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“An animal may have holes”

An animal may have holes.” So says the report:

Animal enumerations on regular tilings in Spherical, Euclidean, and Hyperbolic 2-dimensional spaces,” Tomás Oliveira e Silva, July 25, 2013. The author, at the Departamento de Eletrónica, Telecomunicações e Informática, Universidade de Aveiro, Portugal, writes:

“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…. 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.”

BONUS: “Animal enumerations on the {4,4} Euclidean tiling,” which begins:

The animals on the {4,4} regular tiling of the Euclidean plane are called polyominoes, a name coined by Solomon Golomb in 1953 [1]. An alternative name would be square animals, since the {4,4} regular tiling is composed of squares. Each polyomino may be equal to its mirror image (achiral) or different from its mirror image (chiral). The vast majority of the polyominoes do not have any kind of symmetry

BONUS: polyominoes

BONUS (unrelated): Slobodchikoff on the imminence of cross-species chit-chatter

 

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