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The geometry of the quadratic formula

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Tags: calculusgeometryquadratic-formulacurvature

There's a beautiful connection between the quadratic formula and the Gaussian curvature of surfaces. I aimed to explore this connection, starting from defining curvature of curves, all the way until computing and discussing the Gaussian curvature of surfaces in R^3. I tried making the video accessible to a general audience. Most of the video is a discussion of geometric concepts, but someone with a basic understanding of calculus should be able to follow most of the computations. There's just one computation that requires multivariable calculus.

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6.76 Overall score*

36 Rank

12 Votes

6 Comments

Comments

3.3

Got lost in the 3d part

This was exceptional! The varied styles of physical props, Geogebra elements and mathematical working out kept everything feeling fresh and engaging. At the same time the pacing of this video was very nicely done, and you didn’t spend too long on any particular subtopic. Genuinely a wonderful video which I would definitely show to students interested in geometry or topology.

The math here is really interesting, and the visuals are pretty strong. I can follow all the steps along the way. The big issue is that I don’t see where those steps come from. What insight would lead someone to take directional derivatives, or to relabel all the variables and set y to 1? It would help to lay out the big picture before filling in the details.

7.6

Very nice. Style and production quality is solid. Just an overall very good video. I can’t help but be sad you didn’t throw in a little mention of Gauss-Bonnet, but of course it might be a little out of the scope of the video.

7.6

I really loved how this provided an algebraic understanding for gaussian curvature; I really learned a lot, especially the visualization of principal curvature.

I’ve learned the quadratic formula in school and it’s been nice to see it applied like this. This video works really well as a mix of digital animation and analog filming. It’s well balanced in that regard. Of course, I couldn’t do the math myself after watching just this video, but I now understand how it can be used and why it’s important to know. The explanations are very clear, as well are the visuals (partly because your hand-writing is readable, which sadly isn’t the norm). At least subjectively, there’s not much novelty in this video, but I think its clarity makes up for that. What will stick with me is the simple but effective visuals.