Isoperimetric Formulas for Hyperbolic Animals

An animal is a planar shape formed by attaching congruent regular polygons along their edges. In 1976, Harary and Harborth gave closed isoperimetric formulas for Euclidean animals. In our paper, we provide analogous formulas for hyperbolic animals. We do this by proving a connection between Sturmian words and the parameters of a discrete analog of balls in the graph determined by hyperbolic tessellations. This reveals a complexity in hyperbolic animals that is not present in Euclidean animals.

Below you can use our calculator, which has these formulas integrated, to know the exact minimum perimeter of hyperbolic {p,q} animals with n tiles.​

*Click here for cool visualizations of these extremal hyperbolic animals.