Saturday, September 5, 2020

More Cubic Fun

As sort of a quick follow-up to the last post, I decided that it was high time to start exploring (of course) the configuration space of cubics. More explorations on this forthcoming, but I'm testing out a new graphing calculator interface called Desmos. Let's see how well it can be shared.

(click the lower right to go to an interactive graph. What may be fun is playing around with the sliders. If I can figure out how to save it to a movie, I'll update it here)

Basically what's going on here is I'm trying to visualize the phase space of depressed cubics $x^3 + px + q$ (In our half-tablespoon example, $p = -3$ and $q = 1$) by seeing what happens when you map the unit disk in the complex plane. (If $p=0$ it's basically the cubing function which triples angles and makes things wind around the origin 3 times as often). It stands to reason that if $p$ and $q$ pass through certain interesting values, a qualitative change may occur (to relate this to some math that may be familiar, think the duscrumminanmanaent, I mean, discriminant (that word is just so annoyingly hard to say), of the quadratic: $b^2 - 4ac$ in $ax^2 + bx+c$: when it passes through 0, the nature of the solution changes). One day I may make a post detailing the connection between this kind visualization and topological understanding of the concept of solvability by radicals. It might be far in the future, though!

Here's a 3D rendering of the image of the unit disk as $p$ varies (with a vertical dimension rendered to distinguish distinct domain points that map to the same point (because the mapping is not 1-1. The mapping is not perfect, as the surface still self-intersects in places, but the direction the surface is facing as one traverses an intersection is distinct enough so you stay on the right path). (This is generated using SageMath, not Desmos, which doesn't yet have 3D stuff, oh well)

$r \leq 1, -3 \leq p \leq 3$

$r \leq 2, -6 \leq p \leq 6$, 

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