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?