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Visualizing Complex Functions

77 点作者 hyperific9 个月前

9 条评论

bmitc9 个月前
A good book is <i>Visual Complex Functions: An Introduction with Phase Portraits</i>: <a href="https:&#x2F;&#x2F;www.amazon.com&#x2F;Visual-Complex-Functions-Introduction-Portraits&#x2F;dp&#x2F;3034801793" rel="nofollow">https:&#x2F;&#x2F;www.amazon.com&#x2F;Visual-Complex-Functions-Introduction...</a>
vankessel9 个月前
Oh wow, I wasn&#x27;t expecting to see this on Hacker News again!<p>This remains my most popular post. I&#x27;m very glad about the interest in mathematics it continues to generate!<p>---<p>To that one criticism, yes, there is no real &quot;why&quot; to the animations other than I thought they looked cool.<p>The post is not meant to be comprehensive, or teach anything more than bare basics meant to enjoy the visualizations.<p>I disagree that math visualizations must have clear pedagogical goals. Math visualizations can be purely exploratory.<p>The curves the poles trace out over time, are they significant somehow? Perhaps. Perhaps not. That&#x27;s the exciting part of exploring new concepts. And part of the reason I chose linear over geometric interpolation.<p>Exploring those curves and alternate interpolations&#x2F;animations was going to be part two, but it never happened.<p>I try to make posts accessible to as many people as possible. There is plenty of rigorous content already out there for learning more.<p>The focus for my blog is exploration and curiosity.<p>---<p>Perhaps I&#x27;ll get around to part 2, and make it interactive with a compute shader.<p>Apologies for the code, it was never meant to be reused. I&#x27;m sure you can improve it!<p>Thank you for reading :)
azeemba9 个月前
The animation looks awesome! Looks like the author used matplotlib (as they mention in a comment on the website): <a href="https:&#x2F;&#x2F;github.com&#x2F;vankessel&#x2F;sandbox&#x2F;blob&#x2F;master&#x2F;graph&#x2F;interp2.py">https:&#x2F;&#x2F;github.com&#x2F;vankessel&#x2F;sandbox&#x2F;blob&#x2F;master&#x2F;graph&#x2F;inter...</a><p>In the past, I have used manim to make mathematical animations: <a href="https:&#x2F;&#x2F;www.manim.community&#x2F;" rel="nofollow">https:&#x2F;&#x2F;www.manim.community&#x2F;</a> Manim is more flexible but that comes with some overhead of complexity and learning. Example of some animations using manim:<p>- List of videos using manim: <a href="https:&#x2F;&#x2F;www.manim.community&#x2F;awesome&#x2F;" rel="nofollow">https:&#x2F;&#x2F;www.manim.community&#x2F;awesome&#x2F;</a><p>- A blog post I made: <a href="https:&#x2F;&#x2F;azeemba.com&#x2F;posts&#x2F;degenerate-matter.html" rel="nofollow">https:&#x2F;&#x2F;azeemba.com&#x2F;posts&#x2F;degenerate-matter.html</a>
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liminal9 个月前
This is very cool, but would greatly benefit from a perceptually linear color space. I think the author is using HSL -- HCL would be a better choice.
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fauria9 个月前
One of the best explanations on imaginary numbers I&#x27;ve seen is Kalid Azad&#x27;s from Better Explained: <a href="https:&#x2F;&#x2F;betterexplained.com&#x2F;articles&#x2F;a-visual-intuitive-guide-to-imaginary-numbers" rel="nofollow">https:&#x2F;&#x2F;betterexplained.com&#x2F;articles&#x2F;a-visual-intuitive-guid...</a>
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simonmysun9 个月前
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.<p>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 <i>Visual Complex Functions: An Introduction with Phase Portraits</i> which is mentioned by another comment here.<p>[1]: <a href="https:&#x2F;&#x2F;blogs.hrz.tu-freiberg.de&#x2F;mathekalender&#x2F;english&#x2F;" rel="nofollow">https:&#x2F;&#x2F;blogs.hrz.tu-freiberg.de&#x2F;mathekalender&#x2F;english&#x2F;</a><p>Edit: Navigate to the german page if you want to buy it
mkaic9 个月前
If you want to mess around with these sorts of visualizations yourself, I recommend checking out David Bau&#x27;s little web app for it: <a href="http:&#x2F;&#x2F;davidbau.com&#x2F;conformal" rel="nofollow">http:&#x2F;&#x2F;davidbau.com&#x2F;conformal</a>
ttoinou9 个月前
Plug of my own work on the topic :<p>Using images as input to show conformal deformation <a href="https:&#x2F;&#x2F;www.youtube.com&#x2F;watch?v=CMMrEDIFPZY" rel="nofollow">https:&#x2F;&#x2F;www.youtube.com&#x2F;watch?v=CMMrEDIFPZY</a><p>Better phase portraits with a grid, zeroes, poles <a href="https:&#x2F;&#x2F;www.shadertoy.com&#x2F;view&#x2F;Ms2Bz3" rel="nofollow">https:&#x2F;&#x2F;www.shadertoy.com&#x2F;view&#x2F;Ms2Bz3</a>
hyperific9 个月前
This was also posted previously in 2019.<p><a href="https:&#x2F;&#x2F;news.ycombinator.com&#x2F;item?id=19423278">https:&#x2F;&#x2F;news.ycombinator.com&#x2F;item?id=19423278</a>