Strongly Normal Sets of Tiles in N Dimensions

Abstract

The first and third authors and others [2,8,9,10,11,12] have studied sets of “tiles” (a generalization of pixels or voxels) in two and three dimensions that have a property called strong normality (SN): For any tile P, only finitely many tiles intersect P, and any nonempty intersection of these tiles must also intersect P. This paper presents extensions of the basic results about SN sets of tiles to n dimensions. One of our results is that if SN holds for every n + 1 or fewer tiles in a locally finite set of tiles in Rn, then the entire set of tiles is SN. Other results are that SN is equivalent to hereditary local contractibility, that simpleness of a tile in an SN set of tiles is equivalent to contractibility of its shared subset, and that deletion of a simple tile in an SN set of tiles preserves the homotopy type of the union of all the tiles.