What pushes something towards fractal-ity?
In: 0
The most common fractals you see are the ones that repeat themselves forever. This means a part of it, like one of the three triangles making up a larger triangle in the Sierpinski triangle is just like the larger shape. However, this infinite repetition is not what makes something a fractal. What being a fractal actually means is it is a fraction of a dimension. To understand this think of a triangle made out of just the edges and a triangle completely filled in. The one with only edges is made of lines so you can think of it as one dimensional. The filled in triangle is on a flat plane so it can be thought of as two dimensional. Now think of the Sierpinski triangle. It’s made out of only lines however it seems to fill up the space inside the triangle somehow. This is not measurable normally since no matter how far you zoom into the Sierpinski triangle there is more to measure. So a mathematician (filix hausdorff?) thought of a way to assign a number to indicate how much space is filled up by these never ending lines and called them fractals. So what makes something a fractal is no matter how far you zoom in to measure there is always more to measure.
TLDR: what makes a fractal a fractal is that no matter had far you zoom in you can not measure it because it goes smaller forever.
There is a very nice video by 3blue1brown that goes over this. It’s worth a watch! [fractals are not typically self similar](https://youtube.com/watch?v=gB9n2gHsHN4&pp=ygUVRnJhY3RhbHMgc2VsZiBzaW1pbGFy)
Fractals are objects that are “rough” however far you zoom in on them. The easiest way to see this is in ‘traditional’, self-similar fractals such as the Sierpinski triangle or the Koch curve. However, this “rough”ness can also be seen in things like the coastline of Britain or Norway. The “rough”ness of a fractal can be quantified in a number called the Hausdorff dimension, which extends the concept of integer dimensions to fractional dimensions. For example, the Hausdorff dimensions of the Sierpinski triangle and the Koch curve are 1.585 and 1.262, respectively; the coastlines of Britain and Norway are 1.25-dimensional and 1.52-dimensional, respectively.
Latest Answers