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

Abstract

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′⊆ 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 R 3: components, tunnels and cavities) in N ∗(P, P) .