Shape as Memory Storage - Ambient Intelligence for Scientific Discovery - page 86

Fig. 6. The processes inferred by the rules.

Under this analysis, processes are understood as creating the curvature ex-
trema; e.g. the processes introduce protrusions and indentations etc., into the
shape boundary. This means that, if one were to go backwards in time, undoing
all the inferred processes, one would eventually remove all the extrema. Observe
that there is only one closed curve without extrema: the circle. Thus the impli-
cation is that the ultimate starting shape must have been a circle, and this was
deformed under various processes each of which produced an extremum.

7 Corroborating Examples

To obtain extensive corroboration for the above rules, let us now apply them to
all shapes with up to eight curvature extrema. These are shown as the outlines
in Figs 7-9. When our inference rules are applied to these outlines, they produce
the arrows shown as the inferred histories. One can see that the results accord
remarkably well with intuition.
Further considerations should be made: Any individual outline, together with
the inferred arrows, will be called a process diagram . The reader should observe
that on each process diagram in Figs 7-9, a letter-label has been placed at each
extremum (the end of each arrow). There are four alternative labels,

M

+ ,

m − ,

M − , and these correspond to the four alternative types of curvature
extrema. The four types are shown in Fig 10 and are explained as follows:
The first two have exactly the same shape: They are the sharpest kinds
of curvature extrema. The difference between them is that, in the first, the
solid (shaded) is on the inside, and, in the second, the solid (shaded) is on the
outside. That is, they are figure/ground reversals of each other. The remaining
two extrema are also figure/ground reversals of each other. Here the extrema are
the flattest points on the respective curves.
Now notice the following important phenomenon: The above characteriza-
tions of the four extrema types are purely structural. However, in surveying the

m

+ ,and