No perfect circles in space...

[Jehanne]

I can imagine a 3D sphere just fine; a 3D sphere that is curved through a (technically, 5th dimension) hyperspace, no, I can not imagine that. Trying to do so only gives me a headache. If you have access to a 3D printer, feel free to construct a model. Please post pics.

[polymath257]

I can imagine a 3D sphere just fine; a 3D sphere that is curved through a (technically, 5th dimension) hyperspace, no, I can not imagine that.  Trying to do so only gives me a headache.  If you have access to a 3D printer, feel free to construct a model.  Please post pics.

OK, let's be clear in our use of language.

The sphere in 3D is a *2D* sphere: locally it is a two dimensional thing. That's why people can be confused that the Earth is flat.

The sphere in 4D is a *3D* sphere. it consists of all points in 4D that are some fixed distance from the center in 4D.

In both cases, distances *in the sphere* are different than distances *in the larger space*.

Thinking of our 3D space as actually being like the 3D sphere in 4D is similar to those who thought the Earth to be flat coming to grips with it actually being a 2D sphere in 3D.

So, on the earth, if you head off in any direction and keep going in a 'straight line', you will eventually come back to your starting point. And this is true no matter which direction you go.

The same is rue for the 3D sphere in 4D.

[Jehanne]

If you want to code a Matlab simulation, I will be happy to watch and enjoy.