"Clouds are not spheres, mountains are not
cones, coastlines are not circles, and bark is not smooth, nor does lightning
travel in a straight line."
- Benoit
Mandelbrot, introduction to The Fractal Geometry of Nature
Introduction
Possibly
the most remarkable thing about the study of fractals is that there are fractal
patterns all around us! Even if you think you don’t know anything about
fractals
-yet-, you've
still grown up in a world full of fractals.
Many people –including me- are fascinated by the beautiful images that fractals
present. Extending beyond the typical perception of mathematics as a body of
complicated and boring formulas, fractal geometry mixes art with mathematics to
demonstrate that equations are more than just a collection of numbers. What
makes fractals even more interesting is that they are the best existing
mathematical descriptions of many natural forms, such as coastlines, mountains
or parts of living organisms.
A fractal
is a pattern that is repeated at different scales, and examples are all around
us. Technically, they call these kind of shapes "Self-similar"
because a little piece of the shape looks similar to itself. For example, the
fern shows a rough self-similarity, as it’s
made of little copies of the same general shape.
Although fractal geometry is closely connected with computer techniques,
some people had worked on fractals long before the invention of computers. Some
of these were British cartographers, who encountered the problem of measuring
the length of Britain
Properties
of fractals
Two of the most important properties of fractals are self-similarity
(as I’ve said above) and non-integer dimension.
What does self-similarity mean? If you look carefully at a fern leaf, you
will notice that every little leaf - part of the bigger one - has the same
shape that the whole fern leaf has. Thus, you can say that the fern leaf is
self-similar. Fractals work in the same way: you can enlarge them many times
and after every step you will see the same shape.
The non-integer dimension is more difficult to explain. Classical
geometry deals with objects of integer dimensions: zero dimensional points, one
dimensional lines and curves, two dimensional plane figures such as squares and
circles, and three dimensional solids such as cubes and spheres. However, many
natural phenomena are better described using a dimension between two whole
numbers, and one of them is fractals.
Fractals in the body
Not only
are fractals in the world all around us - they are even inside us! In fact,
many of our internal organs and structures display fractal properties.
- The lungs
are an excellent example of a natural fractal organ. If you look carefully, you
can see that the lungs share the same branching pattern as trees. And it is for
good reason! Both the trees and lungs have evolved to serve a similar function
- respiration. As they perform a similar function, it should not be surprising
that they share a similar structure.
Lungs
This is a common concept in science, and it’s known as the Structure-Function Relationship. Many of the fractals in the biological systems have modified their structures in order to perform specific functions.
In the case
of lungs and trees, both breathe. In animals, the lungs breathe in oxygen and
breathe out CO2. In plants, the process happens inversely and animals
and plants are two halves of the same respiratory cycle. Fractal geometry provides an incredibly useful
way to make a very large surface area extremely compact.
-Blood Vessels: There are many
more examples of fractal branching patterns in our bodies, and blood vessels
are one of the most impressive. Every cell in the body must be close to a blood
vessel in order to
receive oxygen and nutrients. The only way this is possible is through a
fractal branching network.
- Fractal Neurons: Our
brains are full of fractals! In fact, they couldn't function if not for fractal
geometry. The human brain contains approximately 100 billion neurons.
Amazingly, there are about 100 trillion synapses, or connections, among these
brain cells. That's an average of 1000 connections for a given cell, though
some neurons may only make a single connection, while others may have hundreds
of thousands of synapses. The axons reach out to make synaptic connections with
the dendrites of other neurons. It is the fractal branching pattern of the
neuron's axons and dendrites that allows them to communicate with so many other
cells.
Neurons
Fractal Rivers
"All the branches of a watercourse at every
stage of its course, if they are of equal rapidity, are equal to the body of
the main stream”. - Leonardo da Vinci
Leonardo
anticipated the discovery of fractal geometry by his intuitive analysis of natural
systems. Similar to the circulatory system in the body, the planet Earth has
fractal river networks that transport rainfall from the land to the oceans.
Like all fractals, self-similar patterns are formed by the repetition of a simple process over and over.
In the case
of a river network, it is formed by rainfall which flows down the hills. In
this process, erosion occurs, which forms small channels, and then the next
time rain falls it will follow the same path, and make the channels a little
bit deeper. Erosion is very powerful, but very slow.
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River network from Shaaxi province (China)
One of the remarkable properties of a river network is that it collects a huge amount of rainfall from a very large land area and condenses it into a small area. This explains why a river can keep flowing, even when it hasn't rained recently.
The
processes that create rivers are not restricted to the Earth. A similar
drainage network has also been identified on the planet Mars.
River network on planet
Mars
Spirals in nature
- Galaxies
are the largest known examples of spirals. A single spiral galaxy may contain a
trillion of stars. Interestingly, there is a relatively uniform distribution of
stars in a spiral galaxy. The spiral arms are also brighter because they
contain many short-lived, extremely bright stars, formed by a rotating spiral
wave of star formation. The waves of star formation are made visible because
they contain many young and very bright stars that only live a short time,
perhaps 10 million years, as compared to the more common stars, such as our sun
which live for several billion years.
Spiral galaxy
-
Hurricanes and typhoons are the largest spiral example here on the Earth. The
largest of these storms on record was Typhoon Tip, which measured 2170 km in its diameter. In
the Northern hemisphere , they spin counter-clockwise, while in the Southern
they spin in a clockwise direction.
Hurricane
Katrina
- The plant
kingdom is also full of spirals, as is evidenced in many cacti, flowers, fruits,
pinecones, etc.
Agave cactus
Agave cactus
- A
nautilus shell serves to illustrate the simple, repetitive process that creates
a spiral. The organism keeps expanding by adding sections to its shell. Each
section is a little bigger than the one before, and a little bit rotated. The
scaling factor and the rotation angle remain the same at every step in the
process. It is this simple combination of rotation and expansion that creates
the spiral and explains the ubiquity of the spiral.
Nautilus shell
Fractal
fictions
Fractal
geometry has “leaked” into many areas of science, such as astrophysics and the biological sciences, and has become one of the
most important techniques in computer graphics.
But the
biggest use of fractals in everyday life is in computer science. Many image
compression schemes use fractal algorithms to compress computer graphic files
to less than a quarter of their original size. Computer graphic artists use
many fractal forms to create textured landscapes and other intricate models.
Fractal graphic
It is
possible to create all sorts of realistic "fractal fakes"; i.e.
images of natural scenes, such as lunar landscapes, mountain ranges and
coastlines. We can see them in many special effects in Hollywood
movies and also in television advertisements. The "Genesis effect" in
the film "Star Trek II” was created using fractal landscape algorithms,
and in "Return of the Jedi" fractals were used to create the
geography of a moon, and to draw the outline of the dreaded "Death
Star".
Fractal Planet
Fractal patterns can also be used to model natural objects, allowing us to define our environment mathematically with higher accuracy than ever before.
Many
scientists have found that fractal geometry is a powerful tool for uncovering
secrets from a wide variety of systems and solving important problems in
applied science. The list of known physical fractal systems is long and growing
rapidly.
Fractals have
improved our precision in describing and classifying organic objects, but they
are perhaps not perfect. For many people, fractals will never represent
anything more than beautiful pictures.
Scissor Sister
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