Hopf fiber bundle topology is taught as simply as possible. Physicist Roger Penrose called the Hopf fibration, "An element of the architecture of our world." Essential in at least 8 different physics applications, the Hopf fibration is a map from a hypersphere in 4D onto a sphere in 3D. Many visualizations are displayed herein. Mathematician Eric Weinstein commented on the structure on Joe Rogan's podcast as, "The most important object in the entire universe."

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Chapters
0:00 Intro
1:08 Defining the Hopf fibration
1:42 Stereographic projection
3:24 Mapping the Hopf fibration
5:33 Hopf facts
7:11 Rotating in 4D
8:30 3D magic eye stereogram
For full length magic eye, Hennigan (2014):    • Hopf Fibration Stereogram (Divergent)  

If you're not yet familiar with higher dimensional shapes, you may want to first watch my video explaining a 4D hypercube known as the tesseract:
   • 4th Dimension Explained ► Tesseract H...  

The initial outline for this video was over 25 minutes, so I trimmed details of nspheres for the sake of brevity. Including here for those interested:

S0 0sphere | Pair of points | Bounded by lines
S1 1sphere | Circle | Bounded by pairs of points (S0)
S2 2sphere | Sphere | Bounded by circles (S1)
S3 3sphere | Hypersphere | Bounded by spheres (S2)

So the pair of points at the ends of a 1D line segment is considered a 0sphere, or S0. It's hard to visualize, but a straight line is an arc of a circle whose radius is infinite.

Now, a circle is bounded by those pairs of points. We say a circle is S1, or a 1sphere, sitting in 2D space.

A sphere is bounded by circles. We say a sphere is S2, or a 2sphere, sitting in 3D space.

You are probably noticing an important pattern here. Each of these structures are one dimension lower than the Euclidian space they are embedded within. This is because we are only concerned with the boundaries of each shape.

So for a circle, we look at just the 1dimensional circumference. Thus, S1.
For a sphere, the surface is actually 2dimensional. Thus, S2.

Now, we are navigating beyond the limits of human perception.
A hypersphere is bounded by spheres. We say a hypersphere is S3, or a 3sphere, sitting in 4D space. This is technically impossible to visualize.

At 2:58, I've included two visualizations of a hypersphere. The first is the shadow of a wireframe surface of a hypersphere, projected in 3D. A perfect model would be an opaque object, so this cage gives you a sense of the hypersphere composed of spheres. The second is a highly polished version with a few vertices in view. Neither version is perfect, but they are the next best things compared to Hopf maps:

https://commons.wikimedia.org/wiki/Fi...
Seemann (2017) https://vimeo.com/210631891

Nerd Alert
The interactive Hopf map visualizer by Nico Belmonte (@philogb)
http://philogb.github.io/page/hopf/#

Read this paper about the 8+ physics applications
https://www.fuw.edu.pl/~suszek/pdf/Ur...

Works Cited
http://dimensionsmath.org/ "Dimensions" series by Jos Leys
   / josleys  

Niles Johnson (2011), www.nilesjohnson.net
   • Hopf fibration  fibers and base  

Guido Wugi, Wugi's 4D World Series (2020)
   • Wugi's 4D world The 3sphere and its...  

Joe Rogan Experience No. 1203 (2018) | Eric Weinstein

Hennigan (2013)    • Hopf Fibration  

Dror BarNatan
http://www.math.toronto.edu/~drorbn/G...

Azadi (2020)    • Hopf Flower  

Roice Nelson http://roice3.org/h3/isometries/

https://www.joergenderlein.de/stereo...

3Blue1Brown (2018): Visualizing quaternions (4d numbers) with stereographic projection    • Visualizing quaternions (4d numbers) ...  

WBlut (2020) https://wblut.com/images/hopftubesa...

NonEuclidean Dreamer (2019)    • Train Ride through the Hopf Fibratiom...  

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