Symmetry - Line Symmetry and Rotational Symmetry

Line Symmetry: A figure has line symmetry if there is a line about which the figure can be folded so that the two halves coincide exactly. This line is known as the axis of symmetry or line of symmetry. For example, a human face or the letter 'A' has one vertical line of symmetry, whereas the letter 'H' has both vertical and horizontal lines of symmetry.

Regular Polygons and Symmetry: A regular polygon has all its sides and angles equal. Every regular polygon of $n$ sides has exactly $n$ lines of symmetry. For instance, a regular pentagon has $5$ lines of symmetry, each passing through a vertex and the midpoint of the opposite side, while a square has $4$ lines of symmetry (two diagonals and two lines connecting the midpoints of opposite sides).

Rotational Symmetry: If a figure, after being rotated about a fixed point through a certain angle, looks exactly the same as the original figure, it is said to have rotational symmetry. The fixed point is called the 'center of rotation'. A classic visual example is a ceiling fan or a windmill, which looks the same after a partial turn.

Angle of Rotation: The minimum angle through which an object must be rotated to look identical to its initial position is called the angle of rotation. For example, a square looks the same after a $90^\circ$ rotation, so its angle of rotation is $90^\circ$. A full turn is always $360^\circ$.

Order of Rotational Symmetry: This is the number of times a figure fits onto itself during a full rotation of $360^\circ$. For a regular hexagon, the figure looks the same at $60^\circ, 120^\circ, 180^\circ, 240^\circ, 300^\circ,$ and $360^\circ$. Therefore, its order of rotational symmetry is $6$.

Point Symmetry: A figure is said to have point symmetry (or symmetry about the center) if it looks the same when rotated by $180^\circ$ around its center. This is a special case of rotational symmetry with an order of $2$. Examples include the letter 'S', the letter 'Z', and the center of a parallelogram.

Combination of Symmetries: Some figures possess both line symmetry and rotational symmetry. For example, a circle is the most symmetrical figure; it has infinitely many lines of symmetry (any diameter) and an infinite order of rotational symmetry as it looks the same after rotation through any angle about its center.