Concave Hexagons

Paul Gailiunas
Proceedings of Bridges 2009: Mathematics, Music, Art, Architecture, Culture (2009)
Pages 243–250 Regular Papers


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 ( 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.