Fractals: the Koch Snowflake
Executive Summary
"Clouds are not spheres, mountains are not cones, coastlines are not circles and bark is not smooth, nor does the lightning travel in a straight line."
- Benoit Mandelbrot
Beautiful patterns surround us. You can see them on trees, clouds, on bodies of water. You can even see them on plants, on animals and on our very skin. The very tips of our fingers prove just that. There is also no doubt that patterns are just as mysterious as they are beautiful. In fact, there are some patterns that are so perfect that they self-replicate. To be technical, some patterns are fractal in nature. Fractal or not, patterns give us something more to admire and wonder about.
Introduction
Fractals never fail to fascinate. If you aren't just gazing at their unearthly beauty, you ponder the mathematics behind them... and then you can't help but wonder how such prosaic, unsensational mathematical formulae can give rise to such intricacy. What is it that makes it possible for (to some) a short, ugly equation to generate the exuberant beauty of the Mandelbrot set? Or is it all just in the way our brains are wired?
Fractals are objects with infinite lengths that occupy finite volumes, resulting in a "fractional dimension" that is not 1-, 2-, or 3-D, but a combination of all three, depending on its spatial configuration.
The Koch snowflake is the repetitive procedure of dividing the image into three equal parts and replacing the middle piece with two similar pieces.
Hypothesis
Fractals mimic nature. (true or false)
This is the basic belief of fractals, and a common concept among those who study fractals. In nature, symmetry is often remarked upon. To mimic is to be similar in to a certain object, and in this case, of a lesser proportion. Thus, we would like to propose that fractals may mimic nature.
Definitions
Fractals
1. A curve or geometric figure, each part of which has the same statistical character as the whole. 2. Any of various extremely
"Clouds are not spheres, mountains are not cones, coastlines are not circles and bark is not smooth, nor does the lightning travel in a straight line."
- Benoit Mandelbrot
Beautiful patterns surround us. You can see them on trees, clouds, on bodies of water. You can even see them on plants, on animals and on our very skin. The very tips of our fingers prove just that. There is also no doubt that patterns are just as mysterious as they are beautiful. In fact, there are some patterns that are so perfect that they self-replicate. To be technical, some patterns are fractal in nature. Fractal or not, patterns give us something more to admire and wonder about.
Introduction
Fractals never fail to fascinate. If you aren't just gazing at their unearthly beauty, you ponder the mathematics behind them... and then you can't help but wonder how such prosaic, unsensational mathematical formulae can give rise to such intricacy. What is it that makes it possible for (to some) a short, ugly equation to generate the exuberant beauty of the Mandelbrot set? Or is it all just in the way our brains are wired?
Fractals are objects with infinite lengths that occupy finite volumes, resulting in a "fractional dimension" that is not 1-, 2-, or 3-D, but a combination of all three, depending on its spatial configuration.
The Koch snowflake is the repetitive procedure of dividing the image into three equal parts and replacing the middle piece with two similar pieces.
Hypothesis
Fractals mimic nature. (true or false)
This is the basic belief of fractals, and a common concept among those who study fractals. In nature, symmetry is often remarked upon. To mimic is to be similar in to a certain object, and in this case, of a lesser proportion. Thus, we would like to propose that fractals may mimic nature.
Definitions
Fractals
1. A curve or geometric figure, each part of which has the same statistical character as the whole. 2. Any of various extremely