Fractal topology of hand-crumpled paper

We study the statistical topology of folding configurations of hand folded paper balls. Specifically, we are studying the distribution of two sides of the sheet along the ball surface and the distribution of sheet fragments when the ball is cut in half. We found that patterns obtained by mapping of ball surface into unfolded flat sheet exhibit the fractal properties characterized by two fractal dimensions which are independent on the sheet size and the ball diameter. The mosaic patterns obtained by sheet reconstruction from fragments of two parts (painted in two different colors) of the ball cut in half also possess a fractal scale invariance characterized by the box fractal dimension DBF=1.68 ± 0.04 , which is independent on the sheet size. Furthermore, we noted that DBF, at least numerically, coincide with the universal fractal dimension of the intersection of hand folded paper ball with a plane. Some other fractal properties of folding configurations are recognized.