2016/01/24

The fractal dimension of crumpled paper

This is my homework for the class "Complex Fuids" I take during this quarter:

You see six paper balls of different sizes starting approximately with a diameter of 0.5 cm up to 4.5 cm (the last one at the bottom). They are all made of the same paper.

Why do we care about paper balls of different sizes of the same material? Well, of course it's just a demonstration how to determine the "fractal dimension" of a sample. Indeed, the ball lives in three dimensions as we look at it, but in which direction can you walk if you're on the ball itself? And how complex is the paper crumpling compared to the pattern of a snow flake?

To measure this degree of complexity we use the fractal dimension which is e.g. 3 for a normal filled sphere (until you leave the sphere, then there's no shape any more) or 2 for a flat piece of paper etc. For fractals it happens to be a non-integer value, a fraction. Mathematically the fractal Dimension D is just m~r^D with the mass m and the distance to an origin point r. I obtained the fractal dimension of the balls by comparing the increase in diameter to the size of the paper I used for it, plotted the graph and fitted the data. The fractal dimension turned out to be around 2, with an uncertainty of 0.5 ;)