As far as I know, the following fractal has a self-similar fractal dimension of

$D = -\log(3) / \log(1/2) = 1.5850$

quarter of fractal

But what is the fractal dimension of the following fractal (4 times the fractal of above)

full fractal

Does the fractal dimension change when I include the fractal in each sqare, like partially drawn in the following graphic?

fractal iteration

(The other squares, e.g. the yellow ones will also get the fractal inserted. I didn't draw it yet, since I only use Photoshop for the self-similar inclusion)

This inclusion of the fractal in each squares seems to be self-similar, but it cannot described with the self-similar fractal dimension formula, since the stretch-constant is not the same, since the squares, where the fractal is included, have different sizes.

If the self-similar fractal dimension cannot be applied, which method should I use to determinate the fractal dimension? Box counting?


Here is my non-formal construction description:

  • The central square has a size of $1^2$.

  • All subsequent squares have the half of the edge length. The distance is also half of the edge length of the previous square.

Using the distance scale down factor of 0.5 and the square scale down factor of 0.5, the resulting fractal has a perimeter of $4v$ where $v$ is $5*\sqrt{2}/2$ . Consequently, when each square is replaced by the whole set (like it is done with graphic #3), then the scale down factor of the whole fractal is $1/v$.

Here is a graphic for better explanation of the construction:

construction graphic

In the next level, each soldit square is replaced by the whole set itself, so that a true fractal is created.


1 Answer 1


It would be nice to have a precise description of the set that you are dealing with, rather than just a picture. Judging from the picture, though, I'd bet you're working with an IFS that is geometrically similar to the IFS consisting of the following three functions: \begin{align} f_1(x,y) &= \frac{1}{2}R\left(\frac{\pi}{2}\right) \left( \begin{array}{c} x \\ y \end{array} \right) + \left( \begin{array}{c} 1/2 \\ 0 \end{array} \right) \\ f_2(x,y) &= \frac{1}{2} \left( \begin{array}{c} x \\ y \end{array} \right)+ \left( \begin{array}{c} 0 \\ 1/2 \end{array} \right) \\ f_2(x,y) &= \frac{1}{2}R\left(-\frac{\pi}{2}\right) \left( \begin{array}{c} x \\ y \end{array} \right)+ \left( \begin{array}{c} 1/2 \\ 1 \end{array} \right), \\ \end{align} where $R(\theta)$ is the $2\times2$ rotation matrix through the angle $\theta$. The image of a set $A$ under this IFS is simply $$ \bigcup_{i=1}^3 f_i(A). $$ An oriented unit square, together with it's image under this IFS, looks like so:

enter image description here

If you start with a solid unit square and apply the IFS iteratively, you generate a sequence of images that looks like so:

enter image description here

If we iterate this 10 times and color the pieces, we get an image like so:

enter image description here

Let's call this set $S$, which is, indeed, a self-similar set, without overlap, consisting of 3 copies of itself scaled by the factor $1/2$. As such, its similarity dimension is, as you say, $\log(3)/\log(2)$.

However, it appears to me that your images are generated using a technique called condensation. This simply means that, rather than just applying the IFS to generate a new set, we retain the previous set as well. If we do this starting with a small, rotated square centered at $(1/2,1/2)$, we generate a sequence of images like so:

enter image description here

And, after a few more iterates:

enter image description here

Lets' call this set $T$, which I guess is your picture??

Now, some answers. Though, $\dim(S)=\log(3)/\log(2)$, this is not the case for $T$. $T$ contains solid squares and, as such, it has full dimension 2. Your 4-fold versions should have the same dimension as the set you start with, as they are just the union of the original set with a rotated copy of that set.

Finally, computation of the fractal dimension of your last set is a non-trivial matter. I believe that it is most easily described using a digraph iterated function system. Assuming so, I generated the following image:

enter image description here

Note that the central square is a copy of the whole scaled by the factor $1/4$. There is overlap between the pieces, which makes the computation of the dimension difficult. Assuming the central scaling factor is a power of $1/2$, the dimension can be computed using the techniques defined in this paper though, again, this is not particularly easy. Using those techniques, though, I computed the dimension of the last picture to be $$\frac{\log{\lambda}}{\log{4}} \approx 1.75829,$$ where $\lambda\approx11.44$ is the largest, positive root of $x^3-11 x^2-5 x-1$.

  • $\begingroup$ Thank you very much for your great and detailled answer! I did create this fractal/set $T$ using a non-formal construction description, while I learning about fractals; I did manually draw it using a small self developed program. I am sorry that I forgot to tell the non-formal construction description in the question. I didn't hear the terms IFS and condensation before; I will need to learn much more about this topic. $\endgroup$ Dec 31, 2014 at 2:03
  • $\begingroup$ In my version, the central square has a size of $1^2$. All subsequent squares have the half of the edge length. The distance is also half of the edge length of the previous square. (you have used a distance reducation of approximately 0.75, while I used 0.5). Using the distance 0.5, I had a scale down factor of $1/v$ where $v$ is $5*\sqrt{2}/2$ , for the nesting of fractals. If I am not mistaken, the perimeter of the whole set $T$ is $4v$, if the central square has the area of $1^2$. $\endgroup$ Dec 31, 2014 at 2:04
  • $\begingroup$ I have a few additional questions: 1. You said, there is overlap between the pieces. Where is the overlap exactly, and does it only affect the digraph iterated function system, or also my non-formal definition? In the rendering of set $T$, I did not notice that the squares will touch each other. In my last picture, each square has been fully replaced by the whole fractal (using Photoshop, due to lack of software), so there should also be no overlapping. 2. Please tell me, which software did you use to create such great graphics? $\endgroup$ Dec 31, 2014 at 2:05
  • $\begingroup$ 3. will the calculated dimension of $1.75829$ differ, if a different distance scale down factor is used? (since you probably used 0.75, while my version has a scale down factor of 0.5 between the squares) $\endgroup$ Dec 31, 2014 at 2:14

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