Recursive Voronoi diagrams (RVDs) use Voronoi concepts to tessellate a space with respect to a given set of generators and repeat the construction every time with a new generator set consisting of objects selected from the previous generator set plus features of the current tessellation. In this paper, we investigate the behaviour of three variants of a single recursive Voronoi construction involving point generators. We describe how these structures may be constructed in Arc/Info. Although we suggest that RVDs have a number of potential uses in GIS, we focus on a spatial modelling perspective and examine some of their statistical characteristics. The RVDs examined are found to be fractal in nature.