A Decomposition of Menger's Sponge

# Menger's Sponge (depth 3)

A model of a level 3 Menger Sponge could be built out of business cards, one cube at a time, with many folders helping by precreasing the business cards. As the structure got larger there would be room for perhaps as many as 4-8 people to work on it simultaneously, but they would still be the major bottle neck, since assembly and panelling take more half the construction time. It would take such a group working together around 50-100 hours from start to finish. I doubt that I could find enough dedicated volunteers to do it this way.

Instead, I would like to try build the sponge with the help of many volunteers building subassemblies that could be brought or shipped to a central location for final assembly. It is not immediately obvious what form the sub-assemblies should take, since they must be joined together after they are built. Building level 1 sponges is easy and takes about an hour to ninety minutes. But you cannot just pretend that they are simple cubes and link their faces to make a level 2 sponge, because each face of a level 1 sponge has 8 cubes which would have to be linked simultaneously with their neighbors. Trying to link level 2 sponges would be even worse.

It must be possible to join the sub-assemblies together by building linking cubes between them. After several false starts, I have devised such a decomposition of the sponge based on a sub-unit that I call a "tripod".

A tripod is a block of four linked cubes, where a corner cube has three cubes attached to three adjacent faces. A sponge of any level can be broken into pieces that are combinations of one, two, three, four, six or eight tripods joined together by additional cubes.