The T-square Fractal

Figure 1: source

Yesterday (when I was young) I came across the T-square fractal in the context of turning 26598 days old. Figure 1 shows a screenshot of my Twitter tweet for the day. The OEIS entry runs as follows:

A227621

The nearest integer of perimeter of T-square (fractal) after n-iterations, starting with a unit square.

The initial members of the sequence are:

4, 8, 14, 23, 37, 57, 87, 133, 201, 304, 457, 688, 1034, 1553, 2331, 3499, 5251, 7878, 11819, 17731, 26598, 39899, 59851, 89778, 134669, 202005, 303010, 454517, 681778, 1022668, 1534004, 2301009, 3451515, 5177275, 7765914

Figure 2, taken from the Wikipedia entry, shows the initial steps in the creation of this fractal:

Figure 2: source

Figure 3 shows further details of the process:

Figure 3: source

Starting from an initial unit square, the fractal is bounded by the square with a side of two units since:$$ \sum_{n=0}^{\infty} \frac{1}{\,2^n}=2$$As the area of the fractal gets closer and closer to 2, the perimeter gets longer and longer. After 20 iterations, the perimeter is 26598 units in length. As this source explains:

The fractal dimension is the ratio between the "size" of the object and its "area". For example, a simple area has a fractal dimension of 2, which means that if you make it growth by \(x\), its area will be multiplied by \(x^2\). A fractal with a dimension of 1.5 will have its area multiplied by \(x^{1.5}\).

In the case of the T-Square, it has a dimension of 2 because its area nearly entirely fills the inner space within it. However, the dimension of its boundary is \( \frac{\log 3}{\log 2} \approx 1.58\).

OnlineMathTools enables the creation of a variety of fractals, including the T-square fractal, that can be customised in various ways. Figure 4 shows an example of such a fractal.

Figure 4: T-square fractal after 4 iterations