What are Fractals? In addition to being really cool pictures/videos to look at, fractals are mathematical sets that demonstrate a repeating pattern at every scale. Each scale can also be almost the same as other ones. They may not be exactly the same, but they are the same type of structure. Although a fractal could also include a design that repeats itself. Fractals have what's called a fractal dimension. This is not easy to explain but here goes. When you double the edge lengths of a polygon, the area of the polygon increases by four times. If the radius of a sphere is doubled, the area volume by 8 times. When side lengths of a fractal are doubled, the power at which volume is increased is not necessarily an integer. That power is called the fractal dimension. So a fractal dimension is essentially a ratio that can indicate how detail in the fractal pattern changes with the scale at which it is measured or looked at. The closer you look at a fractal, the amount of detail changes. That amount depends on the fractal dimension. The higher the fractal dimension, the more detail/ the fractal gains when side lengths are increased. Knowing that, you can imagine what happens when the fractal dimension is decreased.
Why did you choose fractals for your final project? I chose fractals mainly because they're cool to look at and it seemed very interesting to me how they can continue and have the patterns they have. The fact that this is mathematically possible is intriguing to me.
What are some real life applications of fractals? Fractals are found in many places in nature, including certain plants, sea shells, lightning, and even in human blood vessels and lungs. Humans are using fractals to create things as well. Currently, engineers are working on using fractals to solve the problem of liquid transport. They're being designed into computers to distribute coolant throughout them.
A brief history of fractals? Humans began to partially understand fractals in the 17th century. Gottfried Leibniz initially made some discoveries that weren't far from accurate. It wasn't until 1872 that Karl Weierstrass presented the first definition of a function with a graph that would today be considered fractal. In 1883, Cantor sets were invented, which are now recognized as fractals as well. "self-inverse" fractals were also introduced around that time period. Modern computer technology has given us a helping hand in understanding fractals. This is something these people didn't have when this was discovered. Little did they know these things could create some sweet images.
I used two of Archimedes' spirals to create a visual "background" of sorts. I just manipulated a couple of constants along with the slider speed to create a sort of ambiance in the background for the other equation. For the other equation it got a little more complicated. r=a*sin(b/ctheta) was the equation I used. I then let the slider go and it did all sorts of fun things. The combination of the three really makes for a show. This allowed me to better understand the effect adding certain constants can have on a graph. It's good to know how you can manipulate certain aspects of the graph by adding certain numbers or constants in certain spots.