I have some trouble with the fourth spatial dimension

[TubbaBlubba]

I suppose it might be appropriate here. I'm not really looking for an answer in pure maths (Topology is waaaaaaaaaaay beyond me), but rather in concept.

Now, I can appreciate the properties of a tesseract, how it relates to and derives from the cube (I think of it as either putting 8 cubes together, or "pushing" the vertices of a cube into a fourth dimension), at least from a projective view.

I also have a "feeling" for the properties of a Klein bottle and I can accept the way it's constructed.

However, I've been wrestling about the thoughts of a spatially "enclosed" universe for a while. I've been thinking about a 2-d world on a sphere (which would have no way out, but never encounter a border).

Logically, one would move on to the 3-sphere (I think that's what it called?) but I can't quite grasp the concept or implications of it. All projections of it that I've seen simply seem to be an infinite amount of rings superimposed into a torus of some sort? Can anyone attempt to explain what the 3-sphere is trying to imply and what "properties" it would have?

This is purely out of recreation and curiosity on my part, I'm unlikely to ever come in touch with mathemathics of this sort. Regardless, I have a bit of a keen interest in it. While I might not have a good knowledge of the relevant mathemathics, I can PROBABLY wrap my head around the concepts required to it.

I hope someone will make an attempt to enlighten this uneducated lad.

 

[eok20]

One way to get a grip on the 3-sphere is to consider analogies with lower dimensional spheres (a regular 2-dimensional sphere and circle). If you consider a sphere and intersect it with a plane (a 2-dimensional infinite and flat surface in 3-space), you will get circles. If you take the 3-sphere and intersect it with a hyperplane in 4-space (a 3-dimensional infinite flat "surface" in 4-space) you will get spheres. Algebraically, the 3 sphere is the set of all numbers (x,y,z,w) that satisfy x^2+y^2+z^2+w^2 = 1.

Of all the possible 3-dimensional spaces, the 3-sphere has some very nice properties. First of all, it is compact which basically means it is finite in size. Like you mentioned, it has no boundary. It also has the property that if you have a loop in it you can shrink it to a point (this is also happens on the regular sphere but doesn't happen on a torus since you can take a loop that goes around the hole). It was actually shown a few years ago that this is the only 3 dimensional space that has these properties (this is the Poincare conjecture).

 

[TubbaBlubba]

Thanks a lot! I looked a bit more into it, and I think I can get a bit of a grip on it with the analogy of orthagonally superimposing two 2-spheres in a 4-dimensional room.

Definitely an interesting object.