Shape as Memory Storage - Ambient Intelligence for Scientific Discovery - page 94

shape, splits into two copies of itself, shown at the ends of the two branches in
the right shape.
There is also another feature. In the center of the top of the right shape, a new
extremum has been introduced,

+ . Note that the process at this extremum is a
squashing , as predicted in Table 1 on p87. This process explains the flattening in
the middle of the top, relative to the sharpening towards either end of the top.
The

m

+ extremum is a minimum, and is required mathematically, because the
two branching extrema are maxima

m

M

, and two maxima cannot exist without a

minimum in between.
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
the left shape has split into two copies of itself in the right shape, and a new
extremum

M

+ has been introduced. That is, the transition from the left shape to
the right shape is the replacement of

m

+ (left shape) by the triple, + + +

M M m M

(right shape). The transition is therefore:

+ + + +

M −→ M m M .

+ , meaning Bifurcation at + .Thusthe

This transition will be labelled BM M

transition is given fully as:

+ + + + +

BM : M −→ M m M .

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

+ takes + and replaces it by the triple + + + .

Bifurcation at M M M m M

It will also be worth having a simple phrase to summarize the effect of the
transition in Fig 14. The structure formed on the right shape has the shape of a
shield , and therefore, the transition will be referred to thus:

Shield-formation.

m −

13 Bifurcation at

m − extremum branches

Next we examine what happens when the process at a

m − at the top of the left shape
in Fig 15. In accord with Table 1 (p87), the process at this extremum is an
indentation . The effect of bifurcating is shown in the right shape. One branch
goes to the left, and the other goes to the right. That is, a bay has been formed!
Thus one can regard the transition from the left shape to the right one as the
stage preceding Fig 13 on p92.
The structural change, in going from the left to the right shape in Fig 15,
should be understood as follows: First observe that the single

forward in time. As an example, consider the

m − at the top
of the left shape, splits into two copies of itself, shown at the ends of the two
branches in the right shape.