Calculus on the 3-sphere

Take these six steps in order to visualize three different types of path on S^3. First, parameterize a path by taking points from the line of Reals into the Complex plane. Second, use a map to take the path from the complex plane into the double struck C^2, which is the product of two complex planes. This type of path in C^2 is called gamma sub one. Third, solve a system of differential equations on C^2 in order to deform the path in a smooth way, without cutting it. This involves connecting a point in path gamma sub one to another point in the transformed version of gamma sub one. Do this for all of the points in gamma sub one and call the connecting paths gamma sub 2. Fourth, connect the endpoints of all paths of type gamma sub two in order to form the final version of transformed gamma sub one. Call the path made of the endpoints gamma sub three. Fifth, project all three types of path in C^2 to the three-dimensional unit sphere S^3 as a subset of C^2. Finally, by the Hopf map project all three types of path onto the two-dimensional unit sphere S^2 as a subset of double struck R^3, which is the Euclidean three-space.

Rupert Way (2008).

[jonathanyun]

As someone who's been spoiled by the incredible animations done by 3blue1brown, this video is a bit underwhelming... but it does still clearly illustrate an increasing dimensionality and the continious behavior of calculus throughout, so well done! So much time and effort goes into making just a 10 second animation, so I applaud you for that!

Also, 3 years into my undergrad and a lot of the math is still going over my head loll, so that's probably still more my faulth :P