Lines of symmetry: the mirror test

A shape has reflection symmetry (also called mirror symmetry or line symmetry) if you can draw a line through it so that one side is the mirror image of the other. The line is the line of symmetry, or axis. The paper test: fold the shape along the line. If the two halves land exactly on top of each other, the line is an axis of symmetry.

Everyday shapes are full of axes. Take capital letters in a plain sans-serif font:

• A, M, T, U, V, W, Y have one vertical axis.
• B, C, D, E, K have one horizontal axis.
• H, I, X and an oval O have two axes (an O drawn as a perfect circle has infinitely many).
• F, G, J, L, P, R have none. (Fonts differ, so check your own.)

Faces, butterflies and most animals are only roughly symmetrical: a real face is never an exact mirror image, which is why a photo of someone next to its own mirror image looks like two slightly different people.

Counting axes of polygons. A regular polygon (all sides and all angles equal) with n sides has exactly n axes of symmetry:

For odd n (triangle, pentagon) every axis joins a corner to the middle of the opposite side. For even n (square, hexagon) half the axes join opposite corners and half join the midpoints of opposite sides.

Shapes that are not regular lose axes. A rectangle that is not a square has 2 axes (one parallel to each pair of sides, through the midpoints). Its diagonals are not axes: fold a rectangle along a diagonal and the corners do not meet. A rhombus that is not a square also has 2 axes, and these are its diagonals. An isosceles triangle has 1. A scalene triangle has 0. And a parallelogram with no right angles and unequal sides has 0.