We had a fun time with Matt Parker’s Menger sponge video this week:
Here’s the video that inspired this set of projects:
Unluckily, while I was writing up the 2nd project I noticed that we’d made a mistake discussing the 3d modified Menger sponge whose volume is . I published the project with the mistake anyway, and returned to correct the mistake last night.
The error in the last project related to counting the number of small cubes which were removed from the modified Menger sponge at each step. In the normal Menger sponge, you are removing 7 small cubes from each small cube at each step. In the 2nd step of the modified Menger sponge you don’t chop the side lengths into 3 pieces, though. Instead each side length is chopped into 5 pieces. That means that when you removed the small center cubes that you remove 13 cubes rather than 7 cubes. We started the project by discussing that point:
Now that we had the correct formula for how the volume changes at each step in the construction of the modified Menger sponge, we went to Mathematica to see what we could say about the volume at each step (and in the limit):
So, a fun couple of days with Parker’s video. I’m sort of kicking myself for the mistake in the 2nd project, but it was fun to revisit the project and see how the volume of the shape converges to