Inferring History from Shape
Professor Michael Leyton.
The book Symmetry, Causality, Mind (MIT Press) presents a 630 page rule-system by which the shape of an object can be used to extract the sequence of historical events that produced the object. The book systematically elaborates the geometrical sources on a shape that give the historical events, as well as the rules by which the events can be inferred from those sources.
As an example, the page below shows the rules with respect to only one of the sources, the curvature extrema. Curvature extrema are extremes of curvature (bend) on a shape. A theorem by Leyton states that any curvature extremum corresponds to a symmetry axis leading to the extremum, and a further rule says that the historical movement went along the symmetry axis. The diagrams below show the histories inferred by these two rules when applied to shapes with up to eight extrema.
Shapes with 4 extrema:
Shapes with 6 extrema:
Shapes with 8 extrema:
The next page will show how to extract more detailed history from the curvature extrema. But before that, two comments need to be made:
(1) Notice, on the above shapes, that 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
Now we are ready for the next page, which shows how to break the history down into detailed stages: