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

摘要

A convex polygon in R2, or a convex polyhedron in R3, 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 R3: or face) of both. P is called strongly normal (SN) if, for any partial tiling P′⊆P and any tile P∈P, the neighborhood N(P,P) of P (the union of the tiles of P′ that intersect P) is simply connected. Let P be SN, and let N∗(P,P) be the excluded neighborhood of P in P (i.e., the union of the tiles of P, other than P itself, that intersect P). We call P simple in P if N(P,P) and N∗(P,P) are topologically equivalent. This paper presents methods of determining, for an SN partial tiling P, whether a tile P∈P′ is simple, and if not, of counting the numbers of components and holes (in R3: components, tunnels and cavities) in N∗(P,P).