Symmetric designs on hexagonal tiles of a hexagonal lattice

Abstract

In his paper ‘A Class of Pleasing Periodic Designs’, Fernandez utilized the union of several sublattices of a square lattice to generate a design on a square tile of the Cartesian plane such that the design, the tile and the lattice have the same rotational and reflectional symmetries. The union of those sublattices may be called a generating set for the symmetric design because the design is composed of certain rotated and reflected copies of the union restricted to the tile. In this article, we use different kinds of generating sets and develop effective algorithms to create aesthetic designs on a tile of a hexagonal lattice such that the tile, the hexagonal lattice and these designs have the same rotational and reflectional symmetries.

Keywords:

• pattern designs
• symmetries
• hexagonal lattices
• sublattices
• integer matrices

AMS Subject Classifications:

• 68U05
• 65D18

Acknowledgements

The authors thank the reviewers for their corrections and valuable suggestions on the previous versions of this article. This work was partially supported by US National Science Foundation/EPSCoR under Cooperative Agreement No. (EPS-0903795).