How can I "move through a hypersphere?"

Question

A man walking along a 2 dimensional circle will take a periodic path that begins and ends at the same point.

Since he can travel in only a single direction, let's say how far along he is in his walk is represented by a single variable $\theta$. His position is then given in cartesian coordinates by

$$ x = r\cos \theta \\ y = r\sin \theta $$

A man walking along a 3 dimensional sphere will take a periodic path that begins and ends at the same point.

Since he can travel in only 2 (orthogonal) directions, let's say how far along he is in his walk is represented by a 2 variables $\phi, \theta$. His position is harder to find here, but if we always have the axis through his head and the two circles along which he can move lined up with the other 2 cartesian axes, a displacement along the surface of the sphere (from a given starting position) can still be found:

$$ x \approx r\sin \phi \\ y \approx r \\ z \approx r\sin \theta $$

(For this to be correct the axis must always follow him. I don't think the above formula is completely correct! But the point is, he can walk along the sphere by chasing these 2 variables (and by walking along these orthogonal "elastic band" lines that we wrap around the sphere based on his current position)).

So now here's my question. If you can move along the 1 dimension of a circle in 2D space in a periodic fashion by moving one variable, and you can move through (and completely cover) the surface of a sphere in 3D space in a periodic fashion using two variables, what does motion of the same type described above look like for the hypersphere?? What "directions" do the 3 required variables navigate? Where is the "wrap point" for someone standing in a hypersphere? Can someone visualize this at all?

Answer 1

Visualization is really difficult in higher dimensions, generally I find the easiest way is thinking of the lower dimensional cases and just "go" from 3 to 4 dimensions in the same way I go from 2 to 3.

Like, if I'm walking on a one-dimensional line it's fairly easy to visualize what changes when I switch to walking on a two-dimensional plane, there is one more variable or degree of freedom.

But when I start walking on a three-dimensional "plane", it's hard to visualize beacause I can't visualize a four-dimensional space containing the three-dimensional plane. In this case there's also an additional degree of freedom, so one way of visualizing the change from a two-dimensional plane in a three-dimensional space to a three-dimensional hyperplane in a four-dimensional space is to just imagine one dimension is "hidden" from view.

It might not sound as much, but that's what works for me. I look at the two-dimensional plane as a line and the three-dimensional hyperplane as a plane, keeping in mind that there are "hidden" dimensions.

In the case of spheres it becomes more complicated to visualize. But I look at it like this, when I'm moving on the circle, I'm essentially walking on a straight line whith the property that after a while I return to where I started. The two dimensional case is like playing a computergame where moving of the screen makes you enter on the opposite side. So moving on a three-dimensional hypersphere is like moving on a "surface" where you can move in three dimensions (i.e. meatspace), but where if you move in one direction you'll be back where you started. Like som theories suggests the universe could be like.

I don't have a firm grasp of the exact mathematical differences between "flat" and "spheric" surfaces, but they're both surfaces and share many behaviours. For working out the coordinate mappings I suggest taking a look at wikipedia: https://en.wikipedia.org/wiki/Hypersphere#Spherical_coordinates

I hope this makes sense, otherwise feel free to ask for clarifications. :)

Answer 2

A line can be regarded as one specific cut (1D-view/move/degree of freedom...) of a 2D-space (a plane). A plane can be regarded as one specific cut (2D-view/move/degree of freedom...) of a 3D-space. Suppose a 3D space can be regarded as one specific cut (3D-view/move/degree of freedom...) of a 4D-space. ...

Similar it is with your example in spherical coordinates.

A circularity on a circle can be regarded as one specific cut (view/move/degree of freedom...) of a circularity on a sphere... hence suppose a circularity on a sphere can be regarded as one specific cut (view/move/degree of freedom...) of a hypersphere.

In other words you would have different spheres (3D) on which the buddy moves at the same time (as it would, in lower dimmension, do on different circles that embodied in a sphere), which embody into the same hypersphere.

This is why when the buddy moves in the hypersphere it would look to us in 3D that he would appear and then disappear because we remain part of one sphere (3D). So what I am somehow missing in your well thought approach is that he may disappear and appear. Your approach resembles then such a navigation rule for the buddy in order to never disappear and remain constrained to the one sphere; although moving in the hypersphere.

This accords with the fact that there might be a possiblity to achieve a hyperspace by some folding of space. And I am not talking about science fiction, rather strong strong geometric imagination based on mathematical foundation. In your case it could be for instance that the buddy disappears on one side and just abruptly appears on the other side of the sphere. Imagine you would live on a circle and watch a buddy moving on a sphere, which emboddies that circle - that would be like on one of your colored lines, he then may appear and disappear.

Note: by visible I mean that we are also on the sphere and remain as reference in 3D on the sphere, watching the buddy or waiting for him. I think it is usually significant to be aware about the viewers position and restriction as the reference.