11 ms·

Living on a Torus

47 points by kkwteh 4y ago

lproven 4y ago
When this says 3-dimensional, does it really mean 4-dimensional? Or am I just not smart enough to understand this?
- scapp 4y ago
  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.
  - lproven 4y ago
    After much debate on Twitter, I understand this better now. https://twitter.com/lproven/status/1564613978155802631 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.
a_c_s 4y ago
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?
- slopbop 4y ago
  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.
  - soperj 4y ago
    How's this a donut and not a sphere?
    - kkwteh 4y ago
      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.
gfodor 4y ago
This phenomenon is why I built my spatial documents system https://jel.app https://jel.app to be torus worlds.
alanbernstein 4y ago
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.
Random_BSD_Geek 4y ago
> Living on a Torus I can't help but hear this in Bon Jovi's voice.
kkwteh 4y ago
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".
- Lammy 4y ago
  I would go for "Living as a torus" https://www.youtube.com/watch?v=XoacbEcvmYo https://www.youtube.com/watch?v=XoacbEcvmYo :)
- lproven 4y ago
  "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.
debacle 4y ago
> 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.
- bagels 4y ago
  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.
  - lproven 4y ago
    Exactly so. I wondered what the OP was talking about. Some mathematically-inclined friends on Twitter enlightened me. https://twitter.com/lproven/status/1564613978155802631 https://twitter.com/lproven/status/1564613978155802631
- brummm 4y ago
  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 https://en.wikipedia.org/wiki/Torus) that defines an n-dimensional torus in the same way.
  - debacle 4y ago
    True, but "a three-dimensional torus" is a doughnut.
    - tomjakubowski 4y ago
      donut is to torus as ball is to sphere, is that what you mean?

Living on a Torus

47 points by kkwteh 4y ago

lproven 4y ago

When this says 3-dimensional, does it really mean 4-dimensional? Or am I just not smart enough to understand this?

scapp 4y ago
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.
- lproven 4y ago
  After much debate on Twitter, I understand this better now. https://twitter.com/lproven/status/1564613978155802631 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.

a_c_s 4y ago

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?

slopbop 4y ago
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.
- soperj 4y ago
  How's this a donut and not a sphere?
  - kkwteh 4y ago
    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.

gfodor 4y ago

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

alanbernstein 4y ago

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.

Random_BSD_Geek 4y ago

> Living on a Torus I can't help but hear this in Bon Jovi's voice.

kkwteh 4y ago

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".

Lammy 4y ago
I would go for "Living as a torus" https://www.youtube.com/watch?v=XoacbEcvmYo https://www.youtube.com/watch?v=XoacbEcvmYo :)
lproven 4y ago
"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.

debacle 4y ago

> 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.

bagels 4y ago
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.
- lproven 4y ago
  Exactly so. I wondered what the OP was talking about. Some mathematically-inclined friends on Twitter enlightened me. https://twitter.com/lproven/status/1564613978155802631 https://twitter.com/lproven/status/1564613978155802631
brummm 4y ago
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 https://en.wikipedia.org/wiki/Torus) that defines an n-dimensional torus in the same way.
- debacle 4y ago
  True, but "a three-dimensional torus" is a doughnut.
  - tomjakubowski 4y ago
    donut is to torus as ball is to sphere, is that what you mean?