which says that generative operations are introduced to account only for asymmetries (distinguishabilties), not for symmetries (indistinguishabilities). This can be illustrated as follows: A set of equally-spaced points along a line can be generated by the group of integers, constituting one level. Alternatively, this set can be generated by a two-level structure, in which one particular fiber copy corresponds to a particular pair of adjacent points, and the control corresponds to the group of even numbers that moves this pair onto the next pair, and then the next pair, ..., successively along the line. However, since the points are equally spaced, this would be a spurious decomposition into levels, because there is no distinguishability, along the line of points, that justifies this decomposition.

9 Shape Generation by Group Extensions

One can see from the above discussion that the concept of group extension is basic to our generative theory. A group extension takes a group and adds to it a second group to produce a third, more encompassing, group , thus:

where is the extension operation. It is clear, looking back over the examples given so far, that according to our theory:

That is, shape generation starts with a base group and successively adds groups obtaining a structure of this form:

This approach to shape-generation differs substantially from standard shape-grammar approaches, e.g., that of Gips and Stiny (1980), which are based on the application of production rules. In our approach, structural elements correspond to groups, and the addition of structural elements corresponds to group extensions.