(July 22, 2018 at 12:56 pm)Jehanne Wrote:(July 22, 2018 at 12:38 pm)polymath257 Wrote: Through practice, mostly. Start by trying to imagine a sphere in 4 dimensions: it provides an example of a curved *three* dimensional manifold.
I might recommend the book 'The Shape of Space' as an introduction:
https://www.amazon.com/Shape-Space-Chapm...e+of+space
Like seeing "dead people" (from the movie)? I can't imagine that 4th dimension, although, the math (tensors) is completely reasonable and valid.
Like I said, it takes some practice to get a 'feel' for it. I've spent *way* too much time learning how to visualize what is going on in 4D and in 3D manifolds.
One issue here is terminology. The sphere (surface only) in 3D is a 2D manifold. The sphere in 4D is a 3D manifold. The dimension of a manifold is determined *internally*, not via the embedding.
So, just like the 2D sphere in 3D, the 3D sphere in 4D has 'geodesics' that are 'great circles'. But there are also 'great spheres', that are the intersection of hyperplanes with the sphere.
Initially, it can be useful to use time as a fourth dimension for visualization purposes. So, a 3D sphere in 4D would start as a single point at some time (the south pole), expand up to the 'diameter sphere' and then shrink again to a single point (the north pole) and disappear. The full sequence is the sphere in 4D and is shown by successive cross sections. The problem with this visualization technique is figuring out what things look like when doing a cross section from a different direction.
