An empirical regularity is typically limited to a particular range of independent variables. If many such regularities hold in a multi-dimensional space of independent variables, a diagrammatic representation of the set of regularities and their boundaries can help us to understand their distribution, and the completeness of the set of regularities. Furthermore, a diagrammatic representation can guide the search for regularities. Machine discovery systems, such as BACON and FAHRENHEIT, combine automated experimentation and theory construction based on inductive generalization of data. We describe the incremental manner in which FAHRENHEIT builds a diagram of regularities and uses the diagram to set forward further experiments and theoretical goals until a complete theory is found. Because the diagram of regularities and their boundaries reflects the topology of the space under investigation, it is instrumental in search control and in significant reduction of the search size.