Living on a Torus | Hacker News Reader

Living on a Torus

vinequai.com

47 pointskkwteh23 comments

> You might think that life in a three-dimensional torus

The author is talking about toroidal space which is confusing because a normal torus is a three dimensional thing with normal Euclidean geometry.

Thank you for explaining this, something was obviously missing. Living on a 7 meter torus would be challenging for many reasons, but seeing many cylinders or images of yourself wouldn't be some of them.

Exactly so. I wondered what the OP was talking about. Some mathematically-inclined friends on Twitter enlightened me.

https://twitter.com/lproven/status/1564613978155802631

That's not really correct. A torus is a 2-dimensional manifold that can be embedded into 3-dimensional Riemannian space. The author of the article is correct talking about a three dimensional torus as it's defined as S^1 x S^1 x S^1. You can also check the wikipedia page (https://en.wikipedia.org/wiki/Torus) that defines an n-dimensional torus in the same way.

True, but "a three-dimensional torus" is a doughnut.

donut is to torus as ball is to sphere, is that what you mean?

This isn't very clear what it is demonstrating and the explanation doesn't help.

If I lived on a small sphere and saw a pole in the ground, tied a rope to it, and walked all the way around the sphere I would see the same pole and I could tie the other end of the rope to it. I wouldn't see dozens of poles floating in the sky.

So why would I see duplicate copies of the same pole if I lived on a donut-shaped planet instead?

In this example, you're not living on the surface of a donut shaped planet, you're living in a 3D space (not a surface!) that is the 3D equivalent of a donut.

Pacman lives on the surface of an actual 2D donut, when he goes to the left side of the screen, he pops out on the right side, and when he goes to the topmost part of the screen, he comes out from the bottom. (Not convinced this is the same as a donut? Imagine the surface was made of a stretchy film and bend the lefthand side to meet the righthand side, forming a cylinder. Now, to make the topmost side meet the bottom side, you fold the cylinder into a donut shape!)

This is the 3D version of the "Pacman universe", where if you go up enough, you come back around the bottom, and the same for all the cardinal directions.

How's this a donut and not a sphere?

A donut is a way to embed a two-dimensional torus in our three-dimensional space. What we have here is different. It's a visualisation of a three-dimensional torus. On a two-dimensional donut, there are two directions which loop around. In the space shown here, the only difference is that there are three directions.

A three-dimensional sphere also loops around, but it's not quite the same. One way to get the three-dimensional sphere would be to glue each points at the cube boundary to every other point on the boundary. One way to show that this three-dimensional sphere is not the same as the three-dimensional torus is that in the three-dimensional sphere, you could gather up any tied rope by passing it around the cube boundary.

For a sphere, the location of where you land when you go off the screen is a continuous function of where you started from. If two Pac-Men exit the screen next to each other, they will re-enter the screen next to each other.

The real Pac-Man game is a donut because it's discontinuous at the corners. If two Pac-Men are right next to each other near the top-left corner, and one exits via the top and the other exits via the left, they will end up on opposite sides of the map.

There's a mathematical formalization of this, where the thing you look at is closed paths of Pac-Man leaving a point, traveling around, and returning back to that same point. You group such circuits by whether they can be continuously deformed into each other. The discontinuity at the corners makes two distinct families of circuits, which correspond to traveling on a donut around the circumference vs going through the hole.

between your post, and kkwteh's, it makes a lot more sense. Thank you both.

Pac-Man on a Torus

https://www.youtube.com/watch?v=x_KNFEqdd3Q

I'm sorry my explanation wasn't clear. The reason why duplicate copies are seen in is because the light can loop around space multiple times before reaching your eyes. If you launch a long rope to a distant cylinder, look at what you see in the bottom left miniature. That's an indication of the looping path that the light is taking from that cylinder to reach your eyes. Launch a rope at different cylinders to see what some of the different paths are.

As a side note, if you lived in a small three-dimensional sphere, you would be able to see an object located at the antipodal point smeared out in every direction, because following a geodesic in each direction leads to the antipode. I've seen this visualised in the video game Hyperbolica.

The game Manifold Garden is structured as 3D torus. It mostly affects the visuals, but it is sometimes an important aspect of the gameplay too.

This phenomenon is why I built my spatial documents system https://jel.app to be torus worlds.

When this says 3-dimensional, does it really mean 4-dimensional?

Or am I just not smart enough to understand this?

Spaces like this (manifolds) are usually classified by which Euclidean space they look like locally. So, for example, the surface of a sphere is two-dimensional since it locally looks like a plane (whence flat-earthers). A two-dimensional torus can be visualized as the surface of a doughnut, but also as a square where going off an edge brings you to the opposite edge.

As explained in the article, the torus in question is locally three-dimensional, and you might mistake it for ordinary 3d space at first. The specific geometry we're visualizing is the 3d analogue of the square from above: it's a cube where going off an face brings you to the opposite face.

After much debate on Twitter, I understand this better now.

https://twitter.com/lproven/status/1564613978155802631

For a post that claims to be explaining something, I have to say, I think it's really quite stunningly poor at explaining itself.

This is not about living on a torus. As others have said, a torus is a perfectly possible 3D object in our universe, and SF has many toroidal habitats, including the original Larry Niven Ringworld and Iain M Banks' "Orbitals" as used in several Culture novels, and as also used in the videogame Halo (I believe -- not played it; not a gamer).

Since the article is called:

> Living on a Torus

Note: on

And begins:

> life in a three-dimensional torus that measured only several metres in each dimension

That to me implies inside an object. A toroidal habitat is an absolutely standard part of SF and has been since before 2001: A Space Odyssey in 1969.

The article is in fact about living inside a spacetime continuum, a universe, which is toroidally shaped, and strangely small. (What you are on is unspecified.) It doesn't say so but it seems to be.

Since I have never really wondered about the shape of spacetime, I could not even begin to parse the essay.

Looking at some of the comments, I wonder if the blog post would have been less confusing if I gave it the more appropriate title "Living in a Torus".

I would go for "Living as a torus" https://www.youtube.com/watch?v=XoacbEcvmYo :)

"In" would have been better, but not much at all and would not have helped significantly, IMHO.

About a paragraph of explanation setting out your axioms and explaining that you are discussing the shape of the universe, of a finite bounded spacetime continuum, was needed.

And ideally, several paragraphs of explanation, building up slowly, was what you really needed. Starting with examples of a linear 1D continuum where you can only move in 1 dimension, and how it would look, then of a 2D one where it's possible to go round the axis, then a planar one, then illustrations of how the planar one's edges could be connected. It also needed explanations of the notions of flatness and curvature, of connectivity, and some basic explanations of the topology involved.

As it was, nothing was explained: just a cryptic title that contradicted the misleading first line. :-(

Sorry to be a harsh critic, but as an educated layman who's read entire books on topology, spacetime, definitions of connectivity and dimension and so on, this was totally opaque and mystifying.

If you'd at least given it that 1-para context, it might have worked. Even a sentence of context. But just jumping into a complex thought experiment? Impenetrable.

> Living on a Torus

I can't help but hear this in Bon Jovi's voice.

Reply on news.ycombinator.com