What are patterns formed by repeating tiles all over a flat surface?

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what are patterns formed by repeating tiles all over a flat surface?

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What are patterns formed by repeating tiles all over a flat surface?

Answer: Patterns formed by repeating tiles all over a flat surface are known as tessellations. Tessellations are arrangements of geometric shapes that cover a plane without any gaps or overlaps. These patterns can be found in nature, art, and various fields of science and mathematics.

Types of Tessellations:

  1. Regular Tessellations:

    • These are made up of congruent regular polygons, where each vertex configuration is identical across the entire tessellation. There are only three regular tessellations:

      • Equilateral Triangle Tessellation:

        Each vertex is the meeting point of six triangles.

        Each vertex is the meeting point of four squares.

      • Regular Hexagon Tessellation:

        Each vertex is the meeting point of three hexagons.

  2. Semi-regular Tessellations:

    • These tessellations are formed by two or more different regular polygons where each vertex configuration is the same. There are exactly eight semi-regular tessellations.

      Examples include:

      • The combination of squares and equilateral triangles.
      • The combination of hexagons and equilateral triangles.

      Each semi-regular tessellation has a repeating pattern of vertices.

  3. Non-regular Tessellations:

    • These can be formed by shapes that are not regular polygons. Non-regular tessellations include a wide variety of shapes, including irregular polygons and curves.
    • Examples include tiling patterns found in art and architecture such as those by the artist M.C. Escher.

Mathematical Foundations Behind Tessellations:

  1. Tilings and Polyominoes:

    • Polyominoes are plane geometric figures formed by joining one or more equal squares edge to edge. These can tessellate on a plane in various configurations.

  2. Symmetry Groups:

    • Tessellations can be categorized based on their symmetry properties. The study of these symmetries involves understanding the different wallpaper groups. There are 17 possible wallpaper groups, which categorize the different ways patterns can be symmetrically repeated on a flat surface.

Applications of Tessellations:

  1. Art and Architecture:

    • Tessellations can be seen in various artistic designs, mosaics, and tiling systems in architecture.

  2. Nature:

    • Naturally occurring tessellations can be observed in the honeycomb structures made by bees or in the patterns on a turtle’s shell.

  3. Science and Engineering:

    • Tessellations are used in materials science and engineering, particularly in the study of crystal structures.

  4. Computer Graphics:

    • Tessellation algorithms are integral in computer graphics for creating complex, repeating patterns efficiently.

Final Answer:

Patterns formed by repeating tiles all over a flat surface are called tessellations. They can be regular, semi-regular, or non-regular, and are used in various disciplines including art, nature, and sciences.

This comprehensive guide to tessellations not only highlights the mathematical classifications but also their wide-ranging applications, making it easy for students to understand and appreciate the beauty and utility of these patterns.

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