Shape and Space - Symmetry and Rotations

Line Symmetry (Reflection Symmetry): A shape has line symmetry if a line can be drawn through it so that one half is a mirror image of the other. For example, a drawing of a heart has one vertical line of symmetry down the center; if you folded it along that line, the two halves would match perfectly.

Rotational Symmetry: This occurs when a shape looks exactly the same after being rotated around its center point by an angle less than $360^{\circ}$. Imagine a three-blade ceiling fan: as it spins, it reaches three distinct positions where it looks identical to its starting position.

Order of Rotational Symmetry: This is the number of times a shape fits onto itself during one full $360^{\circ}$ turn. For instance, a square looks the same at $90^{\circ}$, $180^{\circ}$, $270^{\circ}$, and $360^{\circ}$, so its order of rotational symmetry is $4$.

Center of Rotation: The fixed point around which a shape is turned. During the rotation, every point of the shape moves in a circular path around this central point. Think of a spinner in a board game; the pin in the middle is the center of rotation.

Angle of Rotation: This is the measurement of the turn in degrees. A quarter turn is $90^{\circ}$, a half turn is $180^{\circ}$, and a full turn is $360^{\circ}$. If a shape has rotational symmetry of order $n$, its smallest angle of rotation is calculated as $\frac{360^{\circ}}{n}$.

Symmetry in Regular Polygons: Regular polygons have all sides and angles equal. A key rule is that a regular polygon with $n$ sides will always have exactly $n$ lines of symmetry and an order of rotational symmetry of $n$. A regular hexagon (6 sides) thus has 6 lines of symmetry and an order of 6.

Direction of Rotation: Rotation is defined by its direction. 'Clockwise' ($CW$) follows the direction of clock hands ($12 \rightarrow 3 \rightarrow 6$), while 'Counter-clockwise' ($CCW$) goes in the opposite direction ($12 \rightarrow 9 \rightarrow 6$).