Before Fractal Geometry was officially constructed and defined, it was used in many aspects of the world. Many of these areas that use Fractal Geometry are computer science and computer graphics, medicine, engineering, and architecture.
Most computer scientists apply what is called the Mandelbrot set for their work. This is a type of fractal that was generated by computers and is used in graphics everywhere.
When it comes to computer graphics, scenes on Star Wars and Star Trek were created with Fractal Geometry. Artists could not use any other type of natural graphics because they wanted to create worlds and differenct lands that did not look like what we see every day, this was only possible by using fractals. You can see the roughness that the Star Destroyers in Star Wars have. This is applied Fractal Geometry. 
The Genisis Effect used in Star Trek II- The Wrath of Khan was created with fractals. See the video below.
More well known fractals are the Golden rectangles or the Golden spiral which are derived and showcased in the Fibbonaci sequence. While it is not popularly known as a fractal, it is the most used fractal in architecture and civil engineering. Architects have been using the Golden rectangles and the Golden spiral for centuries11. It can be seen throughout Greek and Roman architecture and even the buildings that we see around us being built today. Architects often add their own fractals in their work. An example is the British Museum. The ceiling is constructed with a fractal like pattern and when the light shines through it, the pattern fills the whole museum. 
Structures build with fractals are known to be the most beautiful buildings. This is because of the symmetry and self-similarity that is vital for fractals.
Use the applet below to construct your own Fibbonaci sequence fractal and Golden Rectangles.
Fractal Geometry is also used in the observation of soil samples, bacteria growth, anteni, and more things that possibly have not been discovered yet. One way that modern medicine applies fractals is to see if something is malfunctioning in the body or not. The body is full of fractals so with using fractal geometry, doctors can quantify, describe, diagnose, and have the hope to help cure diseases. The picture below is of different blood vessels and if they are normal or not. You can see that the normal blood vessel is the most fractal like. This is what doctors look for.
Now, there is only one question left. What can you create using fractals?
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