Gluing a Torus
Why Mathematics Stack Exchange recommends it
Mathematics Stack Exchange answerers link it to show visually how a square with identified edges becomes a donut, as in questions about R^2/Z^2 and wrap-around grids.
Recommended by 3 people across 3 different months, 2013 to 2020.
What it is
A 55-second animation showing a rectangle being glued along its edges to form a torus. It was made for a topology seminar at Leibniz Universitaet Hannover and mentions the Seifert-van Kampen theorem for computing fundamental groups of glued spaces.
What people said
Recommends Jared · 2013-04 · Mathematics Stack Exchange · on “torus by identifying two equivalent points (mod $\mathbb{Z^2}$)”
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 [this 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, be
Recommends Arthur · 2015-05 · Mathematics Stack Exchange · on “Total no of closed loop paths in 3-by-3 grid”
No, it's a torus, watch this little animation: [this video] That's what you get when you wrap the edges around to meet.
Recommends Xi Wu · 2020-12 · Mathematics Stack Exchange · on “understanding $\mathbb{R}$/$\mathbb{Z}$”
Just try to add some intuitive answers: The essential intuition that $\mathbb{R}/\mathbb{Z}$ is isomorphic to unit circle in the complex plane is that we can map a real number $x$ to the radian so we get $e^{2\pi i x}$ , and so if two number differ by integers they map to the same radian (we circle back). The same intuition carries over to the isomorphism that $\mathbb{R}^2/\mathbb{Z^2}$ is isomorphic to the torus: We just need to tweak how to generate the coordinates: It turns out that any coordinate on a torus can be indexed by two radians. See this video at 0:33 [this video] , there is one horizontal circle, and one vertical circle, so you can think of the first real number as a radian that ma