Quasi-fractal sets in space

Let a be a positive real number, a < 1/2. A standard construction of a self-similar Cantor set in the plane starts with the unit square [0, 1]× [0, 1], replaces it with the four corner squares with sidelength a, then replaces each of those squares with their four corner squares of sidelength a, and so on. At the nth stage one has 4 squares with sidelength a, and the resulting Cantor set has Hausdorff dimension log 4/(− log a). The limiting case a = 1/2 simply reproduces the unit square, which has Hausdorff dimension 2. Suppose that we keep the boundaries of the squares at each stage of the construction, to get a kind of quasi-fractal set consisting of the Cantor set and a countable collection of line segments. The sum of the lengths of these line segments is finite exactly when a < 1/4. The Cantor set may be described as the singular part of this quasi-fractal set, which is compact and connected. Of course, one can consider similar constructions in higher dimensions. For the sake of simplicity, let us focus on connected fractal sets in R with topological diimension 1, for which the corresponding quasi-fractal set is obtained by including a countable collection of 2-dimensional pieces. As in the case of higher-dimensional Sierpinski gaskets or Menger sponges, these additional 2-dimensional pieces could be triangles or squares. Just as the endpoints of the line segments were elements of the Cantor set in the previous situation, the boundaries of these two-dimensional pieces would be loops in the fractal. For that matter, one could do the same for connected fractals in the plane like Sierpinski gaskets and carpets, where the additional 2-dimensional pieces are simply the bounded components of the complement. The resulting quasi-fractal would then be an ordinary 2-dimensional set, such as a triangle or a square. In each of these situations, the fractal set is the topological boundary of the rest, which is disconnected but smooth. One could also look at the bounded components of the complement of the quasi-fractals in