3. Repeating Patterns and Regular TessellationsA repeating pattern of the Euclidean plane, the hyperbolic plane, or the sphere is a pattern made up of congruent copies of a basic subpattern or motif. For instance, a black halfdevil plus an adjacent white halfangel make up a motif for Figure 1. An important kind of repeating pattern is the regular tessellation {p,q} , of the plane by regular psided polygons, or pgons , meeting q at a vertex. The values of p and q determine which of the three "classical" geometries, Euclidean, spherical, or hyperbolic, the tessellation lies in. The tessellation {p,q} is spherical, Euclidean, or hyperbolic according as (p2)(q2) is less than, equal to, or greater than 4. This is shown in Table 1 below. Note that most of the tessellations are hyperbolic. In the spherical case, the tessellations {3,3}, {3,4}, {3,5}, {4,3}, and {5,3} correspond to versions of the Platonic solids (the regular tetrahedron, octahedron, icosahedron, cube, and dodecahedron respectively) "blown up" onto the surface of their circumscribing spheres. One can interpret the tessellations {p,2} as two hemispherical caps joined along p edges on the equator; similarly {2,q} is a tessellation by q lunes. Escher's only use of these latter tessellations appears to be the carved beechwood sphere with 8 grotesques (Schattschneider 1990, p. 244) based on {2,4}. The tessellations {3,6}, {4,4}, and {6,3} are the familiar Euclidean tessellations by equilateral triangles, squares, and regular hexagons, all of which Escher used extensively.
Table 1: The relation between the values of p and q, and the geometry of the tessellation {p,q}. 

