Fractals
“The word fractal was coined by Benoit Mandelbrot. Fractal originates from the Latin word fractus, which also comes from frangere meaning “to break”. The literal translation of fractal is “weakened, weak, feeble, brittle, faint” (Wahl 3,4).
“Fractal geometry and the chaos theory are providing us with a new perspective to view the world. For centuries we've used the line as a basic building block to understand the objects around us. Chaos science uses a different geometry called fractal geometry. Fractal geometry is a new language used to describe, model and analyze complex forms found in nature.
A few things that fractals can model are:
plants
weather
fluid flow
geologic activity
planetary orbits
human body rhythms
animal group behavior
socioeconomic patterns
music
This is how nature creates a magnificent tree from a seed the size of a pea.
Fractal dimension can measure the texture and complexity of everything from coastlines to mountains to storm clouds. We can now use fractals to store photographic quality images in a tiny fraction of the space ordinarily needed. Fractals win prizes at graphics shows and appear on tee - shirts and calendars. Their chaotic patterns appear in many branches of science. Physicists find them on their plotters. Strange attractors with Fractal turbulence appear in celestial mechanics. Biologists diagnose dynamical diseases. Even pure mathematicians such as Bob Devaney, Heinz-Otto Peitgen and Richard Voss go on tour with slide shows and videos of their research. Fractals provide a different way of observing and modeling complex phenomena than Euclidean Geometry or the Calculus developed by Leibnitz and Newton. An arising cross-disciplinary science of complexity coupled with the power of desktop computers brings new tools and techniques for studying real world systems.”
“Fractal geometry derives rules from natural objects like trees, river, and mountains. We see that natural objects are made from self-similar patterns just like fractals” (Wahl 2). “Many objects in nature aren’t formed of squares or triangles, but of more complicated geometric figures. Imagine the coastline of Africa. You measure it with mile-long rulers and get a certain measurement. What if on the next day you measure it with foot-long rulers? Which measurement would give you a larger measurement? Since the coastline is jagged, you could get into the nooks and crannies better with the foot-long ruler, so it would yield a greater measurement. Now what if you measured it with an inch-long ruler? You could really get into the teeniest and tiniest of crannies there. So the measurement would be even bigger, that is if the coastline is jagged smaller than an inch. What if it was jagged at every point on the coastline? You could measure it with shorter and shorter rulers, and the measurement would get longer and longer. You could even measure it with infinitesimally short rulers, and the coastline would be infinitely long. That's a fractal. Fractals are broken into irregular fragments, in the same way rivers’ tributaries form together to make a river. Brooks flow into creeks, creeks into streams, and streams into rivers (Wahl 4). Fractals also demonstrate percolation. Percolation is the act of striving for equilibrium. Empty spaces, filled spaces, containment, and opposing forces of growth are all constraints and boundaries on the fractals” (Wahl 4).
Why should people study fractals? “Most of the math you have studied in school is old knowledge. For example, the geometry you study about circles, squares, and triangles was organized around 300 B.C. by a man named Euclid. Fractal Geometry, however, is much newer. Research on fractals is being carried out right now by mathematicians.” more on fractals “Fractal geometry is not a straight “application” of 20th century mathematics. It is a new branch born belatedly of the crisis of mathematics that started when duBois Reymond 1875 first reported on a continuous nondifferentiable function constructed by Weierstrass. Much research in mathematics is currently being done all over the world. Although we need to study and learn more before we can understand most modern mathematics, there's a lot about fractals that we can understand.” more on fractals
Look at the Koch curve, sometimes called the Koch snowflake. As you keep dividing and dividing and dividing this shape....the perimeter keeps getting infinitely larger, yet the area is finite! This is one of the mysteries of the Koch curve.
“Linear Fractals are exactly self-similar. If you look at a very small part of a fractal’s overall shape, it looks exactly like the original fractal, only smaller. We can call this size difference the scalability factor of scale. These fractals begin with a seed, the set of lines that form a basic structure. Next, you make duplicate copies of the original seed and you use them to replace the lines found in the original seed. You continue this process at greater levels by replacing line segments with seeds, whose lines in turn get replaced by seeds, so on and so on forever. Since many of these can be easily drawn they were the primary types of fractals generated before computers.” (Wahl 22).
Life Application
“Scientists use fractals. By finding the fractal ratio of growth and containment (or percolation), scientists have been able to explain, among other things. The processes by which tree populations and human populations grow and decline are also predicted. When using the fractal ratio the higher the ratio is the higher the density is. Fractals are also found within the human body. The nervous system for instance exhibits fractal patterns at both visible and microscopic levels. On a more cellular level, neurons and astrocytes display fractal patterns“. Another system of the body which demonstrates fractals is the cardiovascular system. The blood forms from the heart and travels in arteries, which eventually branch into smaller arterioles, which in turn branch into smaller arterioles, and so on, until finally reaching the capillary network. Then the process is reversed, and eventually the blood makes its way back to the heart. (Wahl 4,8).
“The process of achieving ratio percolation was first demonstrated in a fifteenth century orchard and an ancient Chinese game. In the French orchard, the monks experimented with orchard formations to get the most fruit from the trees. The first formation was in straight rows to allow the maximum amount of trees per area. However, this formation made it easy for fruit eating insects to travel from one tree to the next. The monks continued to move the trees, trying different patterns. The best pattern turned out to be a fractal shape. That way if one cluster of trees became infested, not many other trees would be affected. The object of the Chinese game was to capture as much area and as many stones on the board as possible before there is no more room left on the board. The side with the most was declared the winner. The player’s moves turned out to be percolated structures. Both the orchard and the game model territorial conquest in the real world. This demonstrated the principle of growth and containment which would be later used in science to improve the efficiency of the light bulb, monitor epidemics, model the spread of fire, and describe lightning” (Wahl 5,6).
Wahl, Bernt. Exploring Fractals on the Macintosh. Dynamic Software, 1995.