Does it Look the Same? - Rotational Symmetry

Rotational Symmetry: A shape or object has rotational symmetry if it looks exactly the same as its original position after being rotated around a fixed point by an angle less than $360^\circ$. Imagine a pinwheel spinning; as it turns, there are moments where it looks like it hasn't moved at all.

Center of Rotation: This is the fixed point about which the shape is rotated. For most geometric shapes like squares or circles, this point is the exact center of the figure. If you place a pencil tip at the center of a cardboard triangle and spin it, the pencil tip is the center of rotation.

Angle of Rotation: The minimum angle a shape must be turned to look identical to its original position is called the angle of rotation. For example, a square looks the same after a $\frac{1}{4}$ turn, which corresponds to an angle of $90^\circ$.

Order of Rotational Symmetry: This represents the number of times a figure looks exactly the same during a complete $360^\circ$ rotation. If a star looks the same 5 times during one full circle turn, its order of rotational symmetry is $5$.

Half-turn ($\frac{1}{2}$ turn): This refers to a rotation of $180^\circ$. Certain letters like 'H', 'I', 'N', 'S', 'X', and 'Z' look exactly the same after a half-turn. Visually, this is like flipping an object upside down.

Quarter-turn ($\frac{1}{4}$ turn): This refers to a rotation of $90^\circ$. Shapes with high symmetry, like a square or a plus sign ($+$), look identical after this small turn. A rectangle, however, does not look the same after a $90^\circ$ turn, as its long side would become vertical.

Regular Polygons: For regular polygons (shapes with all sides and angles equal), the order of rotational symmetry is always equal to the number of sides. A regular hexagon, which has 6 equal sides and looks like a honeycomb cell, has an order of symmetry of $6$.