How to visualize the quotient space $\mathbb{R^2}/ \mathbb{Z^2}$ to be a torus? you may also refer me to some books or websites. Because I want to see how the knot torus winds in this case. thank you!


1 Answer 1


Think of turning a square into a donut as follows. Start with a (stretchy) square sheet of paper (the unit square $[0,1]\times [0,1]$). Identify the top and bottom edges to form a cylinder. Now imagine you stretch the cylinder into a donut by bringing the two circles forming its boundary together.

Here is a nice video illustrating what I've just said.

To answer the question in your comment, think of the line $y=\frac{p}{q}x$ in the plane. In $\mathbb{R}^2/\mathbb{Z}^2$, we are identifying all lattice points with each other. Notice that on our line, beginning at $(0,0)$, we next reach a lattice point at $(q,p)$. This portion of the line corresponds to the closed curve on the torus, because the portion of the line from $(q,p)$ to the next lattice point $(2q,2p)$ is just retracing the same curve. That is, any point $(x,y)$ on this second portion is identified in $\mathbb{R}^2/\mathbb{Z}^2$ with the point $(x-q,y-p)$ which lies on the first portion.

Here is a video illustrating the case when $p=3$ and $q=4$. As the line travels around the torus, think of the corresponding picture in the plane where we begin at $(0,0)$ traveling towards the lattice point $(4,3)$. When we reach this point in the plane is exactly when the curve closes itself in the video.

  • $\begingroup$ how the line with slope $\frac{p}{q}$ becomes a closed curve ..? $\endgroup$
    – Ronald
    Apr 13, 2013 at 17:16
  • $\begingroup$ thanks a lot, this is very useful :) $\endgroup$
    – Ronald
    Apr 14, 2013 at 6:04

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