Hopf fibration -- fibers and base
Why Mathematics Stack Exchange recommends it
Mathematics Stack Exchange users link it when someone needs a concrete picture of Hopf fibers, such as nested tori, or of how a 3-sphere decomposes into circles.
Recommended by 5 people across 3 different months, 2013 to 2023.
What it is
A short animation by mathematician Niles Johnson showing how circular fibers of the Hopf fibration over points on the two-sphere sit inside the three-sphere, projected into 3D space. It is a purely visual piece with no narration, with details and source code on the author's website.
Cited in 1 Wikipedia article, including Hopf fibration.
What people said
Recommends Jorge Antonio Faya Torres · 2013-04 · Mathematics Stack Exchange · on “What curve is this?”
Your earring is an example of fibers in the Hopf fibration [LINK] . Watch this [this video] . Also check this [LINK] . A video of Niles talking about the Hopf fibration [LINK] can be found here [LINK] .
Recommends Sammy Black · 2013-04 · Mathematics Stack Exchange · on “What curve is this?”
My first guess was that it was a collection of fibers (the circles) of the Hopf fiber bundle $S^3 \to S^2$, which presents the $3$-sphere as collection of circles, parametrized by the $2$-sphere. This animation by Niles Johnson attempts to help you visualize this. [this video]
Recommends Heitor Fontana · 2013-04 · Mathematics Stack Exchange · on “Hopf Fibration in Local Coordinates”
notice that the fibers (the circles) are all disjoint and fill $S^3$. So by $S^3\rightarrow\Bbb{R}^3$ we obtain a decomposition of $\Bbb{R}^3$ into circles! Then, if you take the fibers along all points on a *parallel* of $S^2$ you get a torus (which is mapped on a torus by projection). Moving the parallel up and down you get *concentric* torii. Actually you get a torus if you take the fibers above any circle in $S^2$ I think. In this video [this video] you can see how this is done. In the bottom-right you see the point $p\in S^2$ on which the fiber is taken.
Recommends Son Gohan · 2020-12 · Mathematics Stack Exchange · on “How to dynamically visualize $3$-spheres?”
You can look at this nice video of the Hopf fibration to visualize a sphere in 4 dimension. [this video] Of course to understand it you need some knowledge of what a bundle is, but see it in this way: in the video you see different parts of the 4-sphere.
Recommends D.R. · 2023-03 · Mathematics Stack Exchange · on “Visualizing the Poincare homology sphere”
I know that past a certain point, one should graduate from the view that homology/homotopy groups "count holes" in any realistic, grounded, real-life meaning of the word "hole". However, I still want to see if there is any visual intuition to be had regarding the Poincare homology sphere, which is sort of the first example of Is there a non-simply-connected space with trivial first homology group? [LINK] (also the first example in this MO thread [LINK] ). I did come across [LINK] , which presents some nice but difficult (for me) to make-sense-of pictures [LINK] or another picture here [LINK] . I searched up "visualization of poincare homology sphere", and there was only one Youtube video [LI
Cites Arthur · 2015-11 · Mathematics Stack Exchange · on “What software can draw pictures like this?”
The image information page [LINK] claims it is done with Sage. It is part of this animation [this video] , where your image appears at timestamp 2:27.