Rotation symmetry and order

A shape has rotation symmetry if you can turn it about a point (the centre) by some angle less than a full turn and it looks exactly the same as before. Think of a pinwheel, a bicycle wheel with equally spaced spokes, or a starfish.

The order of the rotation symmetry is the number of different positions in which the shape fits its own outline during one full turn, counting the starting position. If the smallest turn that works is θ, then

order n = 360° ÷ θ and θ = 360° ÷ n.

A shape with order 1 has no rotation symmetry: it only matches itself after a full turn, which is no turn at all. The turn by 180° has its own name, the half-turn, and it is the reason a parallelogram, and the letters N, S and Z all have a kind of symmetry even though they have no mirror line.

Rotation without reflection. Most of the shapes above also have mirror lines, but not all of them do. A three-bladed pinwheel (a propeller with curved blades) can be turned by 120° and fit itself, so its order is 3. But if you hold it up to a mirror, the reflection spins the other way, so the reflection is a different object. The pinwheel has no axes of symmetry. The same is true of the triskelion, the three-legged spiral, and of the recycling symbol’s arrow loop.

This gives us two counts for every shape, and they do not have to agree:

• the number of axes (mirror lines), and
• the order of rotation symmetry.

For regular polygons they match (a hexagon has 6 axes and order 6). For a pinwheel they do not (0 axes, order 3). For a rectangle they happen to match (2 and 2). For a generic parallelogram they do not (0 and 2). Mathematicians keep the two apart: a shape with order n and no mirror lines belongs to the family Cₙ, and one with order n plus n mirror lines belongs to Dₙ (the dihedral family). A square is D₄, an equilateral triangle is D₃, a pinwheel with 3 blades is C₃.