Cell Growth Problems

Abstract

The square lattice is the set of all points of the plane whose Cartesian coordinates are integers. A cell of the square lattice is a point-set consisting of the boundary and interior points of a unit square having its vertices at lattice points. An n-omino is a union of n cells which is connected and has no finite cut set.The set of all n-ominoes, R_n is an infinite set for each n; however, we are interested in the elements of two finite sets of equivalence classes, S_n and T_n, which are defined on the elements of R_n as follows: Two elements of R_n belong to the same equivalence class (i) in S_n, or (ii) in T_n, if one can be transformed into the other by (i) a translation or (ii) by a translation, rotation, and reflection of the plane.