Question about a torodial universe

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In summary: The original question was about higher dimensional analogs and the same is true, one can for instance start with a cube and make the same "Pac Man" identifications. Going out thru the right side is the same as coming in thru the left. Ditto front and back, ditto top and bottom.
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Does the fact that on 2D torus's, one dimension on the torus always has to be 'smaller' than the other dimension hold on the higher dimension versions?
 
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You can have a 2d torus?
 
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I mean 2D in the way you could call the surface of a sphere a 2-sphere, I guess I was trying to sound 'mathy'. I just mean a doughnut surface by 2D torus.
 
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I am not sure what you mean...even in 2D, you can identify the opposite sides of a *square* to have torus topology. So what do you mean by "smaller"?
 
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yenchin said:
I am not sure what you mean...even in 2D, you can identify the opposite sides of a *square* to have torus topology. So what do you mean by "smaller"?

Good point, thanks. 2D torus so constructed is often given as an example of a flat differential manifold. Zero intrinsic curvature. No boundary. Not embedded in any surrounding space.
Sometimes people talk about the "Pac Man" game screen as a square with left edge identified with right edge, and top with bottom.

It is not true that one circumference or girth has to be bigger or smaller than the other. The construction works just as well with a square as with a rectangle.

The original question was about higher dimensional analogs and the same is true, one can for instance start with a cube and make the same "Pac Man" identifications. Going out thru the right side is the same as coming in thru the left. Ditto front and back, ditto top and bottom.

that's a 3D torus and it does not have to "live" in any higher dimensional space, and it is boundaryless. Standard differential geometry.

Thanks for making the same point in the other thread about the 1D torus analog---the 1D "ring" made by taking a line segment and identifying the endpoints. Does not require a 2D surround. Need not be immersed in any higher dim'l space.
 
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marcus said:
It is not true that one circumference or girth has to be bigger or smaller than the other. The construction works just as well with a square as with a rectangle.
Right. This limitation only comes into play when you embed the torus in three dimensions. I believe you can get around this limitation by embedding it in four dimensions instead. Or just by not embedding it at all and only dealing with the two dimensions inherent to the torus.
 

1. What is a toroidal universe?

A toroidal universe is a theoretical concept in cosmology where the shape of the universe is that of a torus, similar to a donut. This means that the universe would be finite but have no boundaries, and would eventually lead back to its starting point.

2. How is a toroidal universe different from a flat or spherical universe?

A toroidal universe is different from a flat or spherical universe in terms of its shape and curvature. In a flat universe, the shape is infinite and has no curvature, while a spherical universe has a positive curvature and is finite. A toroidal universe has a negative curvature and is finite but has no boundaries.

3. Is there any evidence for a toroidal universe?

Currently, there is no definitive evidence for a toroidal universe. It is a theoretical concept that is still being studied and explored by scientists. However, some theories and observations, such as the topology of the cosmic microwave background radiation, suggest that a toroidal universe is a possible explanation for the shape of the universe.

4. Can we travel in a toroidal universe?

In theory, it would be possible to travel within a toroidal universe. However, due to its negative curvature and finite size, it would be difficult to travel in a straight line, and one would eventually end up back at their starting point. It is also important to note that this is purely hypothetical, as we do not know for certain if our universe is toroidal.

5. What implications would a toroidal universe have for our understanding of the universe?

A toroidal universe would have significant implications for our understanding of the universe and cosmology. It would challenge our current models and theories about the shape and size of the universe, and could potentially lead to new discoveries and insights about the nature of the universe. It would also have implications for concepts such as the Big Bang theory and the ultimate fate of the universe.

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