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 Posted: Thu Dec 13, 2007 6:41 am
Everyone has their own image of what a high dimensional vector space looks like in their mind. My abstract algebra TA loved to use his hand to make the cartesian axes on his head. I am wondering how other people view these things in their mind.
Personally, I view them sorta like a 3D video game. As you move down an axis, you can turn around and view perpendicular axes and just rotate about so you can eventually simultaneously see the axes that you care about. Basically it's just seeing more than 3 axes that are perpendicular to each other, but not trying to conceive of what perpendicular means in 4+D.
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Posted: Fri Dec 21, 2007 11:00 pm
ensignhotpants Everyone has their own image of what a high dimensional vector space looks like in their mind. My abstract algebra TA loved to use his hand to make the cartesian axes on his head. I am wondering how other people view these things in their mind. Personally, I view them sorta like a 3D video game. As you move down an axis, you can turn around and view perpendicular axes and just rotate about so you can eventually simultaneously see the axes that you care about. Basically it's just seeing more than 3 axes that are perpendicular to each other, but not trying to conceive of what perpendicular means in 4+D. heh.. I stop thinking that way altogether and my "conception" kind of regresses. A 129 dimensional vector space means I have a set of vectors that is spanned by a set of 129 linearly independant vectors. That's about as far as the picture goes for me.
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Posted: Sat Dec 22, 2007 9:38 am
I don't really have a visualization as such, but I do have somewhat of a visual. It's really a juggling act; I can only think of three dimensions at a time, but if I keep throwing the extra dimensions into the air, it doesn't matter quite so much as long as I concentrate.
I tried using the color thing for visualizing a fourth dimension; didn't get very far when my synaesthesia started interfering.
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Posted: Sat Dec 22, 2007 9:25 pm
I do the colour thing, sometimes. Like Morberticus says, there comes a time when you just do the calculations and forget the visualisations.
But the colour thing goes like this. We can visualise three dimensions (or a two dimensional projection of three dimensions, but whatever...) and then you add blue to the mix. So points have three regular position coordinates, and the "blueness" coordinate, so that if two things have a differing amount of blueness they're in different 3D slices of the space.
This works okay for 4 (good for Klein bottles), and you might be able to get 5 and 6 to work (using other colours), but the thing is you can prove facts about the plane with diagrams (Behold!) and you might be able to do that for three space, but with anything more you might be able to get an idea for a proof or a calculation, but you can't 100% trust it, because you are just a monkey on a rock, and monkeys on rocks only think well in three dimensions tops.
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Posted: Wed Dec 26, 2007 1:36 pm
Layra-chan I don't really have a visualization as such, but I do have somewhat of a visual. It's really a juggling act; I can only think of three dimensions at a time, but if I keep throwing the extra dimensions into the air, it doesn't matter quite so much as long as I concentrate. I tried using the color thing for visualizing a fourth dimension; didn't get very far when my synaesthesia started interfering. Yeah, the color thing doesn't really work for me, either. I do something almost like Layra's juggling act. I use a small section of 2- or 3-dimensional space (which contains the origin, of course). If that's what I'm focused on, it's in the center of my mind. But combinations of the other dimensions are nearby, as if you were looking at several different printed graphs in front of you, with markers and arrows showing how they relate to each other. Then I just scroll through them as needed. This trick also works when trying to envision higher-dimensional graphs and immersions (but is much less useful).
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Posted: Wed Jan 16, 2008 8:13 pm
I also don't really like the color thing because it's good for 4D to maybe even 6D but it just extends the bound of problematic visualization instead of removing that bound.
I do like Layra's method though. I think that when thinking about infinite dimensions, you just need to stop thinking about everything in particular. Being able to juggle or throw up an extra dimension is exactly that. You don't care that you don't see the extra dimensions but you know that they are there.
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Posted: Thu Jan 17, 2008 10:38 pm
Thanks to good ol' Bott Periodicity, it really stops mattering after so many dimensions anyway. There's only so much extra structure you can hide away in spaces isomorphic to F^n.
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Posted: Wed Feb 27, 2008 5:32 am
When I try to visualize a fourth dimension I just visualize a changing figure rolleyes The fourth dimension is time. The exact appearance of the figure at the beginning is with a t=0 (fourth dimension value of 0). As t increases and as time passes, the figure bends and twists in whatever way it is affected by the fourth dimension.
For more dimensions I just turn it all into numbers. I don't bother with mental images.
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Posted: Mon Apr 28, 2008 8:36 pm
Since vector spaces are isomorphic to the direct sum of some subspaces, I picture a bunch of 3-dimensional Cartesian axes side by side. I can't really describe how I deal with the sums of two vectors not in the same subspace, but I manage.
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Posted: Wed Aug 27, 2008 10:32 pm
geodesic42 Since vector spaces are isomorphic to the direct sum of some subspaces, I picture a bunch of 3-dimensional Cartesian axes side by side. I can't really describe how I deal with the sums of two vectors not in the same subspace, but I manage. Somewhat similarly, I sometimes think of a bunch of "visualized" spaces stacked on top of each other. But more often than not, I don't bother with a mental image of anything. Everyone knows the real fun begins when you hit infinite dimensions!
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Posted: Sun Aug 31, 2008 10:05 am
spider_desu When I try to visualize a fourth dimension I just visualize a changing figure rolleyes The fourth dimension is time. The exact appearance of the figure at the beginning is with a t=0 (fourth dimension value of 0). As t increases and as time passes, the figure bends and twists in whatever way it is affected by the fourth dimension. For more dimensions I just turn it all into numbers. I don't bother with mental images. This is sort of what I do as well, and my images for things higher than 4-D don't include other axes, but rather other ideas of dimensions such as time, kind of a "time moving sideways" sort of a thing.
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Posted: Wed Sep 10, 2008 3:19 am
4D is easy to visualize using time as a dimension; just image it changing. As for anything else, I just ignore the fact that the members of the n-tuple are part of an infinite set and create some flatter representation in E^3. I don't usually bother though unless I'm worried about orthogonality. And anything over 6 dimensions isn't worth the trouble to take out of notation.
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Posted: Fri Dec 05, 2008 11:16 pm
As some other people have said in this thread. I also like to visualize separate spaces next to each other. If I'm working in 4 and feeling adventurous though, sometimes I'll take bunch of close together 3 dimensional chunks of the shape and picture them all lined up. to try and get an approximation.
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Posted: Tue Mar 24, 2009 1:22 pm
New question:
What's easier to visualize:
RP^{2n} or CP^n? I find CP^n easier at least for 1 and 2, but for many people the S^{2n}/Z_2 for RP^n seems to be more natural.
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Posted: Fri Mar 27, 2009 2:14 pm
Layra-chan New question: What's easier to visualize: RP^{2n} or CP^n? I find CP^n easier at least for 1 and 2, but for many people the S^{2n}/ Z_2 for RP^n seems to be more natural. S^{2n}/Z_2 for RP^n is usually pretty easy for me. I've never had to visualize CP^n, so I can't really answer that question.
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