Mathematics in Tessellation

The value of the internal angle of polygon=m-polygon is defined by the equation

Assume n-number of m-gons arond each vertex in the tiling so that all of n interior angles starting at the vertex make a compete circle and; therefore, all angles add up 360 degrees as shown in the figure below

 Thus, an equation that illustrates this idea will be:

 Simplifying this equation by dividing both sides of the equation by 360 gives

Equating the equation to 0 and solving for mn gives

Adding 4 to both sides of the equation gives

Factor the equation by assuming n for m and replacing m for n again at the end  

Case a)  m=6 and n=3 it means that we have 3 regular hexagons (m=6) meeting at each vertex

Case b)  m=4 and n=4 it means that we have 4 regular squares ( m=4) meeting at each vertex

Case c) m=3 and n=6 it means that we have 6 equilateral triangles (m=3) meeting at each vertex