A process-grammar for shape

Abstract

Inference rules are developed by which process-history can be recovered from natural shapes such as tumors, clouds, and embryos, etc. We argue that the inference of history arises from a newly discovered duality between curvature extrema and symmetry structure. We also develop a formal grammar by which someone, who has two views of an entity at two developmental stages, can infer the processes that produced the second stage from the first. More specifically, we find that a grammar, of only six operations, suffices to express the relationship between any two smooth shapes such that one shape is described as the extrapolation of processes inferred in the other under the above inference rules. In fact, a deformation is expressed as a transformation of process-records—a technique reminiscent of Chomsky's description of linguistic transformations in terms of transitions between phrase-structure trees. In the present case, our process-grammar has the psychological role of explaining the curvature extrema in terms of a sequence of psychologically meaningful deformations. Finally, we compare a process-based symmetry analysis, that we introduce in this paper, with other symmetry analyses in the literature; and we compare our process-based grammar with another grammar based on curvature extrema.