A Galilean experiment to measure a fractal dimension

Abstract

A simple and pedagogical kinematic experiment to measure the mass-size fractal dimension D of crumpled surfaces is presented. The experiment is inspired in the well-known Galilean experiments with bodies rolling on inclined planes. The experimental results for D agree with those obtained from static means.

Fractal dimension analysis with popcorn grains and popped popcorn grains

“…The fractal dimension is different from the Euclidean dimension in the way that it indicates, at which level, the object shape is far from the uniformity. Different methodologies have been proposed to introduce the concept of fractal to students such as, two dimensional picture analysis with open source software [2], analysis of structures found in the nature [3], analysis of man-made procedures [4][5][6] and analysis of dedicated [7][8][9] and sample modeling experiments [10]. In this sense, one can imagine that by adapting the simple methodology used to analyze the fractal dimension of a cauliflower [3] and different types of breads [4,5] as well as the pore space in sponges [6] and even, the fractal dimension of different types of experiments involving crumpled paper balls [7][8][9] and aggregates of tennis balls [10], it is also possible to obtain the fractal dimension of the popcorn grains before and after the popping process.…”

Moments of inertia of spheres without integration in arbitrary dimensions

“…Discs and rings are special cases of such objects with D = 2 while ordinary solid spheres and spherical shells are with D = 3. An example of non-integer dimensional spherical objects is the fractal of crumpled surfaces (2 < D < 3) obtained by random and irreversible compaction of paper sheets or aluminium foils [8]. In arbitrary dimensions, a method alternative to direct integration is highly desirable because multiple integrations in D-dimensional spherical coordinates are too difficult for introductory students.…”

Almost rolling motion: An investigation of rolling grooved cylinders

“…A few years ago an article appeared in this journal 1 which described how the fractal dimension of a randomly crumpled surface might be determined by rolling it down an inclined plane and measuring its moment of inertia I. In this paper, the claim is made that the moment of inertia of a hypersphere is given by…”