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.
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.
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.
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.
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.