Can Anyone Explain How to Visualize a 4D Structure Like a 3-Sphere?

[surajt88]

I have been reading about http://en.wikipedia.org/wiki/3-sphere" and found it was impossible for me to visualize it. I tried visualizing a 4d structure with 3 space-like and one time-like dimensions by imagining a sphere (with 3 dimensions) growing with time, with successive layers (formed every second) coloured differently (a bit like tree rings). Any point inside the solid sphere would represent 4 values: 3 spatial dimensions and the fourth being the time taken for the sphere to reach the point. Please correct me if I'm wrong here. Coming back to 3-Spheres, with all four space-like dimensions, can anyone guide me in layman terms to visualize this structure? Thanks.

 

[Zula110100100]

What about a cone growing over time by smearing that growing sphere away from you at a constant velocity not sure if that would help, it would be like each point was described by where it would be if you didn't move the sphere and the 5th represented by the extra displacement from it moving? there is room for the same point being described by more than one coordiante

 

[surajt88]

What about a cone growing over time by smearing that growing sphere away from you at a constant velocity not sure if that would help, it would be like each point was described by where it would be if you didn't move the sphere and the 5th represented by the extra displacement from it moving? there is room for the same point being described by more than one coordiante

That's a good way to visualize. If it indeed represents 4 spatial dimensions, then by colouring the growing cone with respect to time, we can get a 5 dimensional structure. or can we? Also, wouldn't the growing sphere be moving in one of the three dimensions? Then the sphere's movement along an axis, say x and the growth of the sphere along that axis can be transcluded into a single dimension right?

 

[ghwellsjr]

I have been reading about http://en.wikipedia.org/wiki/3-sphere" and found it was impossible for me to visualize it. I tried visualizing a 4d structure with 3 space-like and one time-like dimensions by imagining a sphere (with 3 dimensions) growing with time, with successive layers (formed every second) coloured differently (a bit like tree rings). Any point inside the solid sphere would represent 4 values: 3 spatial dimensions and the fourth being the time taken for the sphere to reach the point. Please correct me if I'm wrong here. Coming back to 3-Spheres, with all four space-like dimensions, can anyone guide me in layman terms to visualize this structure? Thanks.

The 4d structure that your link is talking about are four dimensions of space, not 3 space-like and one time-like dimensions. And I don't know why you are calling them "space-like" and "time-like". Space-time is three ordinary dimensions of space and one ordinary dimension of time. You can have a 3-sphere in four dimensions of space because it is defined with all four dimensions contributing the same way to its definition (see your link). But similar formulas in space-time treat the time coordinate differently than the three space coordinates, usually by multiplying the time squared coordinate by negative c².

Besides, nobody can visualize four spatial dimensions.

 

[pervect]

Well, if you color-code a 2d map by making assigning different terrain heights different colors, you can represent a 3d surface with a 2d map.

It's probably more common to use contour maps to represent 3d surfaces with a 2 d map, it's similar to the above technique where you draw only the outlines of the differently colored regions. It's a bit easier on the eyes, and you only need a monochromatic map/display, but also a bit ambiguous without adding in some other notes that give you the elevation information of the contour.

You could apply either of these techniques to representing a 4d object via a 3d one. Exactly how useful it is, I'm not sure, but you can do it.

 

[robphy]

One can start with Abbott's Flatland http://www.geom.uiuc.edu/~banchoff/Flatland/
then move on to visualizations by Thomas Banchoff at Brown.
http://www.geom.uiuc.edu/~banchoff/projects.html
http://www.math.brown.edu/TFBCON2003/art/welcome.html
http://www.math.brown.edu/TFBCON2003/mathematics/welcome.html

As mentioned, one can also use color-coding to replace some "spatial dimensions".
However, I think one may lose some of the "geometrical content".
(In some cases, like visualizing complex-valued functions, color-coding may be useful
http://www.maa.org/pubs/amm_complements/complex.html )

A google [including image and video] search for "hypersphere" and "hypercube" may help.

 

[genius01]

i think it would be quite a complex task to visualize 4 dimensions

1. We are barely able to visualise 3 dimensions
2. Since the basis of space/time is motion then the mere movement of an air molecule should be recognized and it would need perception much greater than ours
3.Since time is linear it will be harder for us to visualise 4 dimensions on both scale sides