Line and Rotational Symmetry

A shape has symmetry when it looks the same after a certain move. Two kinds matter here: folding it over a line, and turning it about a point.

Theorem Line symmetry

A shape has line symmetry if folding it over a line makes the two halves match exactly. That line is an axis of symmetry.

The more regular the shape, the more axes of symmetry it has — a circle has infinitely many.

Theorem Rotational symmetry

A shape has rotational symmetry if it lands on itself after turning by less than a full circle. The order is how many times it matches in one full turn.

The smallest turn that maps the shape onto itself is:

\frac{360^\circ}{n}

where n is the order. A higher order means a smaller angle.

Reference Common shapes

Shape	Lines	Order	Angle
Square	4	4	90°
Rectangle	2	2	180°
Equilateral triangle	3	3	120°
Five-point star	5	5	72°
Circle	∞	∞	any

Example Three worked cases

Axes of an isosceles triangle

Only the fold through the apex and the middle of the base matches the two halves.

⟹ 1 axis of symmetry.

Order of a five-point star

Turning it by

\tfrac{360^\circ}{5} = 72^\circ

lands it on itself, and this happens 5 times in a full turn.

⟹ order 5.

A cylinder in 3-D

Spun about its central axis, a cylinder looks the same at every angle.

⟹ infinite rotational symmetry about that axis.

Summary

Line symmetry: folding over an axis matches the two halves; the number of axes depends on the shape.
Rotational symmetry: the order is how many times a shape matches itself in a 360° turn.
The smallest turn is $\tfrac{360^\circ}{n}$ , where n is the order.
A higher order (or more axes) means a more symmetric shape — the circle is the limit, with infinitely many.