How A Threesphere Would Look or Work?

[Silverbackman]

How A Threesphere Would "Look" or Work?

I have been trying to understand threespheres (4D spheres) for a while and I just can't understand it.

First off is the 4th dimension time (it progressing in time is what makes it 4D)? Or does this sphere involve 4 spatial dimensions and one time dimensions (thus 5D)? Or is time not used?

How can it involve 4 spatial dimensions when we can only imagine 3? You can supposedly "construct" a threesphere according to the wiki article;

http://en.wikipedia.org/wiki/3-sphere

This article tries to demonstrate how to make it but I don't get it;

http://www.dpmms.cam.ac.uk/~etc21/hexlet/hexlet3.html

When I pressed "Animate" I still couldn't get it. What do a few balls rotating under and above two balls inside a bigger ball have to do with when you go in one direction in the universe that you will eventually come back to the point you started from (in a assumed finite universe)?

This site explains it a bit better;

http://www.math.union.edu/~dpvc/math/4D/stereo-projection/welcome.html

Ok that does make more sense of the universe was like a donut shape. And I can see how a sphere can be rotated to a torus (well sort of). But even in this case there a surface you can go outside of!

So in order for a threesphere to work does it have to constantly "move" in order to catch up with the point traveling at the "edge" of its shape before going around? Furthermore if the "outer space line" moves at a further distance from the point would it not violate what makes a sphere or circle a, well, sphere or circle (not at an equal distance from the origin)?

 

[matt grime]

Mathematially dimensions aren't anything physical. Time is only the 4th dimension if you are talking about minkowski space time (and ordering them so that it is the 4th I suppose).

 

[Silverbackman]

Mathematially dimensions aren't anything physical. Time is only the 4th dimension if you are talking about minkowski space time (and ordering them so that it is the 4th I suppose).

Then how is it that you can "construct" a model of a threesphere or "demonstrate" one? All I can imagine is going in one direction and coming around to the point you started from (and somewhat the torus analogy. How do the many balls touching demonstrate a threesphere?

 

[J77]

Doesn't it depend on self intersections - you can't make a three-sphere but you can immerse the three-sphere in our 3d world. However, it will have self-intersections...

See the Klein bottle for these ideas: http://www.kleinbottle.com/whats_a_klein_bottle.htm

I could be wrong here tho'...

A photograph of a stapler is a 2-dimensional immersion of a 3-dimensional stapler. In the same way, our glass Klein Bottles are 3-D immersions of the 4-D Klein Bottle. Our Klein Bottle is a 3-dimensional photograph of a "true" Klein Bottle.

A Klein Bottle cannot be embedded in 3 dimensions, but you can immerse it in 3-D. (An immersion may have self-intersections; Embeddings have no self-intersections. Neither an embedding nor an immersion has folds or cusps.)

We represent a Klein Bottle in glass by stretching the neck of a bottle through its side and joining its end to a hole in the base. Except at the side-connection (the nexus), this properly shows the shape of a 4-D Klein Bottle. And except at the nexus, any small patch follows the laws of 2-dimensional Euclidean geometry.

 

[matt grime]

Then how is it that you can "construct" a model of a threesphere or "demonstrate" one? All I can imagine is going in one direction and coming around to the point you started from (and somewhat the torus analogy. How do the many balls touching demonstrate a threesphere?

By anology, as we can represent a 3d object in 2d space. Some people use shading to indicate the 'size' of the 4th dimension for instance.

A 2sphere is lots of circles touching, if you think about it.

And I don't think you can create a model with intersections in 3space. A klein bottle is locally 2d, where as a 3sphere is locally 3d, which might create problems. There is a theorem about the minimal dimension of a model (with intersections and without somewhere).

 

[DaveC426913]

First off is the 4th dimension time (it progressing in time is what makes it 4D)? Or does this sphere involve 4 spatial dimensions and one time dimensions (thus 5D)? Or is time not used?

There is a 4th physical dimension and there is a time dimension, which is sometimes called the 4th dimension. The numbering has to do with historical precedent and little to do with any actual label of the dimensions. Usually, you are talking about one or the other. In the case of the mathematical hyper-objects, time is not a factor, so the extra physical dimension is referred to as the 4th.

There is no reason why space is not the 4th dimension while time is the 5th dimension, as opposed to time being the 4th dimension while space is the 5th.

How can it involve 4 spatial dimensions when we can only imagine 3?

It's a tough concept.

It might be easier with cubes, since they have edges.

A normal 3D cube casts a 2D shadow on a table. The shadow does not look like a square, it looks like a six-sided polygon (try it). But if we only saw that polygon, we could imagine what that 3D cube looked like. We could even calculate its dimensions (including its 3rd dimension).

While we cannot actually construct 4D cubes or spheres, we can contruct their 3D shadows. We can then imagine what the 4D cube looks like that cast that 3D shadow. we can even measure the length of its sides in the 4th dimension.

A cube is composed of 6 squares, all of which have their edges joined. To do this requires a 3rd dimension. A tesseract is an object composed of 6 cubes, all of which have their faces joined. To do this requires a 4th dimension.

Wiki "tesseract".

 

[Nimz]

Another way to look at a tesseract would be by "unfolding" one. If you unfold a cube, you get a cross shape (well, depending on how you unfold it...) made up from 6 squares. Similarly, with a tesseract, you get a crosslike shape made up from 8 cubes.

Here's another way you can sort of visualize a 3-sphere, again, by analogy to Flatlanders visualizing 2-spheres: A Flatlander could visualize a 2-sphere by looking at two circles. One of the circles would correspond to the Northern hemisphere, while the other corresponds to the Southern hemisphere. The edges of the circles are the equator. To get from one hemisphere to the other, you cross the edge of one circle at some point and enter at the corresponding point of the edge of the other circle.

Now we can visualize two spheres. Objects moving in the 3-sphere can be visualized as moving inside those spheres. The tough part is how they move in the spheres when they cross the equator. I'm not sure whether this would be considered an immersion or an embedding, but there will definitely be some scale distortion. Objects near the poles would look smaller than identical objects near the equator.

 

[SpongeBobRhombusHat]

Topologically, a 3-sphere is the one point compactification of 3-space, so it can be thought of as 3-space with an extra point. This way of thinking completely ignores the metric unless you put a strange metric on 3-space.

A good way to visualize the geometry of the 3-sphere (or other 3-manifolds for that matter) is to imagine you are standing inside it and figure out what you would see. An analogy with the 2-sphere is in order. What follows is non-rigorous and drawing some diagrams might help you visualize what I am talking about.

Imagine you and a friend are 2-d beings living in the 2-sphere. Geodesics on the 2-sphere are great circles. Light will travel along these geodesics. Because of the positive curvature, if you watch your friend walk away from you, she will appear to get farther away for a bit and then seem to be getting closer and closer until she reaches the antipodal point on the sphere to where you are standing. She will appear to be smeared out on the horizon, covering everything you see in each direction, as all geodesics will point to her. If you were alone in the 2-sphere, you would see yourself from behind, smeared out on the horizon.

The situation is analogous in the 3-sphere. All geodesics are closed, so in any direction you look, you will see yourself smeared out on the "background." If your friend had a jetpack, and flew away from you in any direction, her image would shrink and then grow until she was at the antipodal point. If she continued in the same direction, her image would appear to be upside down as she moved away since the geodesics "cross" at the antipodal point. As she approached you from the other side, she would again appear to grow until she was standing on top of you again. It would be quite disconcerting and confusing so suddenly find yourself inside the 3-sphere.

I hope this helps you a bit. It's an interesting thought experiment anyways to think about what the geometry would look like.

-SBRH