For this week’s response, I will be writing more of an informative format about fractals. Wikipedia defines a fractal as “natural phenomenon or a mathematical set that exhibits a repeating pattern that displays at every scale.” Those of who are familiar with the movie Inception, the concept of fractal is similar to the movie’s dream within the dream, as the fractal contains a shape within a shape, which is self-similar and repeats infinitely.
I will write about different types and examples of fractals, the mathematics behind fractals, and applications of fractals.
First, there are four different types of fractals which are Base-Motif, Dust & Cluster, Canopy, and Paper Folding fractals.
For base motif fractals, base is any shape that is composed of line segments. Base usually is line, square, equilateral triangle. Motif is another shape that is composed of line segments. To make base-motif fractals, substitute every line segment in the base with the motif.
Above image is progression of a Base-Motif fractals. Dust and cluster fractal is similar to a base motif fractal with base. The motif and base are the same shape, and can be formed by cutting out parts of a 2-dimensional figure. Below image shows how the dust and cluster fractal is developed.
The canopy fractal is a bit simpler than the other ones as it only takes splitting a line into two or more smaller segments, which resembles a branch of a tree. Below image is an example of canopy fractal.
And lastly, for paper folding fractal, I will attach an image of how it is made.
Next, I will discuss the basic mathematics that is related to fractals. The most fundamental mathematical property of a fractal is that it is not differentiable. In other words, one cannot draw a line which is tangent to the surface of the fractal. Weierstrass function is similar to fractal since it is continuous, but it is not differentiable because zooming into the graph creates its own self similarity. The below image shows the Weierstrass function.
Lastly, I will write about applications of fractals. There are many different fields and disciplines that fractals can be applied to, which includes computer science, fluid mechanics, telecommunications, structural engineering, and so forth. For computer science, fractal is used for the image compression, which allows the picture to be enlarged without having the image pixelated. For fluid mechanics, fractals allow engineers and physicist to model turbulence flow, which is very complex and chaotic. For telecommunications, the fractal-shaped antenna allows its size to reduce greatly by having the fractal parts produce ‘fractal loading’ which makes the antenna smaller for a given frequency of use.
Fanks for reading!
Woongkee (Sean) Min