I saw an incredible tweet from John Baez last night:
The tweet links to a couple of blog posts which I’ll link to directly here for ease:
John Baez’s “Juggling Roots” Google+ post
Curiosa Mathematica’s ‘Animation by Two Cubes” post on Tumblr
The Original set of animations by twocubes on Tumblr
So, I think the path that the animation took to our eyes was from twocubes to curiosamathematica to John Baez to us. Sorry if I do not have the sources and credit correct, but I will make corrections if someone alerts me to an error.
I’d never made any sort of animation before, but since the pictures looked like they came from Mathematica I started to play around a little bit last night to see what I could do. In doing so I learned about Mathematica’s “Animate” and “Manipulate” functions and made some progress, though the animations that I made were not nearly as good as the ones from the above posts. This Stackexchange post was helpful to me in improving the quality of my animations, but still mine aren’t in the same league as the original ones:
Anyway, with that introduction, I thought it would be really fun to share these animations with kids and do a tiny bit of background explanation. I stared this morning by just showing the boys some of the pictures and asking them to describe what they were seeing:
Next I showed them one of the animations that I made and asked them to see if they could see some similarities with any of the previous animations:
Next we went down to the living room to talk about roots of equations. My older son knows a little bit about quadratic equations, but only a little bit. I didn’t want this part of the conversation to be the main point, but I did want them to get a tiny peek at the math behind the animations we were looking at today:
Finally, we went back up to the computer to look at some of the animations for quadratic and cubic equations. My maybe too open-ended task for them here was to compare the animations of the roots of quadratic and cubic equations to the animations of the roots of the quintic equations.
I’ve always wanted to be able to share some of the basic ideas from Galois theory with kids. I’ve never seen anything like these animations previously. They make for a neat starting point, I think, since kids are able to talk about the pictures. I would **love** to know what a research mathematician sees in the pictures. In particular, is there something in the pictures that gives a clue about why the roots of 5th degree polynomials are going to be more difficult to study than 2nd, 3rd, or 4th degree ones?
Mike's Math Page 
Woke up this morning obsessing over visualizing quadratic equations….and when I go on-line I see a bunch of different tweets which indicate that a bunch of different people are thinking in various ways about this same thing. Though I shouldn’t be spending this day off going down these rabbit holes, I went through your post first, as I can’t see any other way to forward towards any understanding about the image in Baez’s tweet. Also, I looked through the links you posted (ah!!! Rabbit Holes everywhere!!!) and noticed something which I wonder if you noticed, because this is something you will really like: It appears to me that the image in Baez’s tweet started with the guy who wrote that book you’ve talked about, about 3D printing, Henry Segerman https://plus.google.com/+HenrySegerman/posts/NdWL2E9Ht2a
http://www.3dprintmath.com/
Have a good day…and please resist posting anything else today that I might be interested in .
There’s a proof by Arnol’d of the unsolvability of the quintic that starts from such juggling of roots. It seems to be written here
https://projecteuclid.org/download/pdf_1/euclid.tmna/1471875703
and in half-hour video here
Thanks Allen – the video is fascinating. I’ve been playing around with some of those ideas in Mathematica to see about creating some of the specific root swaps. Unfortunately some work travel is going to slow down that process, but it’ll be fun to try to find a specific combination meeting the criteria set out in the video.