Manifolds of Tessellations on the Euclidean Plane

The space whose points are the different positions of a regular dodecahedron inscribed in the 2-sphere, with the most natural possible topology, is a closed, orientable 3-manifold known as the Poincaré homology 3-sphere A dodecahedron is a tessellation of the 2-sphere, as is an octahedron or a tetrahedron, for instance. The original examples of tessellations belong to the euclidean plane ℝ2, like the hexagonal mosaics that one can admire in The Alhambra de Granada or in The Aljaferia de Zaragoza The hyperbolic plane H2 is very rich in tessellations. The object of the rest of the book is to describe the 3-manifolds of euclidean, spherical and hyperbolic tessellations.