# Menger’s Sponge

Material: Lots of business cards (or any type of paper of the same size, 4×7)

Goal: Let participants have fun (and learn a thing or two about counting and fractals)

Vocabulary: Menger’s sponge, fractal, surface area, iteration, volume

Activities:

- Introduce the Sponge and its formation
- Instruct how to build the unpaneled and paneled basic cube unit and how to assemble them
- Bring up the problem of calculating the number of business cards needed for a Menger Sponge of arbitrary level, break the problem down to several components first: the number of small cubes in a Menger sponge, the number of business cards for an unpaneled Menger sponge (especially the first 3 iterations)
- If they seem to understand how to carry out step 3, present the solution after some time of thinking
- Gather all the previously-built results to get the surface area and volume of the Sponge, and talk a bit about how counterintuitive fractals can be
- End with a Menger Sponge Race: Divide into teams and see who can build a (preferably paneled) Level 2 Menger Sponge, if you can gather enough material

Details:

- The Menger Sponge is what we get after carrying out the following iterative process infinitely many times:

- Begin with a cube.

- Divide every face of the cube into nine squares, like Rubik’s Cube. This sub-divides the cube into 27 smaller cubes.
- Remove the smaller cube in the middle of each face, and remove the smaller cube in the center of the more giant cube, leaving 20 smaller cubes. This is a level-1 Menger sponge
- Repeat steps two and three for each of the remaining smaller cubes, and continue to iterate.

The shape after n times of this procedure (or iteration) is called a Level n sponge

- Include a step-by-step illustration
- (and 4) Start with Level 0 moving up to Level 1. It’s divided into 27 smaller cubes but 7 cubes in the core are removed, leaving us with 20. This reasoning can be repeated with the ever-smaller cubes. The general formula is then 20^n, and since each cube requires 6 business cards, the total number of business cards for an unpaneled sponge of level n is 6*20^n

Thinking of a Level n+1 sponge as 20 Level n sponges stacked together, and subtracting the hidden areas, is the main idea for calculating the number of panels we need to add to round it out.

- (for 6) We’ll need 3456 business cards for a paneled Level 2 Sponge, and 2400 for an unpaneled one, which is not a small number, and any help will be useful. Calculating the time for certain numbers of participants can be tricky since how quickly they can build the Level 2 sponge depends on how well they collaborate, and I have no concrete data to work out how long this usually takes. This can be left as a fun challenge for those interested.