Determining simplicity and computing topological change in strongly normal partial tilings of R2 or R3

A convex polygon in R 2 , or a convex polyhedron in R 3 , will be called a tile. A connected set P of tiles is called a partial tiling if the intersection of any two of the tiles is either empty, or is a vertex or edge (in R 3 : or face) of both. P is called strongly normal (SN) if, for any partial tiling P 0 P and any tile P 2 P 0 , the neighborhood N(P; P) of P (the union of the tiles of P 0 that intersect P) is simply connected. are topologically equivalent. This paper presents methods of determining, for an SN partial tiling P, whether a tile P 2 P 0 is simple, and if not, of counting the numbers of components and holes (in R 3 : components, tunnels and cavities) in N (P; P).