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Re: Computing pi (or not)
Posted:
Sep 18, 2012 1:28 PM
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On Fri, Sep 14, 2012 at 7:06 AM, kirby urner <kirby.urner@gmail.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.
Kirby
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