🧊 Menger Sponge

The Menger sponge is a three-dimensional fractal formed by dividing a cube into a 3×3×3 grid of twenty-seven equal sub-cubes, removing the center sub-cube and the six sub-cubes at the center of each face, and repeating the same process inside each of the twenty remaining sub-cubes indefinitely. It was first described by the Austrian-American mathematician Karl Menger in 1926.

Starting from a solid cube, it is split into twenty-seven smaller cubes arranged in a 3×3×3 grid. The one cube exactly in the center and the six cubes centered on each face are removed, leaving the twenty cubes at the corners and edges of the grid. The same subdivide-and-remove rule is then applied again inside each of those twenty cubes, and so on for every subsequent level.

After n iterations there are exactly 20ⁿ filled sub-cubes, because each subdivision step keeps 20 of the 27 cubes in the grid and this happens independently inside every cube that survived the previous level. At depth 3, for example, there are 20³ = 8,000 cubes.

Each level keeps 20/27 of the volume that was present before it, so after n iterations the remaining volume is (20/27)ⁿ of the original cube, shrinking toward zero as n grows. At the same time, every removal step exposes new interior faces, so the total surface area of the sponge actually increases without bound, even as its volume vanishes.

Its Hausdorff dimension is log(20)/log(3), approximately 2.7268. This sits between the dimension of a flat plane (2) and a solid filled cube (3), reflecting how the sponge fills space more thoroughly than any flat surface but leaves infinitely many holes so it never fills a complete solid volume.