1. Begin with a cube.
2. Shrink the cube to 1 / 3 of its original size and make 20 copies of it.
3. Place the copies so they will form a new cube of the same size as the original one but lacking the centerparts, (next image).
4. Repeat the process from step 2 for each of the remaining smaller cubes.
After an infinite number of iterations, a Menger sponge will remain.
The Menger Sponge is a fractal particularly appealing to modular origamists, for two reasons: it can be made out of Sonobe modules (the very first “brick” that every origamist learns), and given its fractal nature, it can be expanded forever and ever. What follows is a list of origami websites on this fascinating structure:
-Level 1 sponge by Michał Kosmulski, along with a “level1-1/2” sponge and other origami fractals (the whole site is very interesting).
-Several sponges by Krystyna Burczyk, up to level 3. Notice that the smallest level-1 sponge already needs 72 modules, so this is quite an amusing project 😛
-A 2400 module sponge by Darren Abbey.
-A different approach on the same idea is to use business cards as bricks, instead of sonobe modules. Jeannine Mosely has built a sponge in this way, using exactly 66,048 cards. This sponge was finished in june, 2005 and took some nine years to make in between collecting cards, planning and building. I think that someone should try the same but with the printed sides pointing outwards. That would make a million-dollar sponge!
-The insides of a business card sponge, showing how the pieces interlock together.
-Another business card sponge, along with its inverse, which I believe looks even cooler.
-Finally, the amazing Level minus-one sponge, depicted in a sharp photograph by John Paulsen. This is truly the Zen of all Menger sponges 😀