Tessellation

Stone-Cone Temple mosaics, Pergamon Museum

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Roman geometric mosaic

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Semi-regular-floor-3464

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Tassellatura alhambra

P5-type15-chiral coloring

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2005-06-25 Tiles together

Tessellation refers to the process of creating a two-dimensional plane using the repetition of a geometric shape with no overlaps and no gaps. The concept of tessellation is a fundamental principle in both mathematics and art, particularly in the study of geometry and tiling. Tessellations can be found in nature, in the arts, and in man-made structures, showcasing the intrinsic beauty of repeated patterns.

History

The history of tessellations dates back to ancient civilizations, including the Romans and Islamic cultures, where they were used in decorating floors and walls. The Alhambra in Spain is a famous example, showcasing intricate tessellated patterns that have inspired mathematicians and artists alike. In the 20th century, Dutch artist M.C. Escher brought tessellations to the forefront of artistic exploration, creating complex and imaginative works that play with space, perspective, and the infinite.

Types of Tessellations

Tessellations can be categorized into three main types based on the shapes used and the methods of creation:

1. Regular Tessellations: These are formed by repeating a single type of regular polygon, such as equilateral triangles, squares, or hexagons. There are only three regular tessellations in the Euclidean plane.

2. Semi-regular Tessellations: Also known as Archimedean tessellations, these are created by two or more types of regular polygons arranged in a repeating pattern. Each vertex in a semi-regular tessellation has the same pattern of polygons.

3. Irregular Tessellations: These tessellations do not have regular polygons and can include shapes that are not considered polygons. This category includes non-periodic tessellations that do not repeat in a predictable manner, such as those found in Penrose tilings.

Mathematical Principles

The study of tessellations intersects with several areas of mathematics including topology, group theory, and crystallography. Mathematicians study the properties of tessellations, such as symmetry, angles, and the arrangement of shapes, to explore concepts of infinity, the nature of the universe, and the mathematical principles underlying the visible world.

Applications

Tessellations have practical applications in various fields:

- In architecture and design, tessellations are used for aesthetic and functional purposes, such as floor tiling and the design of structured surfaces. - In computer graphics, tessellation algorithms help in rendering three-dimensional objects more efficiently by breaking down surfaces into simpler geometric shapes. - In quilt making and other crafts, tessellations provide a framework for creative expression through the arrangement of colors and patterns.

See Also

• M.C. Escher
• Geometry
• Tiling
• Penrose Tiling