**Tessellation** is the process of covering the plane with shapes without gaps and overlaps. Tiles on walls and floors are the most common tessellations.

In this post, we are going to prove that the there are only three regular polygons that can tile the plane. Tiling the plane with regular polygons is called **regular tessellation**. These are equilateral triangles, squares, and regular hexagons. This proof only requires simple algebra. We will use the notation $latex \{a,b \}$, where $latex a$ is the number angles of the polygon and $latex b$ is the number of vertices that meets at a point. For example, a square has four angles, and at every point on the tessellation, four vertices meet at a point. So, we can represent square as $latex \{ 4,4 \}$. Another example is the regular hexagon. A hexagon has 6 sides, and at every point 3 vertices meet (see figure below), so a hexagon can be represented as $latex \{ 6,3 \}$. Continue reading…