Exploring the Topology of a Two-Dimensional Torus and a Rolled Cylinder

In summary, the space resulting from the identification (x,y) ~ (x+2piR, y+2piR) is a cylinder rolled up along the diagonal. This is different from the space resulting from (x,y) ~ (x+2piR, y) and (x,y) ~ (x, y+2piR), which is a two-dimensional torus (a donut). Switching to coordinates such as x' = (x + y)/sqrt(2) and y' = (x - y)/sqrt(2) results in a cylinder rolled around the y' axis.
  • #1
ehrenfest
2,020
1
What is the space resulting from the identification (x,y) ~ (x+2piR, y+2piR)? How is it different from the space resulting from
(x,y) ~ (x+2piR, y)
(x,y) ~ (x, y+2piR), which is a two-dimensional torus (a donut)
 
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  • #2
It's a cylinder, rolled up along the diagonal.
 
  • #3
What diagonal?
 
  • #4
I'd say the diagonal indicated on the attachment (up to symmetry :tongue:)
 

Attachments

  • diagonal.jpg
    diagonal.jpg
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Last edited:
  • #5
Sorry, I don't see any attachment. Was that just a joke?
 
  • #6
No, I forgot to click upload. Apparently you posted while I added it.

(BTW: Post [itex]13^2[/itex] for me)
 
  • #7
[itex]13^2[/itex]

Why would you ask me to post that?
 
  • #8
He meant he hit post #169, he did not want you to post anything.
 
  • #9
Take (x,y) to (x+a,y+a) and you move up parallel to the diagonal between the x and the y axis. if you identify after a = 2Pi then you get a cylinder rolled along that diagonal.
 
  • #10
ehrenfest, what happens if you switch to coordinates x' = (x + y)/sqrt(2) and y' = (x - y)/sqrt(2)?
 
  • #11
I see why it is different than the other identification! The R has to be the same for both x and y.

If you switch to light-cone coordinates, then it is a cylinder rolled around the y' axis, right?
 

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