Proceedings of Bridges 2009: Mathematics, Music, Art, Architecture, Culture (2009)

Pages 243–250

The tilings (*n*.3.*n*.3) exist
in the spherical, Euclidean or hyperbolic
plane, depending on whether *n* is less than, equal to, or greater
than 6. In all cases the dual tiling consists of rhombi, which can
be taken in pairs to form "regular" concave hexagons. In the case
of the spherical examples the tilings can be illustrated by colouring
the faces of rhombic polyhedra. In the Euclidean plane "regular"
concave hexagons allow tilings that cannot be constructed from the
dual (6.3.6.3) tiling, some of which allow analogous tilings of
non-"regular" concave hexagons. Some Escher-like designs are derived
from such tilings.

Some of the possibilities in the hyperbolic plane are briefly considered.

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