(July 22, 2018 at 8:18 pm)Jehanne Wrote:(July 22, 2018 at 6:26 pm)polymath257 Wrote: 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.
To get back to my OP, though, let's say that NASA launched a space probe into orbit between Mercury and Venus, such that the eccentricity of that orbit was zero, that is, a circle. Are you saying that the measured radius to the center of the Sun times 2 times pi would give the circumference of the orbit?
I suspect that would depend on how you measure the radius and circumference.
