Frost crystals formed and established their
behavior through fractals
There are some
spatial patterns of nature can be described through classical geometry, but
most of these patterns are irregular and doesn’t have an organized property;
for instance, the leaves of a certain flower. These patterns thus cannot be
interpreted geometrically and thus can be described through the use of a family
of curves which are called fractals.
The word “fractal” can be associated
by the words “irregular” or “fragmented”, according to its Latin etymology –
the adjective “fractus”. Mathematically,
fractals are a set of curves displaying a self-similar pattern, that is, it is
the same pattern as it is enlarged and also as it is reduced. Fractals can be
described through mathematical equations which are nowhere differentiable.
The history of fractals comes from
the work of the mathematicians about “self-similarity”. In 17th
century, Gottfried Leibniz first studied recursive self-similarity and used the
term “fractional exponents”, in which he admitted that it was impossible to
describe this using (classical) geometry. In 1872, Karl Weierstrass studied and
presented some functions whose graphs are everywhere continuous but nowhere
differentiable, which are considered nowadays to be fractals. In the last part
of that century, Georg Cantor, Felix Klein and Henri Poincare developed some
properties of fractals; one of them can be categorized as “self-inverse”
fractals. Later, in the 20th century, more mathematicians became
interested on fractals and they gave their respective versions for it, such as Koch
curve and Levy C curve. The term “fractals” was then coined by Benoit Mandelbrot
in 1975 through the use of computer graphics in his presentations.
From the definition itself, fractals
possess self-similar patterns. Another characteristic of fractals is that they
give detailed patterns for arbitrarily smaller (or larger) scales. It is also
irregular (not differentiable everywhere in mathematical sense), simple, and
can be defined through recursion, that is, the unknown values can be found through
the previous results.
As fractals can be described through
computer graphics, many things in nature can be approximately associated with
fractals, because they appear more irregular in form. Furthermore, fractals are
always intertwined with chaos, because fractals depict chaotic behavior. Some
examples of these things in nature which are fractals are cumulus clouds, the
veins in a hand, the DNA molecule, the oxygen molecule, the leaves in a tree,
and the like.
Fractals have many applications in
science and technology. One purpose of studying fractals is to predict patterns
that are seemingly random. For instance, weather forecasters use fractal
geometry to understand cloud formations, ocean currents and air currents so
that they can predict the weather for a particular place on a particular day.
Fractals can be used to model soil erosion, as well as seismic patterns in
earthquakes. In fluid mechanics, turbulence flows can be described through
fractals, which is helpful for engineers to understand complex flows. In computer science, a certain kind of
compression known as the fractal image compression is used to describe the
world through fractal geometry (which can be seen through JPEG or GIF images). Fractals
can be also used in medicine, geology, geography, and others.
References:
[1] Kluge, Tino
et.al., (2000), Fractals in nature and applications. Fractals and Dynamic Systems. Retrieved from http://kluge.in-chemnitz.de/documents/fractal/project.html
[2] Fractal.
Wikipedia. Retrieved from http://en.wikipedia.org/wiki/Fractal
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