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:

1. Introduce the Sponge and its formation 
2. Instruct how to build the unpaneled and paneled basic cube unit and how to assemble them 
3. 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)
4. If they seem to understand how to carry out step 3, present the solution after some time of thinking
5. 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
6. 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:

1. 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

2. Include a step-by-step illustration
3. (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. 

4. (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.