A tweet last week from John Baez made for a really fun week of playing around. I’ve written several blog posts about it already. Here’s the summary to date, I guess:
(1) The original tweet:
(2) The blog posts:
(3) A video from a comment on one of the posts from Allen Knutson that helped me understand what was going on a bit better:
So, with that as background, what follows are some final (for now at least) thoughts on what I learned this week. One thing for sure is that I got to see some absolutely beautiful math:
Dan Anderson made some pretty neat 3d prints, too:
For this blog post I’m going to focus on the 5th degree polynomial . I picked this polynomial because it is an example (from Mike Artin’s Algebra book) of a polynomial with roots that cannot be solved.
So, what do all these posts about “juggling roots” mean anyway?
Hopefully a picture will be worth 1,000 words:
What we are going to do with our polynomial is vary the coefficients and see how the roots change. In particular, all of my examples below vary one coefficient in a circle in the complex plane. So, as the picture above indicates, we’ll look at all of the polynomials where moves around a circle with radius 8 centered at 10 + 0 I in the complex plane. So, one of our polynomials will be , another will be , another will be , and so on.
The question is this -> how do the roots of these polynomials change as we move around the circle? You would certainly expect that you’ll get the same roots at the start of the trip around the circle and at the end – after all, you’ve got the same polynomial! There’s a fun little surprise, though. Here’s the video for this specific example showing two loops around the circle:
The surprise is that even though you get the same roots by looping around the circle, with only one loop around the circle two of the roots seem to have switched places!
Here’s another example I found yesterday and used for a 3d print. Again for this one I’m varying the “2” coefficient. In this case the circle has a radius of 102:
When I viewed this video today, I realized that it wasn’t clear if 3 or 4 roots were changing places in one loop around the circle. It is 4 – here is a zoom in on the part that is tricky to see:
Next up is changing the “-16” in the x coefficient in our polynomial. Here the loop in the complex plane is a circle of radius 26:
Finally, there’s nothing special about the coefficients that are 0, so I decided to see what would happen when I vary the coefficient of the term that is initially 0. In this case I’m looping around a circle in the complex plane with radius 20 and passing through the point 0 + 0i:
So – some things I learned over this week:
(1) That the roots of a polynomial can somehow switch places with each other as you vary the values of the coefficients in a loop is incredible to me.
(2) The idea of thinking of these pictures as slices of a 3-dimensional space (which I saw on John Baez’s blog) led to some of the most visually striking 3d prints that I’ve ever made. The math here is truly beautiful.
(3) I finally have a way to give high school students a peek at a quite surprising fact in math -> 5th degree polynomials have no general solution.
What a fun week this has been!