Is an Inverse Menger Sponge a fractal?

Is the Inverse of a Menger Sponge a fractal? I know a Menger sponge is fractal in nature, and it seems to me that the inverted form of it would be fractal as well, but I don't know.

• What do you mean by “inverse”? The set complement of the points in the sponge? Commented Oct 6, 2011 at 22:42
• There's a "description" of the Inverse Menger Sponge at minecraftonline.com/wiki/Menger_Sponge but I don't understand it. It may also be the object pictured at flickr.com/photos/friends_of_folding/98165798 Commented Oct 6, 2011 at 23:47
• This seems to be the two-dimensional version of what you want... Commented Oct 7, 2011 at 0:18
• I guess "the inverse Menger sponge" is what mathematicians would call "the complement of the Menger sponge" in the containing cube. If so, then it is an open set in three-space, and thus not a "fractal" in the sense of Mandelbrot. Commented Oct 7, 2011 at 0:37

1 Answer

It is not a fractal.

1. First if we consider the 2-d case(Sierpinski carpet) and the 3-d as a generalization, the precess of construction its take a square, divide it in nine squares, extract the central square and in each of the eight squares left we do the same process, now if we consider the inverse of this, it means only keep the central square and repit the process. The process in the inifinite is equal a square an object of dimention 2 then this is not a fractal. The same occurs in the case 3-d.
2. The case 3-d is not self-similar.
3. The Menger sponge is closed and its complement open then it cannot be a fractal.
• but there are fractals with integer hausdorff dimension Commented Apr 1, 2020 at 18:56