In this paper the aperiodic monotilings discovered by David Smith and his collaborators are analysed according to the vertices of the tilings. It is shown that there are a variety of arrangements of vertices coincident with the monotiles, and when these are connected with straight lines (instead of polylines) sets of polygonal tiles are defined. Tilings based on these polygons suggest an underlying structure for the monotilings. These are presented, along with matching rules and edge modifications that give rise to alternative tilings, that are assumed to be aperiodic.