3 ms·

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,

by 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.
  - lmkg 4y ago
    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.
    - soperj 4y ago
      between your post, and kkwteh's, it makes a lot more sense. Thank you both.
- majou 4y ago
  Pac-Man on a Torus https://www.youtube.com/watch?v=x_KNFEqdd3Q https://www.youtube.com/watch?v=x_KNFEqdd3Q
kkwteh 4y ago
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.

by a_c_s 4y ago

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.
- lmkg 4y ago
  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.
  - soperj 4y ago
    between your post, and kkwteh's, it makes a lot more sense. Thank you both.
majou 4y ago
Pac-Man on a Torus https://www.youtube.com/watch?v=x_KNFEqdd3Q https://www.youtube.com/watch?v=x_KNFEqdd3Q

kkwteh 4y ago

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.