Tessellations are geometrical patterns that can be fit perfectly together and be repeated indefinitely.  All true tessellations fall under one of two categories: regular, and semi-regular. What both of these broader categories of patterns have in common is that the shapes surrounding each vertex, or meeting point, are identical, and it must be possible to repeat the pattern indefinitely without leaving any gaps, or causing any overlaps. What exactly does all that mean, though? Let’s take a look at some examples to find out.

Regular Tessellations

This type of tessellation is created using repeated regular polygons. Regular polygons are 2-dimensional shapes that have identical angles and sides. In general, regular polygons can have any number of sides from three on up. However, there are only three types of regular polygons that can be used to form regular tessellations: triangles, squares, and hexagons.

Semi-regular Tessellations

A semi-regular tessellation is made up of regular polygons as well, but instead of using just one repeated polygon it uses two or more of them to form a more complex pattern. All together there are eight semi-regular tessellations.

Describing Tessellations

Tessellations can be named by listing all of the polygons surrounding a vertex according to how many numbers are in each of them. As noted above, the pattern will be the same no matter which vertex is chosen. So, for example, a regular tessellation that uses only hexagons can be named a “6.6.6” tessellation; one that uses only squares or rectangles would be a “4.4.4” tessellation, and one with only triangles could be labeled a “3.3.3.”

The same principle is used to describe semi-regular tessellations. They may look more complex, but they still follow the same rules. Let’s take a look at a slightly more complicated example to illustrate the point.

A semi-regular tessellation that uses triangles, squares, and hexagons to create a more intricate pattern will still have the same repeating shapes in the same order around each vertex. Take a look at the example above. Choose a starting point, and count the number of sides on each shape that meets up with it. No matter which point is chosen, the description will be the same: “”

When describing a semi-regular tessellation, always start with the shape that has the smallest number of sides. The above pattern would always be described as a “” tessellation, and never as a “”

Demi-regular Tessellations

Although widely used by artists such as M.C. Escher and Robert Fathauer to great aesthetic effect, demiregular tessellations are not considered by most mathematicians to be true tessellations. Similarly, some geometrical artists and mathematicians believe that repetitive patterns involving circles and other curved shapes should be considered tessellations as well. Opponents note that they cannot technically meet the strictest definition as they are not polygons.