Torus Earth (2014) | Hacker News Reader

Torus Earth (2014)

aleph.se

110 pointsmef17 comments

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?

Previous discussion:

https://news.ycombinator.com/item?id=7182822

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

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

”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?

https://en.wikipedia.org/wiki/Ringworld#Errors

The Ringworld is unstable!

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.

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

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

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

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.)

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

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.

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.

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.

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.

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