Lesson 6. How Holey Is the Swiss Cheese?

Objective:
We say a line segment is one-dimensional, a triangle or square is two-dimensional, and a pyramid or cube is three-dimensional. This is because intuitively dimension has something to do with the number of distance measurements needed to specify the size of an object in the Euclidean world we live in. But, alas, what is the dimension of a fractal object that is fractured and scattered in space. We therefore resort to a definition of dimension based on the concept of capacity, that is, how much space an object actually takes up in reality. First, the capacity definition is applied to a line, triangle, and cube to recover the Euclidean dimensions 1, 2, and 3, respectively. We then find that fractal dimension d is not necessarily a whole integer, but can take on any value between the integers, as shown.

Here are dimensions of the objects that we compute in this lesson.

Scary words:  Fractal dimension, Logarithmic function, log-log plot.

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