
Torus Earth (2014) - mef
http://www.aleph.se/andart/archives/2014/02/torusearth.html
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chairmanwow
What a fascinating premise. Kudos to the author for doing such a thoughtful
analysis. His technique for arriving at the cross-sectional shape is
especially interesting. I would have never considered that the optimal cross-
sectional shape to have been anything other than a circle.

I wonder if it would be possible to orbit the ring in the axial plane, around
a single arm of the torus? Or even in a lemniscate, intersecting the inner
Lagrange point?

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lainon
Previous discussion:

[https://news.ycombinator.com/item?id=7182822](https://news.ycombinator.com/item?id=7182822)

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hawski
It would be great to see a simulation how it would look like from the surface.
It has a great game-world potential.

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cronjobma
Exactly my thoughts. Combine that with VR and you'd have an interesting way of
visualising this stuff.

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Someone
_”There is a central unstable Lagrange point at the middle of the hole”_

So, why would Ringworld (an enormous toroidal planet with a relatively small
sun (a sun similar to our sun, which weighs 6E24 kg, Ringworld would weigh 300
times that, at 2E27 kg) at its center) be stable?

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akkartik
[https://en.wikipedia.org/wiki/Ringworld#Errors](https://en.wikipedia.org/wiki/Ringworld#Errors)

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alexozer
I wonder how many millennia it will be before we can trivially terraform one
of these from another planet for fun. Just deploy a hoard of self-replicating
autonomous robots and sit back for ten years. Might need some carefully-
orchestrated nuclear detonations to increase the average gravitational
potential, though.

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macawfish
So bizarre, I was just fantasizing about this possibility the other day. Then
here is this detailed analysis that just pops right up!

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pgreenwood
The ring system of Saturn (and others) aren't a bad start.

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unsignedint
In other word: is a 2D role playing game map feasible or not.

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tgb
But there you usually have a "flat torus" which cannot exist in euclidean
space. I.e. the distances between points in a PacMan style world are such that
the loops around the world are always the same size. That's not the case for a
toroidal planet as in this article: the inner equator is much smaller than the
outer one. Moreover, the flat torus has zero curvature and no surface (of
finite extent) that lies inside our space can have zero curvature everywhere.
The argument for that is remarkably elegant: suppose you have a surface in
space. Pick any point as an "origin". Then since our surface is assumed to be
finitely large, there must be a point on it furthest from the origin. But then
the surface at this point is tangent to a sphere centered at the origin and in
fact must lie entirely on the inside of the sphere. But then it curves at
least as much as the sphere and hence has positive curvature at that point.

So you'll never stumble across a PacMan world in all your travels across all
the galaxies. (Interestingly, PacMan's world can be embedded into a
hypersphere sitting in 4-dimensional space.)

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scrumper
I thought it does have curvature (like a non-flat torus), but that curvature
is in effect compressed into 'singularities' at the corners.

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tgb
It's possible to hide curvature in singularities like that, but that's not the
classical "grid with wrap-around on the top and bottom" you see in video
games. In particular, there's nothing "special" about the corners - you can
still go up/down/left/right at them. In essence, you could recenter the grid
around any point without changing anything so the corners only _look_ special
but aren't.

The torus actually needs total curvature 0 (the beautiful Gauss-Bonnet theorem
says this and is true even for a torus that _cannot_ fit in 3 dimensional
space), which means you'd need some "singularities" with negative curvature
and some with positive curvature to cancel out. But clearly all the corners in
the torus are the same! So there's no way to hide the right curvature in them.

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scrumper
Aha yes I see. I needed to think more carefully about total curvature. I have
been doing a bit more reading on this, really a fascinating area. Thanks for
the pointers.

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jlebrech
and those who live on there would have a head start on knowing the shape of
their planet, well those on the inside at least would.

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mangecoeur
Fascinating, and great to imagine that given the sheer number of start in the
known universe, there is a chance that somewhere out there such a thing exists
right now.

