Freya Holmérhey did you know the roots of a cubic polynomial can be visualized using an equilateral triangle?
🔵 vertices are roots
🔴 the incenter is the inflection point
🟢 the incircle boundaries are the local minima/maxima
Jannik@acegikmo That looks really cool.
Would you mind telling me which library or program you used for the animation?
I recently did one using 3b1b's manim, which was a pleasant but a bit underdocumented experience.
@teraku Unity (the game engine) with my own library (Shapes) for vector graphics and a custom recording tool!
Rob Woolley@acegikmo I really enjoy all the geometric images and animations you share. Thank you for posting them.
Kuba c:@acegikmo Does that mean that if you now where 2 of those points are you can easily calculate the rest of them from the properties of a triangle?
Dag Ågren ↙︎↙︎↙︎@acegikmo Wow, that is great!
That makes it much more clear why solutions to cubic polynomials involve trigonometric functions!
@acegikmo Like, this otherwise absolutely mysterious code: https://github.com/erich666/GraphicsGems/blob/master/gems/Roots3And4.c#L120
Lydia Luka
Ailantd Sikowsky🔻@acegikmo Witchcraft!
DJ Devon3 
@acegikmo this is the kind of nerdy shit I like. Moar.
gws@acegikmo What in the mathematical, how does that...it's because of complex numbers, isn't it?
Gremriel@acegikmo I understood some of these words.😅
Marc B. Reynolds@acegikmo Nice viz!
Niles Johnson*let me in meme*
MAKE THE TRIANGLE SPIN!! 😱
(I mean: make the local max go below the x-axis; the blue dots will still be on the graph; do they relate to the complex roots??)
Also, make the max/min pass through eachother and swap places !!?? ⭕ 💥
BrokenTripod@acegikmo every time you post I learn something that I would have never learned in a million years. Thanks for sharing so many interesting visuals!
Twinfox@acegikmo Truly mesmerising! 🤩
ZhaoLiang@acegikmo @quantum A generalization to complex cubic polynomials is Marden's theorem.
There is an ellipse, called Steiner's ellipse, inscribes the triangle formed by the roots of \( f(z)=0 \) at the middle points of the edges, and its two focuses are the roots of 𝑓'(𝑧)=0.
See Albert Chern's Twitter post
https://twitter.com/theAlbertChern/status/1395468792788967428?s=20
And this blog article by @johncarlosbaez :
https://johncarlosbaez.wordpress.com/2021/05/24/electrostatics-and-the-gauss-lucas-theorem/
Eduardo Maschio
Ricky Reusser@acegikmo 👏 👏 I liked this diagram and was chipping away at improving my plotting setup, so I made an interactive version of it. https://rreusser.github.io/cubic-roots/