Tessellations are created from a repeating shape that can infinitely cover a plane without gaps or overlapping.
One regular shape, a triangle, square, or hexagon, can be used to to create a regular tessellation. An equilateral triangle and a square are most commonly seen, but any triangle can be used to tessellate, and any quadrilateral.
A semi-regular tessellation is composed of two or more regular polygons. To determine if a tessellation works, you add up the interior angles at the vertex (the point where the shapes meet). All the vertex’s in a tessellation should be identical, and should add to 360 degrees.
To name a semi-regular tessellations, name the number of sides on the shapes surrounding the vertex, starting with the least number of sides.
Demi-regular tessellations are the most abstract. They can be any shape that will tessellate. These were most commonly seen when we were in grade school, and colored in tiled shapes or animals. M.C. Escher created the famous Regular Division of the Plane with Birds, shown below.
He was born in Leeuwarden, Holland in 1898. The basis of his tessellations all respect three, four, or six-fold symmetry. The birds on a plane have a triangle basis, and Development I (shown below) has a more obvious square basis.
In other examples of tessellations with abstract shapes, the basis shape was simply changed. For example, you can start with a square, and draw a line through the square, cut that piece off, and slide it to the opposite side, resulting in something like this.
Or, you could take a side of a square, draw a line to the midpoint, and then rotate that shape around the axis. The bottom half will then be a cut out, and the top half will have that shape added on. These new shapes may make it hard to prove the tessellation works, but you can determine if the vertex adds to 360 degrees by drawing tangent lines to the curves at the vertex.
Although it seems as if M.C. Escher simply drew birds that fit together, there is mathematical reasoning behind them to explain why they fit together, and the geometry makes it possible. As a kid, when you were coloring fish on a tile, it seemed like a simple art activity, but you were already practicing geometry.