The Box Fractal can be formed using the cluster fractal generation method or generator iteration.

CLUSTER METHOD

Fractal clusters are formed by repeatedly cutting out parts of a 2-dimensional figure. We can start with a square, divide it into 9 smaller squares, and cut out the center side squares:

We then do the same with every one of the 5 squares, and continue the process infinitely. At the end, we get the Box Fractal:

GENERATOR ITERATION

The Box Fractal can also be formed as a base-motif fractal using generator iteration with the following base and motif:

FRACTAL DIMENSION

The fractal dimension of the Box Fractal can be very easily found using the similarity method. If we look at this fractal, we can see that it is composed of 5 identical figures, the side of each of which is 1/3 of the entire figure. The dimension is thus log 5 / log 3, or approximately 1.46. We can also easily measure the dimension using the box-counting method, since this fractal can be easily placed on graph paper:

PERIMETER

We can very easily find the perimeter of this fractal. When doing generator iteration, during every iteration we substitute every line segment with a figure that is 5/3 its length. After an infinite number of iterations, the perimeter becomes infinitely large, just like all fractal curves.

LENGTH

Since the box fractal is composed of lines, we can find the total length of the line segments. First, you should see that it is possible to form the Box Fractal by continuously crossing lines like this:

We can find the length of each of the long diagonal lines to be sqr(2) using the Pythagorean Theorem. The length of 2 of them together is twice that, or approximately 2.828. You should also see that the lengths of the 4 smaller lines are 2/3 of that. If we form the fractal by crossing lines, we will add a number that is 2/3 the previous one every time. The formula we can use for this is the sum of an infinite geometric sequence formula, S = a / (1 – r), where a is the first number and r is the common ratio. In our case the sum is 2.282 / (1 – 2/3), or approximately 8.485.