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FIGURE 7.
I let
y
, the output of the
D
machine, become in turn the input. I do the
same thing with the
S
machine. This step transforms operational linearity
into operational circularity, a situation that can be represented only in a
plane, therefore in a 2-dimensional manifold. Once again, this can be
expressed in algebraic formalism in two ways: first, in that one makes the
next input,
x
¢, the result of the current operation,
x
¢=
y
, whereby the marked
quantity is to follow the unmarked one:
¢=
(
)
¢=
(
)
x
D x, u , and u
S u, x
whereby the recursiveness of these expression can be recognized in that the
variables
x
,
u
appear as functions of themselves. One gets a “physicaliza-
tion” of this situation, expressing the passage of time, by introducing the
parameter “time” in the form of incremental units:
t
now,
t
+
1
a single incre-
mental unit later:
(
)
=
(
)
x
=
D x , u , and u
S u , x
t+1
t
t+1
t
Those of you who are occupied with chaos theory and with recursive func-
tions will recognize at once that these are the fundamental equations of
recursive function theory. Those are the conceptual mechanisms with which
chaos research is conducted; it is always the same equations over and over
again. And they give rise to completely astonishing, unforeseen operational
properties. Viewed historically, even early on one noticed a convergence to
some stable values. An example: if you recursively take the square root of
any random initial value (most calculators have a square root button), then
you will very soon arrive at the stable value 1.0000....No wonder, for the
root of 1 is 1. The mathematicians at the turn of the century called these
values the “Eigen values” of the corresponding functions. To the operation
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