
Solving 2D equations using color [video] - adamnemecek
https://youtube.com/watch?v=b7FxPsqfkOY?
======
kmill
If you want to play with the winding number proof of the fundamental theorem
of algebra, here's a toy I made a while back to help my linear algebra
students (hopefully) gain some intuition for complex numbers:
[https://math.berkeley.edu/~kmill/toys/roots/roots.html](https://math.berkeley.edu/~kmill/toys/roots/roots.html)

If you want to play with a wider palette of complex functions as well as
domain coloring of the Riemann sphere, there is also
[https://math.berkeley.edu/~kmill/toys/zgraph/zgraph.html](https://math.berkeley.edu/~kmill/toys/zgraph/zgraph.html)

Documentation for each of them is the "Help" link in their respective upper
right corners.

One thing I think would be amusing is to be able to change the texture used
for the domain coloring, from the contoured rainbow to say a cat.

------
gigama
3Blue1Brown @11:40: "Being wrong is a regular part of doing math. We had a
hypothesis and it led us to this algorithm but we were mistaken somewhere.
Being good at math is not about being right the first time. It's about the
resilience to carefully look back and understand the mistakes and understand
how to fix them."

~~~
rimher
This is essentially how I feel also about Computer Science: resilience is the
best quality that an aspiring mathematician can possess!

------
godelski
3Blue1Brown is one of my favorite youtubers. It is a nice format that is
inbetween "explain it to me like I'm 5" and an open courseware. Grant tends to
give enough information to become familiar enough with a subject that you can
do good research on your own.

Their podcast is also great, Ben Ben Blue. Which has Ben Eater, another great
Youtuber.

------
thomasahle
This is nice and pretty, but it still requires evaluating the function in
infinitely many points. Is there an easy fix for that?

~~~
kmill
1\. If you can calculate rough bounds for the terms and their derivatives in
the winding number integral, you can figure out how densely you need to sample
the curve to calculate the integral exactly.

2\. If the winding number is zero, there still might be a root in the region
because poles and zeros contribute opposite winding number, and they might
exactly cancel.

3\. Polynomials are great because the bounds are easy to compute and because
you don't have to worry about cancelation since the only pole is at infinity.

