*Michael Leyton*

Mathematical results are presented that strongly constrain the prototypification of complex shape. Such shape requires local prototypification in two senses: (1) prototypification occurs in parallel at different parts of the figure, and (2) prototypification varies differentially (smoothly) across an individual part. With respect to (1), we present a theorem that states that every Hoffman-Richards codon has a unique Brady Smooth Local Symmetry. The theorem solves the issue of defining units for parallel decomposition, for it implies that a codon is the minimal unit with respect to the existence of prototypification via symmetry, and is maximal with respect to prototypification via non-ambiguous symmetry. Concerning issue (2) above, a further theorem is offered that severely limits the possible shapes that result from the sequential application of prototypifying operations to smoothly varying deformation. This second result explains why considerably fewer prototype classes exist than one would otherwise expect.

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