2. B. Box-Counting Dimension

Box-Counting Dimension of the Product of a Cantor Set and a Line Segment

Now consider a fractal that is a Cantor set in the x-direction and a line segment in the y-direction.
This type of construction is called the product of the Cantor set and the line segment.
To compute the box-counting dimension of this fractal, cover it with smaller and smaller boxes, keeping in mind that we take the boxes to be squares.
Square isn't essential. The important point is that the diameter of the shapes goes to 0.
For example, covering this fractal with increasingly thin rectangles, all of height 1, does not capture the scaling of the fractal.)
Here we count the number of boxes to cover the fractal.
Here we compute the exact value of the box-counting dimension.

Return to Box-Counting Dimension.