Tessellations are the tiling of shapes. The shapes are placed in a certain pattern where there are no gaps or overlapping of shapes. This concept first originated in the 17th century and the name comes from the Greek word “tessares.” There are several main types of tessellations including regular tessellations and semi-regular tessellations.

## Regular Tessellations

Regular tessellations are tile patterns made up of only one single shape placed in some kind of pattern. There are three types of regular tessellations: triangles, squares and hexagons. Regular tessellations have interior angles that are divisors of 360 degrees. For example, a triangle’s three angles total 180 degrees; which is a divisor of 360. A hexagon contains six angles whose measurements total 720 degrees. This also is a divisor of 180, because 180 fits evenly into 720.

## Semi-Regular Tessellations

When two or three types of polygons share a common vertex, a semi-regular tessellation is forms. There are nine different types of semi-regular tessellations including combining a hexagon and a square that both contain a 1-inch side. Another example of a semi-regular tessellation is formed by combining two hexagons with two equilateral triangles.

## Demi-Regular Tessellations

There are 20 different types of demi-regular tessellations; these are tessellations that combine two or three polygon arrangements. A demi-regular tessellation can be formed by placing a row of squares, then a row of equilateral triangles that are alternated up and down forming a line of squares when combined. Demi-regular tessellations always contain two vertices.

## Non-Regular Tessellations

A non-regular tessellation is a group of shapes that have the sum of all interior angles equaling 360 degrees. There are again, no overlaps or gaps, and non-regular tessellations are formed many times using polygons that are not regular.

## Other Types

There are two other types of tessellations which are three-dimensional tessellations and non-periodic tessellations. A three-dimensional tessellation uses three-dimensional forms of shapes, such as octahedrons. A non-periodic tessellation is a tiling that does not have a repetitious pattern. Instead, the tiling evolves as it is created, yet still contains no overlapping or gaps.

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