Analysis of shape grammars: Continuity of rules

The rules in a shape grammar apply in terms of embedding to take advantage of the parts that emerge visually in the appearance of shapes. While the shapes are kept unanalyzed as a computation moves forward, part-structures for shapes can be defined retrospectively by analyzing how the rules were applied. An important outcome of this is that rule continuity is not built-in but it is "fabricated" retrospectively to analyze the computation as a continuous process. An aspect of continuity analysis that has not been addressed in the literature is how to decide which mapping forms to use to study the continuity of rule applications. This is addressed in this paper using recent results on shape topology and continuous mappings. A characterization is provided that distinguishes the suitable mapping forms from those that are inherently discontinuous or practically inconsequential for continuity analysis. It is also shown that certain intrinsic properties of shape topologies and continuous mappings provide an effective method of computing topologies algorithmically.