Uniform Tiling

Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. In geometry, a uniform tiling is a tessellation of the plane by regular polygon faces with the restriction of being vertex-uniform. Uniform tilings can exist in both the Euclidean plane and hyperbolic plane. Uniform tilings are related to the finite uniform polyhedra which can be considered uniform tilings of the sphere. Most uniform tilings can be made from a Wythoff construction starting with a symmetry groups and a singular generator point inside of the fundamental domain. A planar symmetry group has a polygonal fundamental domain and can be represented by the group name reprsented by the order of the mirrors in sequential vertices. A fundamental domain triangle is (p q r), and a right triangle (p q 2), where p, q, r are whole numbers greater than 1. The triangle may exist as a spherical triangle, an Euclidean plane triangle, or a hyperbolic plane triangle, depending on the values of p, q and r.