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