Patterns - Visual and Tiling Patterns

Definition of Patterns: A pattern is a sequence of shapes, colors, or numbers that repeat according to a specific rule. Visual patterns often involve geometric shapes like circles, squares, and triangles arranged in a predictable order, such as $Circle \rightarrow Triangle \rightarrow Square$ repeating continuously.

Tiling and Tessellations: Tiling involves covering a flat surface with repeating geometric shapes, called tiles, so that there are no gaps or overlaps. For example, a honeycomb pattern is a tiling of regular hexagons, while a standard chessboard is a tiling of black and white squares.

Rotational Turns: Shapes in a pattern can change orientation through turns. A $\frac{1}{4}$ turn is a $90^\circ$ rotation, a $\frac{1}{2}$ turn is a $180^\circ$ rotation, and a $\frac{3}{4}$ turn is a $270^\circ$ rotation. If a vertical rectangle makes a $\frac{1}{4}$ turn, it becomes a horizontal rectangle.

Symmetry and Reflection: Many visual patterns are based on symmetry, where one side of the pattern is a mirror image of the other. The line that divides the pattern into two identical halves is called the 'Line of Symmetry'. Imagine a butterfly wings pattern where the left side is perfectly reflected on the right.

Growing Patterns: Unlike repeating patterns, growing patterns increase or decrease in a systematic way. For example, a visual growing pattern might start with $1$ dot, then $3$ dots forming a small triangle, then $6$ dots forming a larger triangle, following the rule of adding one more dot to the base in each step.

Border Patterns: These are linear patterns that repeat along a straight line or the edge of an object, often used in architecture or textile design. A common border pattern might consist of a wave-like curve $(\sim)$ that repeats horizontally across the bottom of a page.

Patterns in Grids: Visual patterns can also be identified in grids where shapes move diagonally or skip squares. For instance, in a $3 \times 3$ grid, a colored square might move one position to the right in each successive frame, wrapping back to the start when it reaches the edge.