6 ms·

Solving 2D equations using color [video]

93 points by adamnemecek 9y ago

kmill 9y ago
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 9y ago
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 9y ago
  This is essentially how I feel also about Computer Science: resilience is the best quality that an aspiring mathematician can possess!
godelski 9y ago
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 9y ago
This is nice and pretty, but it still requires evaluating the function in infinitely many points. Is there an easy fix for that?
- kmill 9y ago
  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.
- EtDybNuvCu 9y ago
  You'd think so, but check out the 'smooth surprise' in Kahan's great writeup: https://people.eecs.berkeley.edu/~wkahan/Mindless.pdf https://people.eecs.berkeley.edu/~wkahan/Mindless.pdf We can approximate quite a bit, but ultimately further analysis is required to show that there are no such surprises on any particular root-finding problem.

93 points by adamnemecek 9y ago

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 9y ago

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 9y ago
This is essentially how I feel also about Computer Science: resilience is the best quality that an aspiring mathematician can possess!

godelski 9y ago

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 9y ago

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

kmill 9y ago
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.
EtDybNuvCu 9y ago
You'd think so, but check out the 'smooth surprise' in Kahan's great writeup: https://people.eecs.berkeley.edu/~wkahan/Mindless.pdf https://people.eecs.berkeley.edu/~wkahan/Mindless.pdf We can approximate quite a bit, but ultimately further analysis is required to show that there are no such surprises on any particular root-finding problem.