Later this month, people will be gathering at museums and schools around the world to build giant Menger sponges as part of a global fractal extravaganza called MegaMenger.

A Menger sponge is a fractal that sits in three-dimensional space. To visualize one, imagine starting with a cube and splitting it into 27 sub-cubes, like a Rubik's cube. Remove the middle cubes from each face of the larger cube and the cube in the very middle. Now do the same thing to the 20 cubes that are remaining, dividing them up into tiny cubelets and removing the middle ones. When you’ve repeated this process infinitely many times, you’re left with a Menger sponge. It has no volume but infinite surface area.

Level 0-3 Menger sponges. Image: Solkoll, public domain, via Wikimedia Commons.

Sadly, we can’t really make Menger sponges because creating a Menger sponge is an infinite process, and we don’t have that long. We finite beings have to settle for stopping at a finite level of the sponge. And during the week of October 20-26, finite groups of finite beings will be creating level 2 and 3 sponges at sites around the world.

Builders working on a level 1 Menger sponge. Image: Manchester Science Festival.

The Menger sponge is a fractal, and one of the defining characteristics of fractals is that they are self-similar, meaning that if you zoom in on a small part of a fractal, it looks just like a larger portion of the fractal. For that reason, instead of taking large cubes and drilling square holes in them to make our sponges, we’ll actually be building them additively. We’ll start by folding business cards into small cubes and then connect 20 of them together to make a level 1 sponge. When we’ve made 20 level 1 sponges, we can put them together to make a level 2 sponge, and so on. The technique we’ll be using was developed by Jeannine Mosely, a software engineer and computational origamist, about a decade ago. (Check out this interview with her in Cabinet Magazine.)

Because there are more than 20 level 3 sites around the world, when we put it all together, we’ll have made a level 4 Menger sponge! That’s no small feat, as the level 4 sponge requires 160,000 smaller cubes made from about a million business cards. The level 3 pieces will be scattered about the globe a bit haphazardly, but we won't be too picky.

MegaMenger is the brainchild of Matt Parker of ThinkMaths and Laura Taalman of James Madison University and the Museum of Mathematics. The dates of the event, October 20-26, are set to coincide with the annual Celebration of Mind events that honor extraordinary mathematics communicator Martin Gardner, whose birthday was October 21. This year marks the 100th anniversary of his birth, so creating what is possibly the largest fractal ever built seems like an appropriate tribute.

For those of you in the Salt Lake City area, there is a level 2 MegaMenger build at the University of Utah, and you’re all invited! The event is being sponsored by the student chapters of the Association for Women in Mathematics and the Society for Industrial and Applied Mathematics, and it will be held on October 22, 23, and 24 at two locations on campus. For more information, head on over to the Association for Women in Mathematics website.

If you can't come build with me, check out to see if there is a build taking place near you!

A completed level 1 Menger sponge. Image: Manchester Science Festival.