Does someone could explain to me (with animation in 2-D if possible) why could we express the torus as $\mathbb{T}= \mathbb{R}^n/\Gamma$ (with $\Gamma = \text{(identity matrix)} \mathbb{Z}^n$)? The problem is pretty clear when I look at the circle $\mathbb{T}= \mathbb{R}/2 \pi \mathbb{Z}$, but when we observe in higher dimension, the problem becomes hard to imagine.
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1If it's clear in 1-D it should be clear in 2-D since a torus is the product of two circles. So the (usual) torus is the plane mod the integer lattice. For higher dimensions you have to use your imagination :-) – user4894 Jun 28 '16 at 21:41
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Look at the coloring a torus on the Torus wikipedia page it shows first getting to a cylinder and then to a 2-torus. – AHusain Jun 28 '16 at 21:46
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When we quotient $\mathbb{R}^2$ by $\Gamma$, I would like to know what it represents on the torus with different matrices $A$ for $\Gamma = A \mathbb{Z}^n$. – AloeVera Jun 28 '16 at 21:46
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What does it mean by $\Gamma=I\mathbb{Z}^n$? – Ziyuan Jun 28 '16 at 22:34
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By the way why is the tag "spectral-graph-theory"? Any background from that field? Looks like topology for me. – Ziyuan Jun 28 '16 at 22:34
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Not an animation, but the image from my answer to “Natural” example of cosets may be of interest. – Andrew D. Hwang Jun 29 '16 at 00:05