Alright, so for my final project for any math class I'll ever take in high school ever, I decided to take a look at fractals and how they relate to math.
First, I guess I should discuss what a fractal is. A fractal, according to the love of my life, Google, "A fractal is a never-ending pattern. Fractals are infinitely complex patterns that are self-similar across different scales. They are created by repeating a simple process over and over in an ongoing feedback loop. Driven by recursion,fractals are images of dynamic systems – the pictures of Chaos."
So essentially, a fractal is a perfect image. When you zoom in on any part of the fractal, a perfect copy of the larger image is in the smaller one. (that doesn't make sense but i'll find a gif for it)
So, how does this relate to mathematics? Basically, all a fractal is, is just a very complex geometric design that seems to repeat itself within itself. But it's not just any normal geometric design we've looked at in Pre-Calculus, the difference between a fractal and a geometric design, is that a fractal has dimensional scaling.
One of the first fractals ever discovered is called the Koch Snowflake (discovered in 1904) which was found by taking an equilateral triangle, dividing the sides into threes and using the middle segment as a base as another equilateral triangle on each side. (shown on the left of the gallery on the bottom) On the right of the Koch Snowflake, is the equation for the snowflake. Because the triangles expanding on each other, the area of this is infinite.
Another cool thing about fractals is how easy they are to make with today's modern technology. By simply googling "Fractal maker" I found a bunch of websites that you can easily begin making a fractal that may never be found again because every fractal is unique.
http://sciencevsmagic.net/fractal/#0220,0162,2,4,0,0,3
First, I guess I should discuss what a fractal is. A fractal, according to the love of my life, Google, "A fractal is a never-ending pattern. Fractals are infinitely complex patterns that are self-similar across different scales. They are created by repeating a simple process over and over in an ongoing feedback loop. Driven by recursion,fractals are images of dynamic systems – the pictures of Chaos."
So essentially, a fractal is a perfect image. When you zoom in on any part of the fractal, a perfect copy of the larger image is in the smaller one. (that doesn't make sense but i'll find a gif for it)
So, how does this relate to mathematics? Basically, all a fractal is, is just a very complex geometric design that seems to repeat itself within itself. But it's not just any normal geometric design we've looked at in Pre-Calculus, the difference between a fractal and a geometric design, is that a fractal has dimensional scaling.
One of the first fractals ever discovered is called the Koch Snowflake (discovered in 1904) which was found by taking an equilateral triangle, dividing the sides into threes and using the middle segment as a base as another equilateral triangle on each side. (shown on the left of the gallery on the bottom) On the right of the Koch Snowflake, is the equation for the snowflake. Because the triangles expanding on each other, the area of this is infinite.
Another cool thing about fractals is how easy they are to make with today's modern technology. By simply googling "Fractal maker" I found a bunch of websites that you can easily begin making a fractal that may never be found again because every fractal is unique.
http://sciencevsmagic.net/fractal/#0220,0162,2,4,0,0,3
Aside from mathematics, surprisingly fractals can appear in nature. They actually do quite often in snowflakes, seashells etc. The reason for this is near accidental but it's all in the cells of what makes up the thing. They naturally repeat, thus causing a fractal in nature. The thing is, that when the cells repeat, they don't all repeat the exact same way for everything. which is why every snowflake is unique and every seashell is the same way.
The reason behind why I chose this topic over everything else I could have, is because the day before this project was assigned, we had a full class discussion about fractal in my novel class, and I thought they were neat. But I didn't learn anything in that class except that they were cool. So when this was assigned, I was like "Ayyyy" and so I spent several days in math on a computer learning about fractals. Ay badda-bing I learned about fractals.