Tesselation on the Hyperbolic plane
Tessellations over the 3 types of geometry
A {k,n} regular tessellation is defined as a tiling by regular n-gons (convex regular polygons that cover all the space) where k polygons of n sides meet at each vertex. k, n 3.
On the Euclidean Plane the only choices are {6,3}, {4,4} and {3,6} regular tessellations by traingles, squares or hexagons.
Theorem of Classification.
For any k,n 3, there exists a tiling on the one type of geometry depending:
- If (1/k) + (1/n) > 1/2, then the tilling is spherical
- If (1/k) + (1/n) = 1/2, then the tilling is euclidean
- If (1/k) + (1/n) < 1/2, then the tilling is hyperbolic.
{7,3} tesellation of the hyperbolic plane
