Symmetry - Order of Rotational Symmetry

Rotational symmetry occurs when an object looks exactly like its original position after being rotated about a fixed point by an angle less than $360^{\circ}$. For example, if you rotate a square by $90^{\circ}$, it will look identical to how it started.

The fixed point around which the object is rotated is called the Center of Rotation. In a rectangle, the center of rotation is the point where the two diagonals intersect.

The minimum angle through which a figure is turned to coincide with its original position is known as the Angle of Rotation. A full turn is $360^{\circ}$, a half-turn is $180^{\circ}$, and a quarter-turn is $90^{\circ}$.

The Order of Rotational Symmetry is the number of times a figure fits onto itself during a complete rotation of $360^{\circ}$. If a shape looks the same at four different positions during a full turn, its order is 4.

Every regular polygon with $n$ sides has an order of rotational symmetry equal to $n$. For instance, a regular pentagon has an order of 5, meaning it looks the same at five positions: $72^{\circ}$, $144^{\circ}$, $216^{\circ}$, $288^{\circ}$, and $360^{\circ}$.

A circle is a special case with infinite rotational symmetry. Because every point on the circumference is equidistant from the center, rotating a circle by any angle around its center results in a shape that looks exactly the same.

It is important to note that every object has rotational symmetry of order 1, as it returns to its original position after a full $360^{\circ}$ turn. However, in geometry, we usually say a figure has rotational symmetry only if its order is 2 or more.