Oh wow, I wasn't expecting to see this on Hacker News again!
This remains my most popular post. I'm very glad about the interest in mathematics it continues to generate!
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To that one criticism, yes, there is no real "why" to the animations other than I thought they looked cool.
The post is not meant to be comprehensive, or teach anything more than bare basics meant to enjoy the visualizations.
I disagree that math visualizations must have clear pedagogical goals. Math visualizations can be purely exploratory.
The curves the poles trace out over time, are they significant somehow? Perhaps. Perhaps not. That's the exciting part of exploring new concepts. And part of the reason I chose linear over geometric interpolation.
Exploring those curves and alternate interpolations/animations was going to be part two, but it never happened.
I try to make posts accessible to as many people as possible. There is plenty of rigorous content already out there for learning more.
The focus for my blog is exploration and curiosity.
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Perhaps I'll get around to part 2, and make it interactive with a compute shader.
Apologies for the code, it was never meant to be reused. I'm sure you can improve it!
In the past, I have used manim to make mathematical animations: https://www.manim.community/ Manim is more flexible but that comes with some overhead of complexity and learning. Example of some animations using manim:
I couldn't disagree more to be honest, about this post. Animations are good when they provide object permanence, and let you track what's changing and how.
This post linearly interpolates complex functions blindly, which doesn't tell you anything useful, unless the thing being interpolated is an affine or projective transform where that makes sense.
e.g. For complex powers, the most natural animation is to animate the exponent, which will show a continuous folding or unfolding. Here the squaring just looks like the extra 360° appears out of nowhere.
For mobius-like transforms, interpolating the inverse might be better.
One particularly good example is e.g. visualizing equally spaced points on a circle, and their various combinations as roots and poles of complex functions.
The goal of math animation should be to highlight and travel the natural geodesics of the concept space, with natural starts and stops too.
It reminds me that each year Freiberg University of Mining and Technology publishes a calendar of Complex Beauties[1], of which I buy several copies as gifts every year.
It includes 12 visualizations of selected complex functions and their background and related mathematicians. I would highly recommend reading them. Prof. Dr. Elias Wegert, the author who actively contributes to this calendar, also wrote Visual Complex Functions: An Introduction with Phase Portraits which is mentioned by another comment here.
If you want to mess around with these sorts of visualizations yourself, I recommend checking out David Bau's little web app for it: http://davidbau.com/conformal
[+] [-] bmitc|1 year ago|reply
[+] [-] vankessel|1 year ago|reply
This remains my most popular post. I'm very glad about the interest in mathematics it continues to generate!
---
To that one criticism, yes, there is no real "why" to the animations other than I thought they looked cool.
The post is not meant to be comprehensive, or teach anything more than bare basics meant to enjoy the visualizations.
I disagree that math visualizations must have clear pedagogical goals. Math visualizations can be purely exploratory.
The curves the poles trace out over time, are they significant somehow? Perhaps. Perhaps not. That's the exciting part of exploring new concepts. And part of the reason I chose linear over geometric interpolation.
Exploring those curves and alternate interpolations/animations was going to be part two, but it never happened.
I try to make posts accessible to as many people as possible. There is plenty of rigorous content already out there for learning more.
The focus for my blog is exploration and curiosity.
---
Perhaps I'll get around to part 2, and make it interactive with a compute shader.
Apologies for the code, it was never meant to be reused. I'm sure you can improve it!
Thank you for reading :)
[+] [-] azeemba|1 year ago|reply
In the past, I have used manim to make mathematical animations: https://www.manim.community/ Manim is more flexible but that comes with some overhead of complexity and learning. Example of some animations using manim:
- List of videos using manim: https://www.manim.community/awesome/
- A blog post I made: https://azeemba.com/posts/degenerate-matter.html
[+] [-] unconed|1 year ago|reply
This post linearly interpolates complex functions blindly, which doesn't tell you anything useful, unless the thing being interpolated is an affine or projective transform where that makes sense.
e.g. For complex powers, the most natural animation is to animate the exponent, which will show a continuous folding or unfolding. Here the squaring just looks like the extra 360° appears out of nowhere.
For mobius-like transforms, interpolating the inverse might be better.
One particularly good example is e.g. visualizing equally spaced points on a circle, and their various combinations as roots and poles of complex functions.
The goal of math animation should be to highlight and travel the natural geodesics of the concept space, with natural starts and stops too.
The rest is cargo culting.
[+] [-] liminal|1 year ago|reply
[+] [-] hyperific|1 year ago|reply
https://github.com/vankessel/sandbox/tree/master
[+] [-] fauria|1 year ago|reply
[+] [-] rramadass|1 year ago|reply
[+] [-] simonmysun|1 year ago|reply
It includes 12 visualizations of selected complex functions and their background and related mathematicians. I would highly recommend reading them. Prof. Dr. Elias Wegert, the author who actively contributes to this calendar, also wrote Visual Complex Functions: An Introduction with Phase Portraits which is mentioned by another comment here.
[1]: https://blogs.hrz.tu-freiberg.de/mathekalender/english/
Edit: Navigate to the german page if you want to buy it
[+] [-] mkaic|1 year ago|reply
[+] [-] ttoinou|1 year ago|reply
Using images as input to show conformal deformation https://www.youtube.com/watch?v=CMMrEDIFPZY
Better phase portraits with a grid, zeroes, poles https://www.shadertoy.com/view/Ms2Bz3
[+] [-] hyperific|1 year ago|reply
https://news.ycombinator.com/item?id=19423278