Symmetry groups on a square grid

Colour some squares on a grid, then ask which moves leave the pattern looking exactly as it was — which turns work, and which folds. That question has remarkably few possible answers: exactly ten. What you will not find among them is a third of a turn. On a square grid that is not merely rare — it cannot happen at all.

This page explains what those ten possibilities are and why there are no others. It is the reference behind Symmetry Scene, an app that reports which of them a pattern has.

What counts as a symmetry

A move that leaves the pattern looking exactly as it did before is called a symmetry. That is a slightly unusual use of the word: in everyday speech symmetry is a property a shape has, but here it names the move itself.

Turn the grid a quarter of the way round: if every coloured square lands where a square of the same colour was, that quarter turn is a symmetry of the pattern. Fold the grid along a line: if the two halves match, that fold is a symmetry too.

The full collection of moves that work is the pattern's symmetry group. It is called a group because these moves behave consistently together: doing nothing always counts, doing two of them in a row always gives another one that works, and every move can be undone.

That last property is the interesting one, because it means you cannot pick symmetries freely. Having some forces you to have others — which is exactly why there are only ten possibilities rather than dozens.

The eight moves a square allows

Those ten answers are assembled from a shorter list. Before asking what a pattern does, ask what the grid itself permits — and a square can be moved onto itself in exactly eight ways:

• Four turns — leaving it alone, a quarter turn, a half turn, and a three-quarter turn.
• Four folds — along the four mirror lines below.

Mathematicians call this collection D₄. A plain square has all eight of them — with nothing coloured in, there is nothing to spoil them.

Colour some squares and that usually stops being true: a pattern keeps some of the eight and loses the rest. The ten are the collections it can be left with. So the two numbers belong to different things — eight is what a plain square has; ten is how many different ways a pattern can turn out. No pattern has ten symmetries. The most any pattern can have is all eight, and most have far fewer.

Why a square pattern can never turn by a third

Turn a triangle by a third of a full turn and it looks exactly as it did. Try the same thing on a square grid and it never works — not for any pattern, however you colour it. Why not?

Because the grid gets in the way. Turning by 120° does not send squares onto squares — the rows and columns sit at right angles to each other, so after a third of a turn the cells no longer line up with cells at all. There is nowhere for the colours to land.

So on a square grid the only possible amounts of turning symmetry are one, two or four: no turn at all, a half turn, or quarter turns. Three is not merely rare, it is unavailable. Order 5 and order 6 are out for the same reason.

This is not a fact about symmetry in general — it is a fact about square grids. Change the grid and the menu changes with it, as the last section shows.

Two families of names

The ten groups have short names, built from two families:

• C groups have turns and no mirror lines at all. C₂ is the half turn; C₄ is the quarter turns.
• D groups have mirror lines, and the number counts them: D₁ has one, D₂ has two, D₄ has four.

Only D₂ and D₄ come with turns as well. D₁ is a single fold and nothing else, which is worth noticing: a pattern can have a mirror line while having no turning symmetry whatsoever.

C₁ — no turns, no folds — is the name for a pattern with no symmetry at all.

The ten possibilities

Every pattern you can draw on a square grid has exactly one of these. Coloured squares show the pattern; yellow lines show its mirror lines.

Four of those are D₁ and two are D₂. As mathematical objects each set of those is the same shape — they differ only in which line you fold along. Counting the variants separately is what gives ten.

Why only ten?

Because symmetries combine, and the combinations are forced:

• A quarter turn plus any single mirror line gives you all four. There is no pattern with quarter-turn symmetry and only one or two folds.
• Any two different mirror lines give you the turn between them. Across and down together force a half turn. A straight line and a diagonal force a quarter turn — and therefore, by the rule above, everything.
• A single mirror line means no turning symmetry at all. There is no "half turn plus one fold" group, because a half turn with one fold hands you the second fold automatically.
• The two-mirror groups come in perpendicular pairs only — across-and-down, or the two diagonals. A mixture is impossible without dragging in all of D₄.

A grid coloured entirely in one colour is a special case: it satisfies all eight symmetries, so it counts as D₄. True, but not very interesting — which is why the app flags it rather than congratulating you.

How they fit together

The ten nest inside one another. Every group sits above C₁ and below D₄; adding a symmetry moves a pattern up the diagram.

C₁ → D₁ → D₂ → D₄, and separately C₁ → C₂ → C₄ → D₄. A pinwheel (C₄) and a butterfly (D₁) have nothing in common at all except doing nothing: one is all turning, the other all folding.

Beyond the square

The impossibility of a third of a turn is a property of the grid, not of symmetry. Change the shape of the cells and the board, and a different menu appears:

A rectangle is the interesting one in reverse: it removes symmetries. Stretch a square into a rectangle and the quarter turn and both diagonal folds stop working, because the shape no longer lands on itself. A pattern's symmetry depends on the canvas as much as on the pattern.

On a hexagonal grid a third of a turn works perfectly well, and so does a sixth. There are also two different families of mirror line, one through opposite corners and one through opposite edges — a distinction with no equivalent on a square.

Try it

Colour some squares and press Check symmetry. It will tell you which of the ten you have made, draw the mirror lines onto your pattern, and — when you are close to one you have not quite got — show you which squares are stopping it.

Open Symmetry Scene

Free, no sign-up, no tracking. Works on a tablet.