Geometry - Symmetry and Patterns

Line of Symmetry: A line of symmetry is an imaginary line that divides a shape or figure into two identical halves. If you fold the shape along this line, the two parts will overlap exactly. For example, a butterfly has a vertical line of symmetry down its body, making the left and right wings mirror images of each other.

Reflection Symmetry: This occurs when one half of an object is a mirror image of the other half. Imagine a mirror placed along the line of symmetry; the reflection in the mirror would complete the original shape perfectly. This is why it is often called mirror symmetry.

Symmetry in Regular Polygons: Regular polygons have lines of symmetry equal to the number of their sides. For instance, a square has $4$ lines of symmetry (one horizontal, one vertical, and two diagonal), while an equilateral triangle has $3$ lines of symmetry passing from each vertex to the midpoint of the opposite side.

Rotational Symmetry: A figure has rotational symmetry if it looks exactly the same after being rotated by some angle around its center point. For example, a ceiling fan with $3$ blades looks the same after a rotation of $120^\circ$. A square looks the same after a $\frac{1}{4}$ turn ($90^\circ$), a $\frac{1}{2}$ turn ($180^\circ$), and a $\frac{3}{4}$ turn ($270^\circ$).

Point Symmetry: This is a specific type of rotational symmetry where a figure looks the same when rotated $180^\circ$ (a half-turn) about its center. An example is the letter 'S' or 'Z', which looks identical even when turned upside down.

Shape and Number Patterns: A pattern is a sequence of shapes, colors, or numbers that repeats or changes according to a specific rule. In a 'Growing Pattern', the elements increase (e.g., $1, 3, 6, 10...$), while in a 'Repeating Pattern', a core set of elements appears again and again (e.g., Circle, Square, Circle, Square).

Tessellations (Tilings): A tessellation is a pattern created by repeating geometric shapes that cover a plane without any gaps or overlaps. Imagine a bathroom floor covered in hexagonal tiles; because the edges meet perfectly without leaving empty spaces, it forms a tessellation.