Tessellation
From The Art and Popular Culture Encyclopedia
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A tessellation of a flat surface is the tiling of a plane using one or more geometric shapes, called tiles, with no overlaps and no gaps. In mathematics, tessellations can be generalized to higher dimensions and a variety of geometries.
A periodic tiling has a repeating pattern. Some special kinds include regular tilings with regular polygonal tiles all of the same shape, and semiregular tilings with regular tiles of more than one shape and with every corner identically arranged. The patterns formed by periodic tilings can be categorized into 17 wallpaper groups. A tiling that lacks a repeating pattern is called "non-periodic". An aperiodic tiling uses a small set of tile shapes that cannot form a repeating pattern. In the geometry of higher dimensions, a space-filling or honeycomb is also called a tessellation of space.
A real physical tessellation is a tiling made of materials such as cemented ceramic squares or hexagons. Such tilings may be decorative patterns, or may have functions such as providing durable and water-resistant pavement, floor or wall coverings. Historically, tessellations were used in Ancient Rome and in Islamic art such as in the decorative geometric tiling of the Alhambra palace. In the twentieth century, the work of M. C. Escher often made use of tessellations, both in ordinary Euclidean geometry and in hyperbolic geometry, for artistic effect. Tessellations are sometimes employed for decorative effect in quilting. Tessellations form a class of patterns in nature, for example in the arrays of hexagonal cells found in honeycombs.
See also
- Types of tessellation
- Aperiodic tiling
- List of regular polytopes
- List of uniform tilings
- Pinwheel tiling
- Tilings of regular polygons
- Uniform tessellation
- Voronoi tessellation
- Mathematics
- Coxeter groups – algebraic groups that can be used to find tessellations
- Girih tiles
- Triangulation (geometry)
- Uniform tiling
- Uniform tilings in hyperbolic plane
- Wallpaper group – seventeen types of two-dimensional repetitive patterns
- Wang tiles
- Related topics
- Jigsaw puzzle
- Mathematics and fiber arts
- Nikolas Schiller – artist using tessellations of aerial photographs
- Patterns in nature
- Polyiamond and Polyomino — figures consisting of regular triangles and squares respectively, often appearing in tiling puzzles
- Quilt block designs and quilt blocks
- Tiling puzzle
- Trianglepoint – needlepoint with polyiamonds (equilateral triangles)