Manifolds of Tessellations on the Euclidean Plane

Abstract

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.

“Digo, pues, que encima del patio de nuestra prisión caían las ventanas de la casa de un Moro rico, y principal; las cuales como de ordinario son las de los Moros, más eran agujeros que ventanas, y aun éstas se cubrían de zelosóas muy espesas y apretadas.“

“Now, overlooking the courtyard of our prison were the windows of the house of a rich and important Moor, which, as is usual in Moorish houses, were more Zike loopholes than windows, and even so were covered by thick and close lattices.”Cervantes,Don Quixote, Part I,Ch. XL, The Captive’s Story continued.