is the step. It has no mirror along the middle. The hatching changes direction from one triangle to the next and would break the mirrors; here we classify the outline, the zigzag.

Copy of the ceiling in the tomb of Qenamun (TT 162), Thebes, Egypt. The Metropolitan Museum of Art, CC0, cropped.

How can a geometry class use this page?

  1. Each pupil stitches a motif that doesn’t resemble itself: not in a mirror, not upside down.
  2. Pupils send the link to the person next to them, who must say, without reading the names, which rule made each band.
  3. At the board, the class tries to switch the switches in an order that gives an eighth band. Nobody manages it, and the reason is the lesson.
  4. Homework: a border photographed on a blouse, a gate, a plate or a fence, classified with the three questions.

Sources and method

A border is treated here as a band without end that repeats exactly, in a single colour. In that form the theorem says there are seven kinds; for walls, 17. If symmetries that swap the colours are counted too, you move to another classification, colour symmetry, with more kinds; the patterns in the gallery are classified by shape, with colours set aside, all by the same rule.

The bands and walls on this page are computed, not drawn by hand: each rule is written as a list of moves on the grid, and a second, independent program searches the resulting pattern for all its symmetries and names it. The page shows the name found by the second program, so if your motif is symmetric itself, you see the extra symmetry too.

The classifications in the gallery are ours and can be checked by eye with the three questions. Where the maker left irregularities, we classified the rule the rest of the band follows.

The step names (hop, step, jump) are John Conway’s; the notation in brackets is the crystallographers’.

Also about shapes and mirrors, on this site