Fractals Everywhere

comes from the Latin word
(translated as ‘fractured’) and it refers to any irregular,
‘fractured’ looking shape. The term was coined by Benoit Mandelbrot,
the IBM mathematician who first produced computer-generated images of
fractals and mathematically interpreted nature in his book
Fractal Geometry of Nature
. The
most famous computer-generated fractal shape is the
related to the Julia
, and its infinite detail and
depth can be described by a very simple equation. A value is fed into
the equation and the result is then fed back into the equation – it
is this recursive nature of the equation which leads to the infinite
nature of the fractal image. 

 Mandelbrot set

It is
interesting that fractals have existed in nature, yet no-one paid any
real, scientific attention to them until the 1980s. This goes to show
that scientific and mathematical breakthroughs heavily depend on
circumstance. Fractals in nature could only really be understood,
mathematically, once computers were powerful enough to generate
fractal images. This is not to say though that an essential feature
of fractals,
had not been understood in the past. Self-similarity is when the
parts of a shape resemble the whole shape or are a copy of the whole shape. The Cantor Set
and the Koch snowflake are prime examples of this.

 Koch snowflake, full of repeating triangles

fact, some artists have even used self-similarity in their art; for
example, the Japanese artist Hokusai used the repeating pattern of
waves in his well-known painting
The Great Wave Off

A lot of religious
art, particularly art from Eastern religions, make use of
self-similarity and bear some similarities to fractals. Tibetan
thangka paintings
would be a good example of this.

Fractal geometry seems to act as a blueprint for many living things. In
nature, fractals are everywhere. The centre of sunflowers, pine
cones, ferns, the shape of lightning and river meanderings, the branching of
trees, veins and blood vessels, lungs, and the countless other forms of other countless plants and animals. It has been said that
nature has exploited fractal geometry because it is the most
efficient way for something to grow or because it is the easiest way
to increase surface area (making it easier for the lungs to absorb
oxygen, for example).

 The lungs of a dog

My favourite example of a fractal in nature is the one that most
resembles a computer-generated fractal, called the Romanesco
cauliflower. It is easy to see the repeating pattern of cones. As a
whole, the cauliflower is a large cone, but at any point is another
cone, which contains cones within cones etc. It’s dizzying. But if you want to appreciate how infinite complexity
can arise from simple rules, watch videos of ‘fractal zooming’ on YouTube. 

 Romanesco cauliflower

study of fractals in the 80s and the later computer-generated images
of the Mandelbrot set spawned a whole new type of art called fractal

 Fractal art

It’s interesting that a kind of mathematics, belonging to a kind
of science (the science of chaos) would lead to a new aesthetic
appreciation of fractals, shapes which have always existed in nature.
Perhaps the psychedelic art of the 60s though did pre-empt this in
some way, since tie-dye and kaleidoscopic art is very similar to
fractal art. It is also supposed to be very common to see images of
fractals under the influence of LSD,
mescaline and DMT. 

 DMT-inspired art

This seems pretty strange. How could a drug make you see an image
of a fractal? Does the drug make your visual system recursive, so
that shapes and colours are fed into themselves, eventually
producing a fractal image? If that’s what happens, how and why does the drug do this?
High doses of these drugs can make users feel connected with nature, so maybe seeing fractals is part of the visual aspect of that experience, since fractals do have their basis in nature itself.

geometry now has a wide-range of applications. They are being applied to
computer chips so that they can store more information and in computer-generated landscapes for films such as
Star Wars and
Star Trek. It was
only because of fractal geometry that computers were able to generate
images of mountains and forests which had that realistic, fractured,
irregular shape. There are many other useful applications in video
game design, engineering, medicine and other areas of technology.
Fractal geometry is a very recent kind of geometry so its
implications for the future are not clear. 

 Computer-generated fractal landscape

It may
even be relevant to physics and our view of reality. Arthur C.
Clarke (author of
2001: A Space Odyssey), in a discussion with Carl Sagan and Stephen Hawking (which is on YouTube) asked Stephen Hawking if the fundamental nature of
reality could be fractal. In other words, if we keep zooming deeper and deeper
into matter, will we reach some smallest, fundamental particle or string
(if string theory becomes verified in the future) or will the zooming go on forever?. Hawking dismissed
the idea by saying that the deepest level of matter is the Planck length
– nothing can be smaller than this length.

But maybe as particle colliders become
more sophisticated, it will become apparent that material objects are
like fractals, in that they are finite in extent, but they are infinite in
depth. In other words, the object takes up a limited amount of space,
but the depth of the object is never-ending. You could zoom into it
forever. Even if this does not turn out to be true, it’s still a
fascinating fact that fractals are both finite and infinite in this


1 Comment

  1. Aaron H
    July 18, 2013 / 3:45 pm

    the recursive nature of the equation hints at the infinite nature of the fractal, but the inverse stumps me. how can there actually be entropy in the universe given that the perceived chaos is susceptible to recursive deconstruction? i am obviously not a scientist so please don't mock me for wondering.

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