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A tessellation is a tiled pattern created by repeating a shape over and over again, with no overlaps or gaps. A classic example of a tessellation is a tile floor in which the floor is covered in square tiles. Tessellations appear in numerous works of art in addition to architecture, and they are also of mathematical interest. These patterns crop up in a variety of settings, and once people start looking for tessellations, they tend to start seeing them everywhere, including in nature.
Tessellations are basically mosaic patterns which are made with a repeating polygonal shape. They can be used to tile a flat plane, or a sculpted surface. In all cases, the tessellation can theoretically be repeated infinitely, with the pattern remaining consistent and the shapes retaining their positions in relation to each other. Certain shapes will not tessellate, or cannot tessellate infinitely because the pattern eventually reaches a point where shapes start to interlock or gaps form.
In regular tessellations, also known as periodic tessellations, a single shape is used to tessellate. Only equilateral triangles, squares, and hexagons can be used in a regular tessellation. Semi-regular or non-periodic versions have two or more shapes. The art of M. C. Escher often includes non-periodic tessellation as a stylistic element, sometimes with very complex shapes, such as interlocking animals. This type of tessellation is also used in geometry and other math classes to introduce students to a number of concepts.
The mathematics background of the tessellation may explain why it is such a popular design element. Many recurring themes in artwork can be described mathematically, suggesting that there is a universal appeal in mathematically-bounded and described concepts. From the cobblestoned streets of Paris to the complex tessellated designs of Islamic art, tessellation can be seen everywhere, in a variety of levels of complexity. Like art, math can be a universal language which may be understood by anyone, and it is interesting to trace commonalities in radically different styles of artwork which can be linked to mathematical concepts.
Exploring tessellation can help children learn about shapes and basic math, and these patterns can make interesting, fun, or engaging projects for students. Students can play with ideas like seeing how many colors they need to ensure that shapes of the same color will not touch, and they can also experiment with visual illusions created with specific shapes and colors in a tessellation.