This model is now disassembled and the 400 tiny cubes will be given away to those who attend my Math Encounters talk at MoMath in NYC next week. The talk is free but you have to register in advance at one of the links below:
Making Mathematics Real: Knot Theory, Experimental Mathematics, and 3D Printing
Here is a close-up picture of the Level 4 assembly:
Thingiverse link: http://www.thingiverse.com/thing:372888
Settings: These cubes were printed using owens' clever external-support-stand method, with no support needed. We used Replicator 2 machines with raft but no supports, and Afinia H-Series machines with as little support as the software allows on a Mac.
Technical notes, assembly flavor: Because of the overhangs in the Level 4 Menger sponge, we had to use a few pieces of paper in the assembly so that things would stay in place without using glue. You can see the papers we used in the time-lapse sequence below. (Shout-out to dayofthenewdan's great Time Lapse Assembler software, which made it very easy to piece our sequence of pictures into a short video.)
Here is another time-lapse sequence, this time from the side: