The Menger Sponge

Definition Start with a solid cube, called stage 0. To go from one stage to the next, cut every cube of the current stage into 27 equal smaller cubes by dividing each of its edges into three equal parts, and keep 20 of them. The cubes removed are the center cube and the six cubes at the centers of the faces, seven different cubes in all. If the small cubes are labeled by positions $(i, j, k)$ with each coordinate 0, 1 or 2, the kept cubes are exactly those with at most one coordinate equal to 1. Stage $n$ consists of $20^n$ solid cubes, each with edge $3^{-n}$ times the original edge, kept together with their faces, edges and corners. The Menger sponge is the set of points that belong to every stage. Its surprise is that it keeps the full outer size of the cube, contains all twelve edges of the original cube and stays in one piece, while the volume of stage $n$ is $(20/27)^n$ times the original volume. These volumes tend to 0, so the sponge itself has zero volume. Every finite stage, and so every physical model or picture, still has positive volume, and zero volume belongs to the limit alone. Karl Menger described the sponge in 1926.

Definition A set in space has zero volume when, for every positive number however small, it can be covered by finitely many boxes whose total volume is less than that number. Each stage of the sponge removes a fixed fraction of what is left, not a fixed amount, because every kept cube keeps 20 of its own 27 pieces. So the volume is multiplied by $\tfrac{20}{27}$ at every stage, and stage $n$ has volume

where $V_0$ is the volume of the starting cube. Every $V_n$ is positive, and since $\tfrac{20}{27} < 1$, the numbers $V_n$ become smaller than any positive number. The sponge lies inside every stage, and stage $n$ is made of finitely many cubes, so the sponge has zero volume.

Definition A fractal is a set that shows detail at every scale, and the fractals in this unit come from one rule repeated inside every piece forever. A three-dimensional fractal is a fractal that lives in ordinary three-dimensional space, so the word describes the surrounding space, not the dimension of the set itself. The Menger sponge is made of 20 copies of itself, each scaled by $\tfrac13$, and two copies meet at most along faces of their small cubes. Its similarity dimension is the number $D$ with $3^D = 20$,

Because the copies overlap only on their boundaries, $D$ is also the sponge’s Hausdorff dimension. A value between 2 and 3 describes how the amount of sponge scales, not a fractional number of directions in which one can move. In the sense of topological dimension the sponge has dimension 1, like a curve, even though it sits in space.

Definition A closed and bounded set is connected when it cannot be split into two nonempty parts that stay a positive distance apart. Every stage of the sponge is connected, because each kept cube shares a face with a neighboring kept cube and these links join all the cubes into one piece. The sponge is connected too, because the points common to a decreasing sequence of closed, bounded, connected sets always form a connected set. More than that, the sponge holds a framework of straight beams. The small cubes along an edge of a kept cube have at most one middle coordinate, so they are kept, and repeating this at every stage shows that every edge of every kept cube, at every stage, belongs to the sponge. These edges form one connected network of segments that comes as close as we like to every point of the sponge, and in fact any two points of the sponge can be joined by a path that stays inside it.