A math/visualization question

[hellfire]

In 2d a line divides my vector space into two partitions:


In 3d a plane (2d space) divides my vector space into two partitions:


Consequently, in 4d a 3d-space divides my vector space into two partitions.

I'm looking for a plausible illustration for a not-so-much-into-math audience.
Any ideas are welcome

[combatking0]

What about illustrating 4D space as a series of cubes? I'm not really in the know about 4D space, or how to project it onto a 2D screen, but that's as close as I can get to understanding it.

Your 3D space could be a highlighted cube amongst the stack.

[Jim]

A toughie! Maybe one three-d environment transitioning to another over time?
Jim

[Kirl]

Not sure if this will be an animation or a picture but I think the best you can do is to visualize it step by step. Like you explained this question to us. Beginning in 1 or 2d, then extrude to 3, and finally 4d. Flatland did a great job of explaining higher dimensional stuff this way, but conveniently left the visuals to our imagination. I have no idea of how to visualize 4 dimensions without it looking like a tangled mess of things...

There is a movie which sort of deals with these kinds of difficulties, like mapping a spherical map onto a 2d plane, it had some excellent visualizations of the various projections etc. Unfortunately I forgot it's name, it might be inspirational. I'll post a link if I can find it.

[edit] Here it is: http://www.dimensions-math.org/Dim_reg_E.htm

[benny!]

That's definately a tough one. Could not think of a good idea atm tbh. Just found this while googling - maybe it is of some help:

http://eusebeia.dyndns.org/4d/vis/01-intro.html

[Shockwave]

A cube with a hole in one of the faces showing the space within?

[rain_storm]

How about a 3D volume called here and now separating two similar 3D volumes called before and after?

Then use that as a spring board to ease them into the idea of 4 spatial dimensions.

[Kirl]

I didn't realize it, but the movie "Dimensions" which I linked above is divided into several parts, so make sure to view the part where it's projecting 4 dimensional objects onto 3.

I just sat through a whole bunch, it's presented so clearly and elegantly, I couldn't pull myself away. Not sure if it helps with the visualization, but it's time not wasted.

[Shockwave]

Here, I just found this nice video of a tesseract 4 D cube projected in 3D with lasers;

[youtube]cpD6atERJ08[/youtube]

[hellfire]

I totally dig the look of those laser lines but the projected hypercube is too much out-of-this world for a serious presentation
I guess I'll stick to the array-of-boxes approach.