Abstract A tessellation of Dirichlet polygons is generated from a pattern of points distributed at random over a rectangular grid, and a simple method is devised to disturb the clumps that are a characteristic feature of such a pattern, by subjecting the pattern to a clustering process. The process consists in applying a regular, periodic variation to the line-spacing of the grid. If the period of the variation is sufficiently small, or sufficiently large, the clumping is found to be scarcely affected by the clustering; but there is a narrow range of scale within which the two aggregative processes appear to interfere with each other. This interference is accompanied by a sudden and marked change in the topological properties of neighboring polygons of the tessellation, and an attempt is made to describe this change quantitatively in terms of previously established parameters. The importance of finding a physical counterpart of this geometrical effect is stressed.