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The Ice Pantheon Some more pics of the Ice Pantheon taken by Dirk Hoeltje 1 2 3 4 5 6 7 8 |
SESSION 1
Friday Morning January 12 2001
to session
2
Introduction
Dynamical systems versus linear systems
Fractal geometry and Chaos theory
Concepts to take note of:
What is a fractal
What is fractal dimension
What is a dynamical or non-linear system?
What is a linear system?
("as predictable as clockwork")
What does the expression " The straw that brake the camel's
back" mean in dynamical or linear terms?
What is a chaotic system?
examples: weather, turbulence, human behaviour, predator-prey
relationship
stock markets, in short most complicated systems are
dynamical systems
readings :
James Gleick,CHAOS, Making a New Science
Assignment for next week: bring in a fractal and discuss
its particular nature
The whole series of articles can be found here:http://www2.rudi.net/ej/udq/59/cover.html
another link Online Planning Journal May 1998 - Articles
A nice quote:http://www.temple.edu/isllc/newfolk/consider1.html
Nice excerpt of the week:
Applying chaos theory to social sciences "conspiracy theory..."
In 1845, P.F. Verhulst introduced a new term to the exponential
growth equation
that described the way a population develops in
a closed area.
He multiplied Xn by a new term (1-Xn) which, in
effect, multiplied
Xn by itself. This procedure introduces feedback
and non linearity
into the equation, which results in yearly
population
growth dependent on what came before, non linearly.
Such factors
as the effect of the various environmental factors on
the way populations
develop could now be calculated.
Verhulst's
equation has significance that extends beyond the
quantitative
measure of populations:
Verhulst's modified equation has a host of applications. It
has been pressed into service by entomologists to
compute the effect of pests in orchards and by geneticists
to gauge the change in the frequency of certain genes in a
population. It has] been applied to the way a rumor
spreads. At first a rumor will expand exponentially until
nearly everyone has encountered it. Then the rate will
drop off quickly as more and more people say "I heard that
one." Verhulst's equation also applies to theories of
learning. What is learned now is related to the amount of
information learned previously. Learning first increases,
but after some time the learner becomes saturated so that
more effort brings only minimal results (Briggs and Peat
1989, 56-57).
Rumor, gossip,
learning, narratives, as well as conspiracies are
iterative
phenomena, recurrent events that move through
populations
as part of the social dynamic of cultural groups. The
rapid rise
and spread of conspiracy theories around the death of
Diana Spencer
is an example of the continuous feedback that
chaotic behaviors
produce in living cultures.
Feedback, or
iteration, using the output of one function as the
input of another,
continually "reabsorbs or enfolds what has come
before" (66)
indicates the presence of chaos. This is not the
chaos of randomness
and disorder. Instead, it is the chaos found
in nonlinear
dynamics, more commonly known as chaos theory.
There are
two distinct streams of chaos theory--the strange
attractor
and the order out of chaos branches. The first
emphasizes
the hidden order that exists within seemingly random
patterns.
It separates the word chaos from its connotations of
randomness
and reveals the deep structures of encoded order
that exist
within chaotic systems.
The second
branch investigates the spontaneous
self-organization
of systems that emerge "far from equilibrium".
Commonly called
dissipative structures, these systems are able
to maintain
their identity by remaining open to the flux of their
environments.
Since the emergence of chaos theory in the
mid-nineteen
eighties, more and more links have been made
between abstract
sciences and physical systems. Culture is one
of those systems.
Culture can be viewed as a complex organic or
living system,
once we recognize the prevalence of chaotic
behavior in
physical and cultural systems. Culture can be
modeled as
a living system that exhibits the characteristics of
chaotic behavior:
non linearity, complex forms, recursive
symmetries
between scale levels, sensitivity to initial conditions
and feedback
mechanisms (Hayles:11-14, 1990).
Nonlinear modeling
of culture is achieved in the introduction of
James G. Miller's
living systems theory (1978, 1990) which
defines all
living entities as complexly structured open systems.
Miller identifies
eight levels of increasing complexity as living
systems: cells,
organs, organisms, groups, organizations,
communities,
societies, and supranational systems (1990). When
we consider
social systems accordingly, the interaction of the
physical and
the cultural becomes an essential element in
understanding
complex phenomena. A system does not exist in
total isolation.
Instead, a living system has boundaries that are
permeable,
which allows the system to import the energy
essential
to maintain itself.
A good link on linear versus dynamic: (an article trying
to relate chaos theory and fractal geometry to the legal definition of
'Obscenity'
http://www.tryoung.com/chaos/025cislo.html
Characteristics of Fractal Geometry
As was mentioned above, fractal objects or images are
characteristically complex objects. Fractals, such as the Mandelbrot set,
are also
characteristically "self-similar" at all levels of magnification.
This refers to an object’s repetitive pattern. A leaf, for example, may
repeat the
same patterns found in the entire plant from which it
came (think of the fern plant). This description, so pervasive in objects
of the natural
world, is also relevant to mathematically generated images,
such as the Mandelbrot set.
Without getting into too much detail, the Mandelbrot set
(among other generated fractals) is created by running a non-linear equation
and
adding the product of that equation to a constant. This
sum is fed back, or iterated, into the equation. After each iteration,
the product of the
equation is plotted on the complex plane. The axes of
the complex plane are graded with coordinates that allow the determination
of inclusion
in or exclusion from the Mandelbrot set to made; the
Mandelbrot set consists in the products of those equations that fall within
the
circumference of two on the complex plane after an arbitrary
number of iterations. This work is carried out on the computer which assigns
a
color, or gradation of gray, given parameters, to each
pixel on the computer screen where the products are plotted. The result
is the
mathematically generated fractal image known as the Mandelbrot
set (Appendix A, Figure 1).
The term "self-similarity" in relation to the Mandelbrot
set (our second characteristic of fractals under consideration) not only
refers to the
similarity noticed at first glance, as described with
the fern plant, it also refers to a more deeply rooted structural similarity
that is found by
"zooming in" on the border area of the image (this is
facilitated by use of the computer). Magnification of this border area
reveals smaller or
"baby Mandelbrots" that are equally complex; this pattern
of similarity goes on ad infinitum.
Another characteristic of fractals is termed "high-sensitivity-to
initial-conditions" [HSIC]. This term applies most conspicuously to
mathematically generated fractals, being their genesis is understood
as coming from the finite mathematical equation; thus, the (system’s)
initial conditions are easily accessible. Briefly, this means that
when running equations for the Mandelbrot set, for instance, two separate
equations that are identical, though have slightly different input
values, can have remarkably different results after iteration. The difference
of
even one digit in the thousandth decimal place can make one equation
part of the set and the other a case of utter divergence after iteration.
The initial conditions in this case are the two equations that we started
with before iteration. High sensitivity to these initial conditions is
the
result of iteration; the difference between the two original equations
is amplified due to each iteration.
So, we have described three characteristics of fractal geometry: (1)
it describes irregular objects or images by an assignment of complexity,
(2) the objects or images described are self-similar at all levels
of magnification, and (3) these objects or images possess
high-sensitivity-to-initial-conditions. Our next tool of DST to consider
is called the "bifurcation diagram."
