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I am preparing a presentation of Differential Geometry aimed to people with moderate knowledge of Mathematics (think about highschool students). I would like to find some concepts or applications of Differential Geometry that are easy to explain and attractive to an audience that may not have heard of the field.

My ideas:

  • Speak about the brachistocrone curve. This is a problem both easy to explain and with an interesting historical context, so I think it is ideal for a presentation like this.
  • Egregious Gaussian theorem. Also easy to explain and surprising (I won't be defining formally the concepts of course)

I would prefer examples about curves if possible, although every idea is welcome. Any reference is great also, I have been consulting Differential equations with historical notes by Simmons, which provides a very beautiful solution to the brachistocrone problem, based in Snell law.

Thanks in advance.

Zanzag
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  • If they know a little bit of probability, what about the Buffon needle problem and Crofton's formula? – Ted Shifrin Apr 09 '22 at 17:14
  • @TedShifrin I think that's a great idea, I will try to introduce it. – Zanzag Apr 09 '22 at 17:22
  • Orthogonal trajectories of a family of circles like here with a figure (which is lacking in the answers there). – Jean Marie Apr 09 '22 at 17:37
  • @JeanMarie I think that's beautiful, but I don't think an audience without a previous interest in Mathematics would be very interested about the orthogonal trajectories to a family of circles. Is there a more natural way to present the problem (in a physical or geometrical context)? In any case, thank you. – Zanzag Apr 09 '22 at 17:41
  • A practical application of these trajectories: geographical (large scale) maps as described here known as Wulff nets (it is stereographic projection which is important here). More generally, building maps that preserve this or that has necessitated (pre-)diffential geometry. In this respect, I advise you to take a look at Mollweide projection for equal area maps. – Jean Marie Apr 09 '22 at 17:49
  • Another track: cycloids (track of the air valve of a bike's wheel) , epi- and hypocycloids. See for the example the beginning of this question of mine here. More generaly, the concept of envelope is very rich (for example the astroid as envelope of the falling ladder...). – Jean Marie Apr 09 '22 at 17:59
  • @JeanMarie I was going to talk about cycloids, so your ideas are absolutely on point. I really like the image of an air valve on wheel for the cycloid. Thank you! – Zanzag Apr 09 '22 at 18:03
  • The answer here may be useful: https://math.stackexchange.com/questions/2390818/curvature-of-the-earth-from-theorema-egregium. Or, the black hole simulations developed for the movie Interstellar (ray-tracing in a curved spacetime) and used to simulate the first colliding black holes detected by LIGO, and/or gravitational lensing? – Andrew D. Hwang Apr 09 '22 at 19:46
  • About envelopes, you can interest people with "string art" giving rise to arc of parabolas (the proof can be given) and more generally helps you to introduce your audience to Bezier curves (and more generally to spline curves) which is so useful in CAD? – Jean Marie Apr 09 '22 at 23:12

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