Sierpinski sponge

"Jello" plot of the Sierpinski sponge animation through recursion steps. To enable better visibility the darker areas are the holes in the brighter cube. Selected sections cutting the largest cubic cavity parallel to the cube walls show up as the full Sierpinski carpets

Sierpinski sponge (or Sierpinski cube) is the exact three dimensional extension of the Sierpinski carpet, a fractal where one central cube is removed at each recursion. It is slightly different from the Menger sponge, where seven cubes are removed at each iteration (one in the center and one on each face). Because in the Sierpinski sponge only interior volume is removed, there are no exterior holes and the internal structure cannot be visualized unless the cube is partially transparent.

Because the number of the filling elements grows by the $26$ while preserving the carpet scaling factor $1/3$ it is near-three dimensional with the Hausdorff dimension:

$d_H=\log_3 26 = \ln 26 / \ln 3 \approx 2.965647$

which is more than for the Menger sponge.

In a similar way one may construct other fractal cubes by removing at the each step arbitrary number $N$ of smaller cubes scaled $1/3$ from bigger cubes e.g. the central cube and $8$ corner cubes and in general asymmetrically. Using the $(1+1)^{27}$ binomial expansion as the sum of the all numbers of combinations one may notice that there is $2^{27}= 134217728$ asymptotic objects while treating each of the smaller scaled removed or left cubes as different. Most of the cubes constructed in such a way are also strange geometrically in three dimensional space having the Hausdorff dimension:

$d_H=\log_3 (27-N) = \ln (27-N) / \ln 3$

which is not the integer. Only recursively removing $26$ cubes and leaving one at the corner converges to one trivial point with the $0$ dimension.

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  Tree-looking variation of Sierpinski sponge obtained by recursively removing the 8 corner, 6 face-centered and the central cubes (totally 15 cubes) with the fractal dimension $\log_3 (27-15) \approx 2.261859$

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  Another tree but Pine-like looking variation obtained by recursively removing 12 edge cubes with the fractal dimension $\log_3 (27-12) \approx 2.464973$