Patterns formed by repeating tiles all over a flat surface are called tessellations or tilings. A tessellation is a pattern that covers an entire flat surface using one or more geometric shapes (called tiles) without any gaps or overlaps

Key Characteristics of Tessellations:

• The shapes fit together perfectly to cover the plane.
• The pattern repeats according to certain rules.
• The interior angles of the shapes meeting at each vertex sum to 360 degrees to avoid gaps

• Tessellations can be made from regular polygons such as equilateral triangles, squares, and regular hexagons, which are the only three shapes that can form regular tessellations by themselves

Types of Tessellations:

• Regular Tessellations: Made using only one type of regular polygon repeated across the surface (e.g., squares, equilateral triangles, hexagons)

• Semi-regular Tessellations: Use two or more types of regular polygons arranged so that the pattern around each vertex is the same

• Monohedral Tessellations: Made from one shape that may be rotated or flipped but is congruent throughout

• Aperiodic Tessellations: Patterns that do not repeat periodically, such as Penrose tilings, which use a small set of tiles that cannot form a repeating pattern

Applications and Examples:

• Tessellations are found in natural structures like honeycombs.
• They are used in art and architecture, notably in Islamic geometric designs and the works of M.C. Escher.
• Practical uses include floor and wall tiling, pavements, and decorative patterns

In summary, tessellations are mathematically rich, repeating patterns formed by tiles that cover a flat surface completely without gaps or overlaps, following specific geometric rules