On Fri, Sep 14, 2012 at 7:06 AM, kirby urner <email@example.com> wrote:
> > I don't see any "algorithm", I see a messy drawing in a notebook, like > something Vi Hart might draw in one of her Youtubes. > > Kirby
On the topic of paradoxes, here's one I like, which I've brought up before:
We have an algorithm for omni-triangulating a sphere with more and more triangles. Even if we toss out the paradox at the end, here's some educational meat, some substance (to go over this topic).
They come together (the triangles) as six around one, or as five around one (just 12 of those, at the vertexes of what started as an icosahedron). But these triangles can't be all equilateral exactly as six of those around one would be flat and all of these vertexes are local apexes i.e. there's a little tiny bit of curvature, and the more overall triangles we have, the less curvature at each vertex, the closer each comes to zero curvature and a full 360 degrees around it, perfect flatness (yet this is a sphere, remember).
And now for the paradox: I have just described a limiting process whereby adding more and more triangles (non-equilateral) to a sphere begets vertexes the number of degrees around which is < 360. How much less than 360? Say v. |360 - v| < epsilon, pick an epsilon. Now add more triangles to shrink the difference to make that true. Standard demo of approaching 0. Except thanks to the sum of all |360 - v| differences adding to 720 (this is something Descartes proved), there's always a delta > 0 at each vertex. The deltas add to 720. So we have both a limit of 0 and a clear proof that the limit can never be 0. Interesting. We can probably hand wave that away, but in the meantime we've covered some interesting geometry + calculus.