FRACTALS

“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.” -Mandelbrot, 1982

What is a fractal?

Fractals are non-standard geometric shapes that do not perfectly fit into the world of Euclidean geometry. Fractals are often used to describe objects in nature, such as coastlines and mountains. A fractal, by definition, is “a rough or fragmented geometric shape that can be split into parts, each of which is a reduced-size copy of the whole”. In traditional geometry, objects are considered to be one-dimensional, two-dimensional, three-dimensional and so on, in integer dimensions. Some examples would be one-dimensional lines and curves and two-dimensional plane figures like triangles and circles. However, many natural objects are better described using a dimension between two whole numbers. So, fractals are considered to have non-integer dimensions. If an object was taken with its linear size equal to 1 in Euclidean dimension, D, and then reduced in its linear size by 1/r in every spatial direction for that dimension, then it takes N=r^D number of self-similar objects to cover the original object. By taking the log of both sides, the equation becomes log(N)=D(log(r)). Then, D = log(N)/log(r). If we use the Sierpinski Triangle (figure shown below) as an example, we can see that D = log(3)/log(2), which is equivalent to 1.585, a non-integer.

All fractals have two major properties: self-similarity and scale-independency. Self-similarity is the idea that as an object is magnified the original shape of the whole is repeated over and over again. The self-similarity of the Sierpinski Triangle (created on Geometer’s Sketchpad) is evident by the repetition of the shape within the original triangle. Scale-independency is the idea that by zooming in on a fractal a person cannot determine whether they are looking at the smallest or largest part of the shape, since the shape is being repeated.

History

17^th century: The mathematics of fractals began to be formed as Liebniz studied the idea of self-similarity.

1872: Much time went by while mathematicians avoided these unfamiliar ideas, until Karl Weierstrass presented a function whose graph could be considered a fractal.

1883: Georg Cantor published the “Cantor Sets”.

1904: Helge von Koch is known for his research involving matrices, but in 1904, he published a paper based on the idea of adding triangles onto the sides of other triangles, creating a Koch curve.

1915: A Polish mathematician named Waclaw Sierpinski, described the fractal that we now know as the Sierpinski Triangle.

1975: Benoit Mandelbrot, also known as the “Father of Fractal Geometry”, first used the term fractal. The word fractal comes from the Latin verb frangere, meaning “to break,” and the Latin adjective fractus, meaning “broken and irregular.”

Fractals in Mathematics

There are many types of fractals, and this essay will seek to group them by the ways in which they were created.

IFS Fractals: IFS stands for Iterated Function System. These fractals are created by applying a function (chosen randomly from the rules set up for the IFS) repeatedly to an initial point, and then graphing each new point. One of the most well known of these fractals is the spleenwort fern (shown below). Each of the leaves is an image of the whole fern.

(Picture Source: http://www.home.aone.net.au/~byzantium/ferns/fractal.html )

Base-motif fractals: These fractals are created by taking any shape, called the base, and another shape, called the motif, and substituting every line segment of the base with the motif.

The simplest of the base-motif fractals is the Cantor Set, introduced by Georg Cantor in 1883. It is formed by removing the middle one third of a line segment, and repeating the process for the resulting lines an infinite number of times. The Cantor Set has the property of self-similarity (even though self-similarity was not defined until 1905), since each horizontal line segment is one-third of the horizontal line segment directly above it.

(Picture Source:http://www.nilesjohnson.net/point-set-topology-topics.html )

Similar to the Cantor Set, the Koch curve is formed by creating an equilateral “tent” after removing the middle third of a line segment. Interestingly, Koch’s curve was not originally thought of in terms of fractals. Koch sought to find a way to prove that functions that were non-differentiable could exist.

Peano Curves are also considered base-motif fractals. The name is given to any fractal whose fractal dimension is 2. In other words, the final shape of these fractals is two-dimensional, occupying area on a plane. Inspired by Cantor, Giuseppe Peano discovered a self-intersecting curve that passes through every point in the unit square. Because he was the first to discover a space-filling curve in two-dimensions, these fractals are named after him.

(Picture Source: http://mathworld.wolfram.com/PeanoCurve.html )

Complex Number Fractals:

Julia Sets:

French mathematician Gaston Julia picked a fixed complex number, c, and studied the function 
f_c(z) = z² + c putting in different values of z. It was named around 1920, but since Julia predated the technology of computers, he couldn’t generate pictures like the one shown below.

(Picture Source: http://www-history.mcs.st-and.ac.uk/HistTopics/fractals.html )

Mandelbrot Set:

Around 1967, mathematician Benoit Mandelbrot, with the aid of computers, came up with the idea of mapping the values of c ∈ C for which the Julia set for the function f_c(z) = z² + c is connected. The function separates the points of the complex plane into points inside the Mandelbrot Set and points outside of the Mandelbrot Set. In the image below, the points of the Mandelbrot Set are colored black. One can see that the Mandelbrot Set is self-similar, by zooming in on the edges.

(Picture Source: http://www-history.mcs.st-and.ac.uk/HistTopics/fractals.html )

Fractals in Nature

Most people encounter a fractal every day without even realizing it. There are a large number of examples of fractals in nature, such as seashells, snowflakes, mountain ranges, coastlines, lightning, clouds, trees and leaves, and broccoli. Mandelbrot wrote an essay entitled “How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension”, in which he linked the idea of previous mathematicians to the real world, specifically coastlines, which he claimed were "statistically self-similar".

(Pictures Source: http://webecoist.momtastic.com/2008/09/07/17-amazing-examples-of-fractals-in-nature/ )

Uses of Fractals

Computer Science:

One of the largest uses of fractals today is in Computer Science. Computer graphic artists use fractal image compression to create textured landscapes. In “Return of the Jedi”, fractals were used to create the geography of the moon.

Bacterial Growth:

Similar to trees, bacteria spread in branch patterns, which can be modeled using fractals. Tohey Matsuyama and Mitsugu Matsushita performed an experiment in which the fractal dimension of a culture of Salmonella anatum was found to be about 1.77

Human Anatomy:

The pulmonary system, which we use to breathe, is made up of tubes through which air travels into sacks called alveoli. The main tube, the trachea, splits into two smaller tubes that leads to different lungs, which are in turn split into different tubes. This continues until the smallest tubes lead into the alveoli. This is only one example of fractals in the body. Other structures include: arteries, the brain and membranes. The human heart beats in a fractal rhythm and doctors can detect medical problems, like heart disease, by abnormal or extreme fractal beating

(Picture Source: http://library.thinkquest.org/26242/full/ap/ap11.html)

Music:

The patterns created in songs can be considered types of fractals. Using fractal patterns distinguishes noise from music. In addition, different patterns make up different types of music, which allow us to hear different styles like rock and country music.