Shape as Memory Storage - Ambient Intelligence for Scientific Discovery - page 96

An extremum E is sent to two copies of itself, and a new extremum e is intro-

duced between the two copies. The new extremum e is determined completely

from E as follows: Extremum e mustbetheoppositetypefrom E ;thatis,it

much change a Maximum ( M ) into a minimum ( m ), and vice versa. Furthermore,

extremum e must have the same sign as E ,thatis,“+”or“-”.

m + 15 Bifurcation at

+ extremum branches

Next we examine what happens when the process at a m

+ at the top of the left shape in
Fig 16. In accord with Table 1 (p87), the process at this extremum is a squashing .
The effect of bifurcation is that

forward in time. As an example, consider the m

+ splits into two copies of itself - the two
copies shown on either side of the right shape. One should imagine the two copies
as sliding over the surface till they reached their current positions.
The other crucial event is the introduction of a new extremum

m

+ in the
top of the right shape. This is in accord with the bifurcation format described
in the previous section. Notice that the upward process here conforms to Table
1 on p87, which says that a

M

+ extremum always corresponds to a protrusion.
Thus the transition from the left shape to the right shape is the replacement
of the

M

+ extremum at the top of the left shape by the triple + + + in the

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, as follows: Notice that the main effect in Fig 16 is that the initial

(a) (b)

Fig. 16. Bifurcation at m + .