Shape as Memory Storage - Ambient Intelligence for Scientific Discovery - page 91

Fig. 12. Continuation at m + .

The structural change, in going from the left to the right shape, should be
understood as follows: First, the

+ at the top of the left shape changes to the
m − at the top of the right shape. Notice that the

m

m − extremum corresponds to

an indentation, as predicted by Table 1 (p87).
An extra feature should be observed: On either side of the m − extremum,

at the top of the right shape, a small circular dot has been placed. Such a dot
marks a position where the curvature is zero; i.e., the curve is, locally, completely
straight. If one were driving around this curve, the dot would mark the place
where the steering wheel would point straight ahead.
With these facts, one can now describe exactly what occurred in the tran-
sition from the left shape to the right shape: The

+ extremum at the top of

m

m − extremum at the top of the right shape,
and two points of zero curvature, 0, have been introduced on either side of the
m − . One can therefore say that the transition from the left shape to the right
shape is the replacement of

the left shape has changed into a

m

+ (left shape) by the triple, 0

m − 0(rightshape).

The transition is therefore:

+

m − 0 m −→ .

0

Cm

+ meaning Continuation at

m

+ .Thusthe

This transition will be labelled

transition is given fully as:

+ +

m − 0 Cm : m −→ 0 .

This mathematical expression is easy to translate into English. Reading the
symbols, from left to right, the expression says:

m − 0 .

Continuation at m

+ takes

m

+ and replaces it by the triple 0

It is worth having a simple phrase defining the transition in Fig 12. Notice that,
since the extremum

+ in the left shape is a squashing, and the extremum

m − m

in the right shape is an indentation, the transition can be described as:

A squashing continues till it indents.