Fractals

Fractals are described as “a geometrical figure in which an identical motif repeats itself on an ever diminishing scale (Lauwerier, xi).” Fractals are both man-made and found in nature, for example snowflakes. Fractals go beyond just patterns and have been utilized in technology and architecture. On that account, fractals can play several important parts in our daily lives.

Although fractal geometry only dates back to 1975, it is important to know that work on fractals started around the 17th century. It began to take shape when Gottfried Leibniz contemplated recurrent self-similarity. However, he made the mistake that a line was self-similar in this light. Famous mathematicians have added to this research like George Cantor. He created sample subsets with unusual characteristics called cantor sets, which are now recognized as fractals.  Pierre Fatou and Gaston Julia added work by mapping complex numbers and iterative functions, but it “led to no illustration in their time (Barnsley, 2).” Without the help of computers, these researchers were limited to manual drawings, impeding greater understanding for its beauty and its implications. This was changed by Benoît Mandelbrot, who helped piece together many years of hard work on the topic. He used computer visualizations to help solidify his definition of fractals, a word he also coined.

Many fractals can be found in nature. In the book “Fractals” by Hans Lauwerier, he provides an excellent example that is easy to perceive. Imagine a tree with a trunk that separates into two branches. In turn, these two branches separate each into two smaller branches, and so on. This can be infinitely repeated in our heads. “Each individual branch, however small, can in its turn be regarded as a small trunk that carries an entire tree (Lauwerier, xii).” This can be illustrated in the picture down below.

This construction has to do a lot with the binary system. The picture above, conceptually, has an infinite number of branches extending. To make a more simplistic-looking, and mathematically based tree, we would use only vertical and horizontal lines. First you start with a “T” shape, and each branch extending is a “T.” At every level the vertical lines would separate into two, scaling down by a ½ each time. These number of these vertical lines would double at each level. Moreover, the horizontal lines would be twice the length of the vertical segment below it. To start, if when began by giving the first line, the trunk, a length one, then the equation to continue on would be: 1x1 + 2x½ + 4x¼ + 8x⅛ +...

Fractals are also found in man-made structures. An example of this being used back in the day are old rope bridges. The Icans wove long lengths of stiff qoya grass by hand. These get woven into larger fibers, which then help create a stronger a sturdier rope. This same idea also applies to modern bridges like the Golden Gate Bridge. The giant steel cables are formed from a bundle of smaller cables, that, additionally, are constructed from smaller bundles. Another instance in where fractals are utilized in buildings and structures is the Eiffel Tower. On page 131 in Mandelbrot’s book “The Fractal Geometry of Nature,” he describes how “the tower that Gustave Eiffel built in Paris deliberately incorporates the idea of a fractal curve full of branch points.” He goes on further to explain that the tower is not made of solid beams but out of “colossal trusses.”  “A truss is a rigid assemblage of interconnected sub-members, which one cannot deform without deforming at least one submember. Trusses can be made enormously lighter than cylindrical beams of identical strength (Mandelbrot, 132).” You can see how the truss below is repeated through the structure. However, this wasn’t a new discovery that Eiffel had discovered. The idea that strength lies in branch points was exercised by designers of gothic cathedrals. As visualized in the Milan Cathedral, one can see the use of large arches interlocked

with smaller arches. The use of fractals helps give these buildings their beauty and strength.

An instance in where fractals are used in technology is in radio and antenna signals. A professor at Boston University, Nathan Cohen became inspired by the Koch Snowflake, a famous fractal construction, to create a more compact radio antenna only using wires and pliers. Today, antennae in cell phones use such fractals as a way to maximize receptive power in a minimum amount of space. The pattern’s compactness is what allows this strong signal.

In Geometry, fractals can generate amazing patterns and works of art. One type of fractal based in geometry is a “ternary” tree. We first start with a point and three protruding branches. Each branch creates a 120° with each other. Then each branch gets another three branches, ⅓ of the length of the previous one. This keeps on going to infinity until it creates this triangular looking snowflake.

In 1913, Polish mathematician Wacław Sierpiński created a nice variation of the ternary tree. It is obtained by starting with a solid equilateral triangle. The next step is to divide this into 4 smaller equilateral triangles by removing one in the middle. With the three remaining solid shapes, we repeat the same step and remove a triangle from each of them, making each triangle have 4 smaller ones. This is repeated indefinitely.

Looking at fractals at a surface level, it may seem that they serve no purpose beyond creating patterns and art. However, they can be exploited to created extraordinary things that help out in our world today.

Lauwerier, H. A. Fractals: Endlessly Repeated Geometrical Figures. Princeton, NJ: Princeton UP, 1991. Print.

Barnsley, M. F. Fractals Everywhere. Boston: Academic, 1988. Print.

Peitgen, Heinz-Otto, Dietmar Saupe, and M. F. Barnsley. The Science of Fractal Images. New York: Springer-Verlag, 1988. Print.

Mandelbrot, Benoit B. The Fractal Geometry of Nature. New York: Freeman, 1983. Print.