Combining moves

Do one symmetry move and then another and you get a third move. This is the key idea that turns a list of moves into an elegant theory. The most important example is two reflections one after the other.

• Two mirror lines that cross. Reflect a shape in line m₁, then reflect the result in line m₂. If the lines meet at a point O and the angle between them is α, the result is a rotation about `O` by `2α`. For example, mirrors at 30° to each other give a rotation of 60°. Mirrors at right angles (90°) give a rotation of 180°, the half-turn.
• Two mirror lines that are parallel. If the lines are a distance d apart, the result is a translation by `2d` perpendicular to the lines. Mirrors 3 cm apart move a shape 6 cm.

Why twice the angle or twice the distance? Reflect once and the shape is as far behind the first mirror as it was in front of it. Reflect in the second mirror and it jumps again by the same rule. The two jumps add up to double the gap between the mirrors.

If you want to work this out with numbers, a reflection in a line through the origin at angle ψ is given by the matrix [[cos 2ψ, sin 2ψ], [sin 2ψ, −cos 2ψ]]. Multiply two of them and you get a rotation matrix for the angle 2(ψ₂ − ψ₁). The Vector Lab explains how matrices act on shapes.

The order of the moves can matter. Take an equilateral triangle with a numbered position at each corner: 1 at the top, 2 bottom left, 3 bottom right. Let r be the rotation by 120° that sends the tile at 1 to 2, the tile at 2 to 3 and the tile at 3 to 1. Let s be the reflection in the vertical axis through position 1: it swaps positions 2 and 3 and leaves 1 alone.

• `r` then `s`: the tile at position 1 goes to 2, and then to 3.
• `s` then `r`: the tile at position 1 stays at 1, and then goes to 2.

The results differ, so r then s is not the same move as s then r. Doing moves in a different order can change the result. (There are special cases where it does not matter: two half-turns about different points, or two reflections in perpendicular lines through the same point, give the same result in either order. In general, the order matters.)

The idea of a group. The symmetries of a shape form a tidy system called a group: (1) doing one symmetry and then another is again a symmetry (the set is closed), (2) doing nothing is a symmetry (the identity), and (3) every symmetry can be undone by another symmetry (its inverse). An equilateral triangle has exactly 6 symmetries: 3 rotations (0°, 120°, 240°) and 3 reflections. A square has 8. A regular n-gon has 2n. Group theory, born from questions like these, is the mathematics of symmetry.