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."
Patreon [ Ссылка ]
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): [ Ссылка ]
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:
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The initial outline for this video was over 25 minutes, so I trimmed details of n-spheres for the sake of brevity. Including here for those interested:
S0 0-sphere | Pair of points | Bounded by lines
S1 1-sphere | Circle | Bounded by pairs of points (S0)
S2 2-sphere | Sphere | Bounded by circles (S1)
S3 3-sphere | Hypersphere | Bounded by spheres (S2)
So the pair of points at the ends of a 1D line segment is considered a 0-sphere, 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 1-sphere, sitting in 2D space.
A sphere is bounded by circles. We say a sphere is S2, or a 2-sphere, 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 1-dimensional circumference. Thus, S1.
For a sphere, the surface is actually 2-dimensional. 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 3-sphere, 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 wire-frame 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:
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Seemann (2017) [ Ссылка ]
🚩 Nerd Alert 🚩
The interactive Hopf map visualizer by Nico Belmonte (@philogb)
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Read this paper about the 8+ physics applications
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🚾 Works Cited
[ Ссылка ] "Dimensions" series by Jos Leys
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Niles Johnson (2011), www.nilesjohnson.net
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Guido Wugi, Wugi's 4D World Series (2020)
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Joe Rogan Experience No. 1203 (2018) | Eric Weinstein
Hennigan (2013) [ Ссылка ]
Dror Bar-Natan
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Azadi (2020) [ Ссылка ]
Roice Nelson [ Ссылка ]
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3Blue1Brown (2018): Visualizing quaternions (4d numbers) with stereographic projection [ Ссылка ]
WBlut (2020) [ Ссылка ]
Non-Euclidean Dreamer (2019) [ Ссылка ]
🚨 Legal
Copyright Disclaimer under section 107 of the Copyright Act 1976, allowance is made for “fair use” for purposes such as criticism, comment, news reporting, teaching, scholarship, education and research. Fair use is a use permitted by copyright statute that might otherwise be infringing. No copyright is claimed herein and to the extent that material may appear to be infringed, I assert that such alleged infringement is permissible under fair use principles in U.S. copyright laws. Content may be amended.
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