dihedral symmetry

Think of a regular polygon — an equilateral triangle, a square, a regular pentagon. You can spin it onto itself by partial turns, and you can also flip it across various mirror lines onto itself. A figure has dihedral symmetry when its symmetries include both kinds: a family of rotations and an equal-sized family of reflections. It is the richest, most balanced kind of finite symmetry a flat figure can have.

For a figure with n-fold dihedral symmetry, written D_n, there are exactly 2n symmetries: n rotations (by 0, 360/n, 2(360/n), ... degrees about the centre) and n reflections (across n evenly spaced mirror lines through the centre). A regular n-gon has precisely this, the group D_n of order 2n. So an equilateral triangle has D_3 with 6 symmetries (3 rotations, 3 reflections), a square has D_4 with 8, a regular hexagon has D_6 with 12. Contrast this with cyclic symmetry, written C_n, where a figure has only the n rotations and no reflections at all — a pinwheel or a swastika-like motif spins onto itself but has no mirror line, so it possesses C_n but not D_n. Dihedral is rotation-plus-mirror; cyclic is rotation-only.

Dihedral symmetry is everywhere shapes are 'regular and reversible': snowflakes (close to D_6), starfish (D_5), many flowers, hubcaps, and company logos designed to look balanced. The honest subtlety is the relationship between the two families: composing two of the n reflections produces a rotation, so the reflections are not independent extras — they interlock with the rotations, and a single reflection together with a single smallest rotation already generates all 2n symmetries. Note also a quirk of naming: some books write the same group as D_{2n} (counting elements rather than the n-fold axis), so always check whether the subscript counts the rotations or the total.