The growth rate and balance of homogeneous tilings in the hyperbolic plane

AbstractLet I be a homogeneous tiling of H2 whose growth from an initial configuration by the accretion of successive layers or coronas can be described by recurrence relations. If tn denotes the total number of tiles of I within n layers of the initial configuration, we define the growth rate λ of I as λ = limn→∞ tn+1tn When this limit exists, we use λ to extend the definition of balanced tilings given by Grünbaum and Shephard to certain normal homogeneous tilings of H2. In E2 all homogeneous tilings are balanced. This is not the case in H2; it is shown that the balance of an isohedral tiling I varies as the corona of a representative tile. Using results of Floyd, we establish that the recurrence associated with an ‘essentially’ isohedral tiling of E2 or H2 is palindromic