## 3. Repeating Patterns and Regular Tessellations

A 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 half-devil plus an adjacent white half-angel make up a motif for Figure 1.

An important kind of repeating pattern is the regular tessellation {p,q} , of the plane by regular p-sided polygons, or p-gons , 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 (p-2)(q-2) 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.

 . . 11 10 9 8 7 q 6 5 4 3 2 1
 . . . . . . . . . . . . . . . . . . . . . . s * * * * * * * * * . . s * * * * * * * * * . . s * * * * * * * * * . . s * * * * * * * * * . . s * * * * * * * * * . . s E * * * * * * * * . . s S * * * * * * * * . . s S E * * * * * * * . . s S S S E * * * * * . . s s s s s s s s s s . .
 1 2 3 4 5 6 7 8 9 10 11 . . p
Legend
Symbol Description
E Euclidean
tessellations
S "Platonic"
spherical
tessellations
s Spherical
tessellations
for which
p=2 or q=2
* hyperbolic
tessellations

Table 1: The relation between the values of  p and q, and the geometry of the tessellation {p,q}.

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