Animal enumerations on regular tilings in
Spherical, Euclidean, and Hyperbolic
2-dimensional spaces
Introduction
Let S be a 2-dimensional space (i.e., a surface), and let {p} be a regular polygon with p sides defined on S. This means that all sides of the polygon are "straight lines" (i.e., geodesics), that they have the same length, and that the inner angles between adjacent sides are all equal. In this page we are interested in regular polygons associated with spherical, Euclidean, and hyperbolic 2-dimensional spaces, corresponding, respectively, to internal angles larger, equal, and smaller than 180(p-2)/p degrees.
Let {p,q} be a regular tiling (or tessellation) of S. This means that the tiling consists of {p} polygons placed edge to edge, with each vertex of the tiling surrounded by exactly q polygons. Thus, the inner angles of the polygons are equal to 360/q degrees. All values of p and q larger than two are allowed. It is possible to confirm that for spherical, Euclidean, and hyperbolic tilings the value of 2p+2q-pq must be, respectively, greater than zero, equal to zero, and smaller than zero. For spherical and hyperbolic tilings, the length of the edges of the polygons is severely constrained (only one value allowed for each curvature of the space).
An animal with area n is any edge-connected set of n polygons (chosen from the polygons of a 2-dimensional regular tiling). An animal may have holes, i.e., it may not be topologically equivalent to a disk. The number of holes of an animal is defined to be one less than the number of edge-disconnected regions of the complement of the animal. (The complement of an animal is of course the set of the polygons of the tiling that do not belong to the animal.) Two animals are said to be distinct if it is not possible to obtain one from the other via translations and rotations without leaving S. The mirror image of an animal is obtained by lifting it from S, flipping it upside down (this requires a third dimension), and putting it back on S. If an animal is not distinct from its mirror image then it is an amphicheiral (or achiral) animal; otherwise it is a chiral animal.
Our results
The enumeration of animals in regular tilings of the Euclidean plane is of some importance in statistical physics, where it provides one way to analyze two-dimensional percolation phenomena. No known formula exists for the number of animals of area n in a regular {p,q} tiling. Our own enumeration efforts are concentrated on the following tiling: {4,4}. Enumeration results for other tilings will be added in the future.