This chapter attempts to reconcile two popular topologies that seem incompatible in two–dimensional cellular automata (CA), namely the two topologies whose base cell is either the hexagon or the square. A detailed study of regular tilings of the infinite plane is first presented although it is a well–known concept. It highlights aspects that are sometimes approached in a confusing manner and defines the relationships between the coordinate systems used. The second part is devoted to the choice of the topology for a finite framework with the appropriate boundary conditions. Two fundamental patterns, the propeller and the bee, emerge among the seven possible tetrahexes. These two patterns induce the generating set of a reproducible tiling. The result is a family of Cayley graphs, the arrowhead and the diamond, isomorphic in their undirected version. Their morphism is the key unifying the hexagon and the square. Such a CA network is a good host for embedding well–known topologies like T–tree or B–tree. It is also suitable for the study of self–similarity in nature and physics, fractal structures and renormalization procedures.