Tessellations

A tessellation is basically a form of tiling. In tessellations, no overlapping or gaps occur and, the pattern is never ending. There are many different types of tessellations with many different rules. One type of tessellations is regular tessellations. In regular tessellations, the shapes must be the same regular polygons, each vertex must look identical and overlapping and gaps should not occur. In regular tessellations, the pattern is never ending. Some shapes that work with regular tessellations are squares, triangles and hexagons. These shapes all have one thing in common that helps this specific type of tessellation occur. All of the shapes in examples of regular tessellations are all similar because all the shapes have internal angles that can factor into 360 degrees. Therefore, when the corners of these regular polygon shapes connect with other ones, a full rotation occurs creating a never ending pattern with no gaps or overlapping.

One shape you could use to create a regular tessellation is a square. In a square, all the internal angles are 90 degrees. Therefore, when 4 different square corners connect, no gaps or overlaps occur because when the four 90 degree angles are connected you get 360 degrees (90+90+90+90=360 exactly therefore no remainder for gap/overlap).

Another shape you could use to create a regular tessellation is an equilateral triangle. In a an equilateral triangle, all the internal angles are 60 degrees. Therefore, when six different triangle corners connect, no gaps and overlaps occur because when the six 60 degree angles are connected you get 360 degrees (60 +60 +60 +60 +60 +60 =360 degrees).

Another shape that can create a regular tessellation is a regular hexagon. A hexagon has six sides and 120 degrees per interior angle. Therefore when three angles of three different hexagons connect, a different type of regular tessellation is created.

In each vertex of this regular tessellation, all three of the hexagons connect their 120 degree angles to make this tessellation without overlapping and gaping to occur. This, just like all other regular tessellations are repetitive with no end. 

Notice that pentagons don’t work. This is because their 108 degree angles do not evenly go into 360 degrees. Some other types of tessellations are semi regular tessellations, non-regular tessellations, simple tessellations, complex tessellations, and many more.

Another example of tessellations is semi-regular tessellations. Semi-regular tessellations are created by connecting two or more different regular polygons. The rules for these are the same as regular tessellations and the most important part is that the configuration must also be the same (must be able to evenly fit into 360 degrees).  

(taken from an eighth grade paper of mine for math research)