Symmetry

A butterfly, a letter, a road sign: many shapes feel balanced because one half mirrors the other. This property is called symmetry. Once you spot it, you understand shapes faster and can even complete figures yourself.

Axis symmetry: the mirror

A figure is axis-symmetric if there is a line along which you could fold it so that both halves match exactly. This line is called the axis of reflection. Every point has a mirror point on the other side: it sits straight across, perpendicular to the axis, and exactly the same distance from it. A square has four such axes, a circle even infinitely many.

Reflecting in the coordinate system

In the coordinate system reflections are easy to compute. If you reflect across the y-axis, the sign of the x-coordinate flips: $(3, 2)$ becomes $(-3, 2)$. If you reflect across the x-axis, the sign of the y-coordinate flips: $(3, 2)$ becomes $(3, -2)$. The other number stays unchanged.

Point symmetry: the rotation

In point symmetry there is no mirror but a centre of rotation. A figure is point-symmetric if it looks exactly the same again after a half turn about that centre, that is after 180°. The letter S and the diamond on a playing card are point-symmetric. In the coordinate system a point reflection through the origin flips both signs: $(6, 3)$ becomes $(-6, -3)$.

Symmetry in everyday life

Symmetry is everywhere: in faces, flowers, buildings and logos. It feels ordered and pleasant to us, which is why designers use it on purpose. Some figures have both: a rectangle has two axes of reflection and point symmetry as well. Others have only one kind: an isosceles triangle is axis-symmetric but not point-symmetric. And many shapes have none at all, like the letter R or a footprint.