I stumbled upon this https://en.wikipedia.org/wiki/Hopf_fibration through a Twitter clickhole and I’d like to know what I’m looking at.
Is it that a circle becomes a sphere by rotating (is that the right word?) in every direction? and then a sphere becomes a hypersphere by… being connected to every other part of its surface? What about a 4-sphere and a 5-sphere etc?
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A Hopf fibration is a way of cutting up a sphere in 4-dimension into circles, such that if you shrink all the circles each into a point you get a sphere in 3-dimension.
(terminology note: a sphere in 3-dimensional space is called a 2-sphere, a 2-dimensional sphere, because the actual surface of the sphere is only 2 dimensions; similarly sphere in 4-dimensional space)
Hopf fibration is extremely limited, only a few dimensions allow it.
An easy way to describe Hopf fibration is using complex numbers. Imagine a pair of complex number (z,w) whose sum of absolute value squared is 1: |z|^2 +|w|^2 =1. This is in the space of 4-dimension because each complex number is 2 dimensions. It’s a sphere because the radius is 1.
Now, cut this into circles as follow. We declare that any 2 pair of numbers (z1,w1) and (z2,w2) belongs to the same circle if there is a complex number t that when you multiply t by both number in a pair you get the other: z1=tz2, w1=tw2. It might be hard to see that they actually form circles, but after doing some algebra, you can check that t must always be a complex number of norm 1, so the possible value of t is indeed a circle. An alternative way of phrasing this is that the proportion z1:w1 and z2:w2 is the same; another way of phrasing this is that z1w2=z2w2 (by cross multiplication).
What if you shrink every circle into a point? You can see what shape it forms by compute z/w. Since every pair of complex numbers that form a circle have the same proportion, when you divide like this, you get a single point. Note that 1/0 is also a point, infinity. When you divide, you obtain every possible complex numbers together with an infinity all around you, and if you think about it (or use stereographic projection), this looks like a sphere.
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