Why Symmetry? Why Asymmetry? by Michael Leyton, Page 3. To go back to: Page 2
Corroborating Examples To see that our two rules consistently yield satisfying processexplanations, let us obtain the processes that these rules give for a large set of shapes. We shall take the set of all possible shapes that have 8 extrema or less. Such shapes fall into three levels: shapes with 4 extrema, shapes with 6 extrema, and shapes with 8 extrema. The reason is that, there cannot be shapes with an odd number of extrema  because maxima have to alternate with minima of curvature  and there cannot be shapes with less than 4 extrema by a theorem in differential geometry. We will refer to the three successive levels of shapes as Levels 1, 2, and 3 respectively. There are a total of 21 shapes, of succesively increasing complexity. Since it takes a while for the internet to mount these shapes, lets make our comments first: (1) When the shapes appear, the reader will notice that, on each shape, each extremum is marked by one of four symbols: M+, M, m+, m. This is because there are mathematically four kinds of curvature extrema: Positive Maxima (M+); Negative Maxima (M), Positive minima (m+); and Negative minima (m). (2) When one surveys the shapes, one finds that there is the following simple rule that relates the type of extremum to an English word for a process: SEMANTIC INTERPRETATION RULE M+ is always a protrusion
Remember, as you look at the figures, the process arrows are inferred by the two simple rules we gave above: The SymmetryCuruvature Duality Theorem, and the Interaction Principle.
The 21 shapes:
Shapes with 4 extrema:
Shapes with 6 extrema: Shapes with 8 extrema: Set 1: Set 2: Let us now go more deeply into the structure of these histories. To continue: NEXT PAGE
