One of the math mountains that I’ve always wanted to try to climb is to find a way to explain to kids why 5th degree polynomials can’t be solved in general.
The “one step closer” came from a comment by Allen Knutson on one of our projects on John Baez’s “juggling roots” tweet. Here’s the tweet:
Here are the two recent projects that we’ve done after seeing that tweet. Knutson’s comment is at the end of the first post:
Sharing John Baez’s “juggling roots” tweet with kids
Sharing John Baez’s “juggling roots” post with kids part 2
The comment pointed me to a video that shows how the “juggling roots” approached can be used to show that there is no general formula for finding the roots of a 5th degree equation:
The neat thing about the combination of this video and Baez’s post is that you can see some of the ideas from the video in the “juggling roots” gifs in the post.
Tonight I used some of the 3d prints of the juggling roots that I’ve made in the last few days to talk about the ideas a bit more and then we watched just a few minutes of the video.
We started with with a print that I accidentally made twice – but luckily the two prints give us a way to view the juggling roots through two cycles:
Next we looked at a different print to see a different juggling roots pattern. Here I was trying to set up the idea that the roots can move around in different ways. The way those different movements interact is the key idea in the video that Allen Knutson shared.
Finally, we went upstairs to watch a little bit of the video. Sorry for the sound issues, I don’t know why I left the sound on in the video. I mainly wanted the boys to see a different view of the juggling roots and I told them that the video gave the explanation for why 5th degree polynomials can’t be solved in general:
So, although I don’t quite have a full explanation of 5th degree polynomials for kids – I feel like I took a giant step towards getting to that explanation today. It is an extra happy surprise that 3d printing is going to come into play for that explanation!