Calculating the Dimension of a Crumpled Paper Ball

A flat sheet of paper is two-dimensional. A solid ball is three-dimensional. But what is the dimension of a crumpled piece of paper? It fills 3D space, but it's mostly empty air. This object is a real-world fractal, and its dimension is typically between 2 and 3.

We can estimate this dimension using the scaling law M ∝ R^D, where M is the mass of the paper, R is its radius, and D is the fractal dimension we want to find.

How to Perform the Experiment

• Step 1: Get Your Materials. You need several sheets of paper of the same type but different sizes (e.g., A5, A4, A3) or you can cut squares of different side lengths (e.g., 10cm, 15cm, 20cm, etc.). You also need a ruler.
• Step 2: Create the Paper Balls. Take each sheet and crumple it into the tightest ball you can make with your hands.
• Step 3: Measure the Radius (R). For each ball, measure its diameter in several different directions with a ruler and calculate the average diameter. The radius is half of this average diameter. Record this value.
• Step 4: Determine the "Mass" (M). Since standard paper has a uniform density, its mass is directly proportional to its area. You can simply use the paper's original area as its "mass." For example, an A4 sheet (21cm × 29.7cm) has an area of 623.7 cm². An A5 sheet is half of that, and an A3 is double. If you cut squares, the area is simply side length². Record this value.
• Step 5: Enter Your Data. Go to the "Data Analysis Lab" tab and enter your pairs of (Radius, Mass/Area) into the table.

No Materials? No Problem!

If you don't have the materials, you can load a sample dataset to see how the analysis works.