# Fractal dimensions of crumpled paper

(1) The Peddie School, Hightstown, New Jersey, (2) Embry-Riddle Aeronautical University, Daytona Beach, Florida

https://doi.org/10.59720/21-261

Fractals are infinitely complex shapes that can be split into two types: mathematical fractals and statistical fractals. Mathematical fractals are infinitely complex and self-similar across multiple scales, whereas statistical fractals are not self-similar but can be applied to real life. Crumpling paper is a mathematical fractal process whereby a sheet of paper undergoes deformation via compression. This yields a three- dimensional structure that introduces isotropic and homogenous spaces of varying sizes inside the crumpled paper structure. There is a hierarchy of spaces: a few large, a moderate number of medium, and many small. Consequently, a crumpled paper ball is a fractal object. Fractal dimension is a non- integer measure of an object’s “roughness” and is useful in studying natural objects consisting of many irregular shapes like crumpled paper. In this study, we determined the fractal dimension of a sheet of paper crumpled into a series of spheres of decreasing radii using continuous Χ2 analysis. The result is a tested model relating mass to radius via fractal dimension for crumpled paper, which in this study, yielded a fractal dimension of 2.40. Using Chi-Squared, we were able to test and validate our model and compare it to other experiments’ results to verify consistency, offering advantages over more conventional least squares.