Re:A Smooth Curve as a Fractal Under the Third Definition
by romwell on Friday March 16, 2018 @06:07PM
(#16604348)
Woah!
Just the other day I've had a discussion with my companion about how one would determine they live on a toroidal[1] planet (assuming the atmosphere is too cloudy, and one can only see a small neighborhood).
She suggested measuring circumferences of circles at various points, as a means of getting curvature; the torus is negatively curved at some places.
"But what if we are on a flat torus?", I asked. "We are talking about a potentially realistic scenario here, not a topological abstraction", she objected. I didn't have anything to say.
But now I do.
On page 66, they even show a 3D-printed model!
I knew of Whitney embedding theorem, but somehow, not of the Nash embedding theorem which proves this is possible - and certainly not of this result, which is delightfully amazing.
Just the other day I've had a discussion with my companion about how one would determine they live on a toroidal[1] planet (assuming the atmosphere is too cloudy, and one can only see a small neighborhood).
She suggested measuring circumferences of circles at various points, as a means of getting curvature; the torus is negatively curved at some places.
"But what if we are on a flat torus?", I asked. "We are talking about a potentially realistic scenario here, not a topological abstraction", she objected. I didn't have anything to say.
But now I do.
On page 66, they even show a 3D-printed model!
I knew of Whitney embedding theorem, but somehow, not of the Nash embedding theorem which proves this is possible - and certainly not of this result, which is delightfully amazing.
[1]...which, according to articles like this one, are possible: https://io9.gizmodo.com/what-would-the-earth-be-like-if-it-w...