Ytukyg

I came up with a self-similar fractal that fits into itself like a jigsaw puzzle. While the surface area is clearly 2-dimensional, it is unclear to me how to compute the Hausdorff dimension of the border. I could use the box counting method to find an approximation. But since I know how to construct the fractal, I was wondering if it were possible to exactly calculate it.

Above you can see the fractal I am talking about. I have divided it into colored areas to make its construction more clear. The red area is just a scaled down version (with scale factor 1/3) of the whole thing. The other colored areas are scaled down versions (with scale factor 1/3) as well, but are also rotated. The blue areas are rotated 90 degrees anti-clockwise. The green areas are rotated 180 degrees. The yellow area is rotated 90 degrees clockwise. The center of the fractal is inside the big gap area, which could be filled with the red area. If this gap area gets filled recursively with smaller and smaller red areas, then the center will remain outside the fractal after any finite amount of iterations. It may not look like it, but the 9 colored areas are placed in such a way that their centers form a grid pattern. To make the grid pattern more clear I've added the image below.

Unfortunately, I don't know how to give a formal mathematical definition of the fractal, so I hope the drawings I made and the description are clear enough. I want to make it clear that it's not important to me to know the Hausdorff dimension for this specific case, but rather how to generally calculate the Hausdorff dimension of the border of well defined closed areas. Is there some sort of technique for this?

$DeclareMathOperatorunit{unit}$Let's start with the dimension $Delta$ of the area, which we already know to be $Delta=2$. Let each of the colored squares have an area of $1text{ unit}^Delta$. Then the area of the template is: $$A_0=9cdotunit^Delta$$ Now let's get more fine grained measuring equipment that is $eta$ times more precise. That is, we will be measuring in $(frac 1etacdotunit)^Delta$ instead of $unit^Delta$. As it is we will find $9$ smaller squares inside each of the $9$ squares, so the total area is: $$A_1 = 9^2cdotleft(frac 1etacdotunitright)^Delta=9^2cdoteta^{-Delta}cdotunit^Delta$$

With the correct dimension $Delta=2$, we must have $A=A_0=A_1$, and therefore:
$$9=9^2cdoteta^{-Delta}quadRightarrowquad eta^Delta=9quadRightarrowquad eta^2=9quadRightarrowquadeta=3$$
We measure on a $3$ times smaller scale in $1$ iteration.

Let $D$ be the dimension of the circumference. The circumference with the same $unit$ is:
$$C_0 = 20cdotunit^D$$
When we measure again with $frac 1etacdotunit$ we find:
$$C_1 = 112cdotleft(frac 1etacdotunitright)^D = 112cdoteta^{-D}cdotunit^D$$

We get with $C=C_0=C_1$:
$$20=112cdoteta^{-D}= 112cdot 3^{-D}quadRightarrowquad 3^D=frac{112}{20}quadRightarrowquad D=log_3 frac{112}{20}approx 1.568$$
As expected this is a bit higher than Koch's curve that has $D=1.268$ and lower than $2$.

I calculate a graph-directed IFS for the boundary (all edges with similarity ratio $frac13$) as:

$$
begin{aligned}
A &to A + B + 2C + D \
B &to A + C + 2D \
C &to A + B + C + D \
D &to 2A + 3B + C + D
end{aligned}
$$

where $A,B,C,D$ are the edges of the red square, clockwise from top edge.

The paper referenced below gives a matrix $M(s)$ (called $A(beta)$ in the paper), and the Hausdorff dimension of the construction is $s$ such that the spectral radius of $M(s)$ is $1$. From the above:

$$
M(s) = begin{pmatrix}
frac13^s & frac13^s & 2frac13^s & frac13^s \
frac13^s & 0 & frac13^s & 2frac13^s \
frac13^s & frac13^s & frac13^s & frac13^s \
2 frac13^s & 3 frac13^s & frac13^s & frac13^s
end{pmatrix}
$$

Using numerical code (implemented in Javascript, see the function hausdorffDimension() in the source code of https://mathr.co.uk/blog/2007-10-03_graphgrow.svg) I get an estimated value for the Hausdorff dimension of the boundary as $1.465ldots$.

However, I'm not entirely sure that my initial graph-directed IFS for the boundary is correct (particularly for $B$ and $C$) - I haven't expanded it another level or two to verify that the numbers of edges of each type is consistent. Nor have I checked open set conditions (too much overlap can ruin everything). But maybe this unverified answer gives some hints.

Reference:

R. D. Mauldin & S. C. Williams, Hausdorff dimension in graph directed constructions, Trans. Amer. Math. Soc. 309 (1988), 811–829.
https://www.ams.org/journals/tran/1988-309-02/S0002-9947-1988-0961615-4/S0002-9947-1988-0961615-4.pdf