Regular tessellation means covering the plane with copies of a single regular polygon in such a way that the polygons meet edge to edge, with no overlap and no gaps. This can be done with triangles, squares, and hexagons. The shapes have N = 3, 4, and 6 sides, and the tessellations have C = 6, 4, and 3 shapes per corner, respectively. In all three cases, we can see that (N-2)(C-2)=4.
Can we make a tessellation with (N-2)(C-2)<4? This is impossible in the plane, but possible on a sphere! There are five such tessellations, each corresponding to one of the five platonic solids.
Finally, what happens if (N-2)(C-2)>4? This is impossible in the plane or sphere, but possible in “hyperbolic” space. In this activity we explore the Poincaré disk model of hyperbolic geometry and show how it can be tessellated in many different ways. This mathematical foundation was used by Escher to make is “Circle Limit” tessellations.