Dark Bit Factory & Gravity
GENERAL => General chat => Topic started by: hellfire on April 18, 2011
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In 2d a line divides my vector space into two partitions:
(http://www.abload.de/img/spacec7nn.png)
In 3d a plane (2d space) divides my vector space into two partitions:
(http://www.electrostatics3d.com/geo/spatial-geometry.gif)
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 :)
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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.
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A toughie! Maybe one three-d environment transitioning to another over time?
Jim
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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
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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 (http://eusebeia.dyndns.org/4d/vis/01-intro.html)
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A cube with a hole in one of the faces showing the space within?
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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.
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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. :)
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Here, I just found this nice video of a tesseract 4 D cube projected in 3D with lasers;
[youtube]cpD6atERJ08[/youtube]
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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.