Juggling roots

As you change the coefficients of a polynomial, its roots move around. This is a surprisingly good source of fun for mathematicians of all ages.

This animation by twocubes shows the five roots of

x⁵ + tx³ + 1

moving around as the number t travels around a circle of radius 2 centered at the origin in the complex plane. See more here:


I think the contours lines are curves where the absolute value of this function is constant, with darker shades where it's smaller, and black where it's zero.

A lot of interesting things in math happen when you have two ways of viewing the same situation: then you can ask how a change in one view corresponds to a change in the other view. The two main ways to view a polynomial are its coefficients and its roots. I can imagine a program where these two views are side by side. You can move the coefficients around in the left side and see how the roots move around at right, or vice versa. For a polynomial of degree 5, dragging one coefficient around a circle in the left-hand view will create this animated image in the right-hand view.

(Be careful: the coefficients are an ordered list of numbers, while the roots are a multiset. Also, to get the roots of a polynomial to determine its coefficients, we should assume it's monic, meaning the first coefficient equals 1. Otherwise you can double all the coefficients without changing the roots.)

Thanks to +Henry Segerman for pointing this out. If any of you haven't seen his posts, it's time to check them out!

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