Envisioning more than 3 dimensions

Wang Chung

One of the troubling things to me is that it seems only useful to model things in two or three dimensions. There are so many things that exist in more than three dimensions (if you don't understand this, think of dimensions as being variables, think of them being things that represent difference) and the fact that we cannot directly plot or convey those things in visually relevant way seems really bad.

I read that there are such things as Rubiks Cube solving in greater than three dimensions: e.g. tesseract/hypercube. I can't do a 3d 3x3 rubix cube. There are people who are good enough at doing that they can do 5x5 rubix or greater, which seems incredibly impressive. But a 4-dimensional cube? I can't even perceive that.

Either way, I feel like if I can't understand 4-d and higher geometry, I'll never know anything about anything.
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Erwin Rommel · Posted: Sat Mar 24, 2012 8:59 pm Post subject:

Just visualize n-dimensions and let n=4. Razz

As someone who has regularly dealt with higher dimensional manifolds and CW-complexes in the context of graduate geometry and topology courses, I can confidently say that our brains are not built to deal with it.

Whenever I need to manipulate such objects, it comes down to very closely following your set of axioms. Any visualization is either of a lower dimensional analogue or very rough.

Higher-dimensional Euclidean space (e.g. R^4) is a good start. You can think of R^4 as a bunch of copies of R^3 glued together, just like R^3 is a bunch of copies of R^2 glued together. You can produce a notion of distance in R^4 from the notion of distance in R^3. With this, you can visualize a ball in 4 dimensions. (Just take all points with distance less than 1 from the origin)

Basically, I feel comfortable with basic shapes in higher dimensions because I have done the basic math surrounding them enough and see ways to generalize it. It's not so much actually perceiving the object itself, as it is completly understanding what it has to do by virtue of its construction. And these basic shapes are usually enough to get a good idea of what's going on in more complicated ones. (e.g. CW-complexes are made up of higher-dimensional balls and manifolds look like a copy of R^n around any given point)

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As for a Rubiks Cube, IMHO the best way to think of it is algebraically and not geometrically:

http://en.wikipedia.org/wiki/Rubiks_cube#Relevance_and_application_of_mathematical_group_theory

That is probably a better way of extending it to higher dimensions, though I know next to nothing about this.
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Last edited by Erwin Rommel on Sun Mar 25, 2012 12:00 am; edited 1 time in total

massive · Posted: Sat Mar 24, 2012 9:26 pm Post subject:

I admire physicists because of their sheer brain horse-power... but that all fizzles away when they start talking multiple dimensions beyond 4 - string theory? really? I'm ashamed to be interested in science when I hear the shit that has to be 'possible' in order to support the 10th dimension.

I believe in intelligent design, not a retarded one:

God to himself: "Im gonna make everything hard to figure out, but sooner or later they'll have it - so ima add this other bullshit that makes no fucking sense in any sense of the history of sense and see how long the brainiacs will douche it out until someone finally kills em all cuz he's sick of it...ahahahhaha"
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Wang Chung · Posted: Wed Mar 28, 2012 8:02 pm Post subject:

Erwin Rommel wrote:

As for a Rubiks Cube, IMHO the best way to think of it is algebraically and not geometrically:

Epic. I always have sucked at the Rubik's cube, pretty horribly actually. I wonder if I could work at understanding it algebraically and could get better that way. I wonder how many subjects you could teach someone by explaining them analogously with a Rubik's cube.

Quote:

I admire physicists because of their sheer brain horse-power... but that all fizzles away when they start talking multiple dimensions beyond 4 - string theory? really? I'm ashamed to be interested in science when I hear the shit that has to be 'possible' in order to support the 10th dimension.

And this is why admiration isn't the best form of flattery in progressive facets of academia. The best form of flattery is challenging ideas intellectually: if you prove them right you can give them a +1 and the world knows they're on to something, and if you prove them wrong you give them a -100 (and probably give yourself a +1 in the long run.). The popularity of string theory is a double-edged sword: popularity will speed up scientific progress in the long run, but can dilute some real scientific progress with hogwash and delay things until the good ideas get sorted from the bad.

String theory does posit a ridiculous number of things, but the highly mathematical thinking that builds up to it is indeed very serious and the foundations of it are understood more and more as you move down progressively and perhaps far more so than any other scientific discipline. It won't be toppled until more people climb the base and we'll build something bigger. When you say multiple dimensions beyond 4, do you mean that the how the universe has four dimensions and none further?

http://www.geom.uiuc.edu/~banchoff/Klein4D/Klein4D.html
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Erwin Rommel · Posted: Wed Mar 28, 2012 8:51 pm Post subject:

Wang Chung wrote:

Erwin Rommel wrote:

As for a Rubiks Cube, IMHO the best way to think of it is algebraically and not geometrically:

Epic. I always have sucked at the Rubik's cube, pretty horribly actually. I wonder if I could work at understanding it algebraically and could get better that way. I wonder how many subjects you could teach someone by explaining them analogously with a Rubik's cube.

I had an algebra professor as an undergrad who talked a little bit about his experience with the rubiks cube . Apparently it was all the rage when he was in grad school. He got into it a little bit. By that I mean he used group theory to come up with an algorithm that could solve the rubiks cube from a generic starting point. At that point, he stopped being interested since he had solved the problem.

Quote:

http://www.geom.uiuc.edu/~banchoff/Klein4D/Klein4D.html

Not sure why you linked this, but there's some interesting stuff there. I don't quite see what's going on in the second video.

It might be interesting to note, however, that when I refer to a generic "Klein Bottle", I'm not referring to something specifically sitting inside of four dimensional space. I am referring to the Klein Bottle as a geometric object unto itself. The Klein bottle is actually a 2-dimensional object (a manifold, to be precise). This is the difference between intrinsic and extrinsic geometry.

The way that mathematicians often look at these sorts of objects is by looking at rectangles and identifying edges together. Here's an introduction to looking at it that way, if anyone is interested:

http://www.jcu.edu/math/vignettes/Mobius.htm
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massive · Posted: Fri Mar 30, 2012 11:30 am Post subject:

I'd like to prove it one way or another, but i'm smart enough to know I shouldn't even be in the room that conversation is happening in...

We start like beings in a 2 dimensional space, who have no perception or concept of up or down. Then some einstein comes up with the 3rd dimension and sells the idea. Im ok with that, and the math proves it. What says tho that math ALWAYS equates to what actually exists - i know it can exist in theory but what about reality?

Do the higher dimensions change reality? If so anything can happen like the last Star Trek movie with time travel - make up your own shit now, cuz you have permission...then we're fcked and i'd like to get back to living
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Vlad Piranha wrote:

I'm offended that a zombie holocaust isn't on the list. It would be terrifying, sure, but it would be pretty sweet if you think about it.

xsupermage · Posted: Tue Apr 22, 2025 11:20 am Post subject: