Paper Ball and Bean Bag Dimensions

Answers

6. The crumpling process generates spaces of a range of sizes, a characteristic of fractals. This can be seen by slicing paper balls in half. Note the wide range of gap sizes in these three pictures. Thanks to Hemesh Rama for these pictures. Click each picture for an enlargement in a new window.

The edge revealed by slicing can be thought of as approximating the intersection of the paper ball and a plane. Consequently, the dimension of the slice cannot exceed 2. In more detail, denote the paper ball by B and the plane by P. Then the slice is B ∩ P. Can we find the dimension of B ∩ P in terms of the dimensions of B and P? Suppose X and Y lie in Eucidean $N$-dimensional space. For typical placements of X and Y we have

N - dim(X ∩ Y) = (N - dim(X) + (N - dim(Y)

Think of two planes in 3-dimensional space, or of a plane and a line in 3-dimensional space, to understand this formula. For the paper ball and plane we have N = 3. Solving for dim(B ∩ P) gives

dim(B ∩ P) = dim(B) - 1

For example, if a paper ball of dimension about 2.5, the slice has dimension about 1.5.

Return to Exercises