‘Fractals—an Organized Chaos’
First Reading Reflection
Self-Organization in Systems and Society (STSH 4580)
Tyler Carrara
March 3rd, 2014
RIN: 661181672
‘Fractals—an Organized Chaos’
Self-organization is a phenomenon of which little is known, yet its presence can be found
affecting all areas of science. What is known is that this peculiar concept relies on a vital set of rules
unanimously obeyed by the individuals within the system for direction, contrasting sharply with the
idea of group reliance on a central-governing leader of who makes every crucial decision. Varying
degrees of complexity exist within each self-organizing system as do multiple methods of measuring
complexity. While it is currently possible to measure complexity with relative accuracy, debate
consistently arises among researchers as to which method should be accepted as the standard method
of measurement. In higher-level mathematics, fractals display self-similar patterns that correlate
significantly with the self-organization frequently found in organic environments. The uniqueness
associated with fractals is the product of exceedingly complex shapes being extracted from a simple
set of rules and a simple initial geometric figure. These derived intricate figures not only offer a
potential method of measuring complexity, but can mimic certain organic systems in which selforganization has been observed. While Euclidean shapes represent perfect order and fractals
represent a perfect balance of order and disorder, the term chaos theory introduces pure disorder into
the equation. Chaos theory is the idea that a dynamic system is extremely sensitive to initial
conditions which causes it to be nearly unpredictable. Strange attractors allow such predictions to be
made as the fractal-like attractors reveal the behavior of chaos only in the immediate future. A
balance between order and disorder in self-organizing systems is required, for without one factor the
systems cease to function effectively.
Self-organizing systems appear in many different forms throughout the world. Army Ants on
the march, neurons firing in the brain, a synchronized flock of European Starlings, the coordination
of the human immune system, and Bigeye Trevally spherical ‘Tornados’ all demonstrate
characteristics of self-organization in which no single entity of the system leads entirely. Members
that are part of a such a system when isolated fail to complete the simplest of tasks and quickly
become obsolete, “yet put half a million of them together, and the group as a whole becomes what
some have called a ‘superorganism’ with ‘collective intelligence’” (Mitchell 3). The description of a
‘superorganism’ with ‘collective intelligence’ closely resembles that of the neurons in our brain
which when isolated have little to no impact but when firing collectively create an anomaly that
humans refer to as ‘consciousness’. The human immune system miraculously “employs randomness
to allow each individual lymphocyte to recognize a range of shapes that differs from the range
recognized by other lymphocytes in the population” (174). Depending on which lymphocytes survive
an encounter with an antigen, more of that lymphocyte is produced by the lymph nodes. This
defensive stance is similar to that of the Bigeye Trevally ‘Tornado’ which aims to form a sphere as a
sphere is optimized to have the least amount of surface area mathematically. This decreases the odds
of an individual Bigeye Trevally being consumed by a predator. Naturally occurring self-organizing
systems may vary in complexity significantly, yet no widely accepted standard method of
measurement for complexity presently exists.
Complexity must first be defined if it is to be analyzed. Nobel prize-winning physicist “GellMann proposed that any given entity is composed of a combination of regularity and randomness…
the effective complexity is defined as the amount of information contained in that description, or
equivalently, the algorithmic information content of the set of regularities” (Mitchell 99). Selforganizing systems are composed of both order and disorder. The flock of European Starlings
demonstrates order by flying as one dense group through the sky with each individual flying in the
same relative direction. Disorders exists within the mass of European Starlings as well, the direction
in which the entire mass moves consistently changes and is unpredictable along with the position of
each individual within the flock. Logical depth is widely considered another potential way of
measuring complexity, where “the logical depth of an object is a measure of how difficult that object
is to construct” (100). What is accepted by most researchers studying self-organizing systems is that
“both very ordered and very random entities have low effective complexity” (99). A balance of these
two factors allows for predictability and unpredictability, this fundamental idea can be related to
music. If a person is listening to a radio and suddenly a song comes on that plays one note at a set
interval, the person listening to the music would grow increasingly bored; the music’s pattern is too
simple and predictable. The same person switches the station and the radio begins to emit whitenoise or static, the person listening becomes bored yet again; this is due to the complete randomness
of the noise in that it contains no differentiable pattern. Desirable music has varying levels of
complexity in which order and disorder attempt to counter one another, the same theory applies to
self-organizing systems.
Geometric mathematical models that associate with this balance of order and disorder are
referred to as fractals. A classic way of describing a fractal is by imagining a coastline:
If you view a coastline from an airplane, it typically looks rugged rather than straight, with
many inlets, bays, prominences, and peninsulas. If you then view the same coastline from
your car on the coast highway, it still appears to have the exact same kind of ruggedness, but
on a smaller scale… The similarity of the shape of the coastline at different scales is called
‘self-similarity.’ (Mitchell 103)
Natural fractals include but are not limited to coastlines, trees, snowflakes, and galaxy clusters. The
concept of a fractal dimension is to relate the complexity associated with fractals to that of the
complexity associated with self-organizing systems in an attempt to measure the intricacies of the
self-organizing systems. A fractal dimension is not an integer, yet ultimately aims to quantify “the
number of copies of a self-similar object at each level of magnification of that object. Equivalently,
fractal dimension quantifies how the total size (or area, or volume) of an object will change as the
magnification level changes” (108). Another way of describing what a fractal dimension represents is
the “‘roughness,’ ‘ruggedness,’ ‘jaggedness,’ or ‘complicatedness’ of an object… quantifies the
‘cascade of detail’ in an object” (108). Fractals maintain details continuously at every level of
magnification; the fractal dimension effectively quantifies how interesting that detail is as a function
of the net magnification required at each level to observe it. Only perfect mathematically derived
fractals—fractals that extend to infinity after undergoing magnification—have a precise fractal
dimension. Natural and finite fractal-like systems can only be measured for an approximate fractal
dimension.
Euclidean geometries attempt to portray matter in an ordered state and can be described as
simple in appearance containing an abundance of distinct edges that meet at clearly defined vertices,
contrasting sharply with the silhouette of a fractal. Euclidean geometries are frequently constructed
of clearly defined patterns that can be easily recognized and predicted. Crystals are a perfect example
of naturally occurring Euclidean geometries while skyscrapers can be referred to as artificially
created Euclidean geometries. The root figure of any given fractal is usually a type of Euclidean
geometry, yet as the figure undergoes transformations each phase becomes increasingly complex
until a fractal has been created. Upon first glance a fractal may appear to be disordered and random,
but having been defined from a Euclidean geometry a complex algorithm does exist. This algorithm
can be applied to self-organizing systems with the goal of defining any underlying complexity—the
fractal dimension. While Euclidean geometries contribute to the ordered aspect of fractals, chaos
theory adds to the disorder of fractals.
Chaos theory is relevant to systems—self-organizing systems—that react in accordance with
their initial conditions. Chaos theory focuses on “the behavior of dynamical systems that are highly
sensitive to initial conditions—an effect which is popularly referred to as the butterfly effect”
(Kellert 32). The ‘butterfly effect’ is described in the following explanation
In chaos theory, the butterfly effect is the sensitive dependency on initial conditions in which
a small change at one place in a deterministic nonlinear system can result in large differences
in a later state. The name of the effect, coined by Edward Lorenz, is derived from the
theoretical example of a hurricane's formation being contingent on whether or n ot a distant
butterfly had flapped its wings several weeks earlier. (Woods 118)
This explanation applies to fractals and thus self-organizing systems alike. If an initial condition is
altered the new fractal produced is completely different; regardless of whether or not the initial
condition affected is the set of rules that governs the phase transformations or is the starting
Euclidean geometry. An attractor is a term that describes a “set of physical properties toward which a
system tends to evolve, regardless of the starting conditions of the system. Property values that get
close enough to the attractor values remain close even if slightly disturbed” (Attractor). An attractor
is referred to as ‘strange’ if it contains a fractal structure. This is often the case when the dynamics
pertaining to it are chaotic. Even though strange attractors are associated with chaos, the attractors
contain a fractal dimension of which is calculable. Numerical analysis of the double-scroll attractor
and the famous Lorenz attractor has shown that geometrical structure of strange attractors is made up
of an infinite number of fractal-like layers. Even though fractals originate from simple Euclidean
geometries and pre-defined transformation rules, their concept can be applied to dynamic systems
incorporating varying amounts of chaos.
Self-organizing systems are composed of both order and disorder. If one aspect weighs too
heavily, the balance is lost and the pattern either becomes too simple or too random. In selforganizing systems, order exists within the simplicity of initial Euclidean geometries and their
transformation rules while disorder presents itself in the form of strange attractors found under the
topic known as chaos theory. This order and disorder combined produces the intricate mathematical
figures termed fractals. These fractals may appear to be disordered upon first glance, but prove to
contain an underlying pattern after magnification. It is this pattern’s complexity which can be
quantified through fractal dimensioning and logical depth. With varying amounts of complexity
being approximated in separate self-organizing systems, the chaotic component may be potentially
increasing the systems’ complexity while hindering its predictability.
Excellent work—I would have said something about phase space regarding your line “An
attractor is referred to as ‘strange’ if it contains a fractal structure”. Grade = A
Citations:
"Attractor." The Free Dictionary. Farlex, n.d. Web. 07 Mar. 2014.
Kellert, Stephen H. (1993). In the Wake of Chaos: Unpredictable Order in Dynamical Systems.
University of Chicago Press. Page 32. ISBN 0-226-42976-8.
Mitchell, Melanie. Complexity: A Guided Tour. Oxford: Oxford UP, 2009. Print.
Woods, Austin (2005). Medium-Range Weather Prediction: The European Approach; The Story of
the European Centre for Medium-Range Weather Forecasts. New York: Springer. Page 118.
ISBN 978-0387269283.