When Healing Becomes Educating, Vol. 2 - Waldorf Research Institute

At the beginning of this article we quoted the physicist Gert
Eilenberger who spoke of the qualitative differences between “natural”
and “manufactured” objects. The latter are characteristically based on
linear forms, natural objects on non-linear principles. The function of
manufactured objects bases on a sequence of one-dimensional causal steps;
in the natural world we have multidimensional steps with feedback creating
an action complex, and this alone makes development possible.
Working with non-linear systems we come to realize that incalculability
and irregularity are an essential part of nature. The linear laws of physics
apply only to the small fraction of the natural world that has ceased to grow
and develop and reached its end-state. Between the calculable end-state and
the irregular and incalculable lies the sphere of periodization, doubling of
periods and finally transition to chaos. It is the sphere where self-similarity
and fractal belong.
Cramer writes: “Fractal dimensions arise wherever chaos appears.” 7
Fractals arise on the borders of chaos. In this realm of self-similarity and
constant repetition we find the images of natural objects that can be
reproduced with the aid of fractal geometry.
It is evident from the universal constant found by Mitchell Feigenbaum
that the road to chaos is governed by a higher law that comes to expression
in the constant proportion and can be expressed as an irrational number.
Irrational numbers, which cannot be expressed as ordinary fractions and
show no periodicity of decimal places, thus are connected with chaos. The
most irrational of all irrational numbers (this can be shown mathematically 8 )
is the golden section d = 1.618..., the divine proportion. Many proportions
which occur in nature and are experienced as particularly well balanced are
in fact based on the golden section.
Friedrich Cramer writes: “The golden section is the most irrational
of all possible irrational numbers and is therefore also connected with
chaos. In certain sequences and mathematical or graphic representations of
complex dynamic systems chaos increases with growing non-linearity. In the
end, areas of chaos are merely separated by a few curves which ultimately
reduce to just one. This can be connected with the golden section, using
the method given above. Is this another indication of harmony existing on
the boundary between order and chaos? The most irrational sequences, i.e.,
sequences based on the golden section, have the greatest chance of survival
if the system is upset. Their resistance to chaos is greatest.”
Later he writes: “Is beauty therefore not just a matter of how we see it
and of convention, but a property inherent in objects and in the world?
Does the world have a fundamentally balanced, harmonious structure
where it borders on chaos?” 9
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