Interesting. I visualize a 3-sphere as a “filmstrip” of 3d film, i.e. a line of 2-spheres of increasing and then decreasing size. This gives up continuity, but something has to give to use the human visual system for higher-D spaces. I find mapping a space dimension to time gives me little idea about the simultaneous spatial structure (even the “visualize a 3d object by viewing its 2d cross-sections one after another” version seems quite hard to me).
Lexicographic ordering is one useful way to collapse 2 dimensions into one, I just prefer to collapse 2 spatial dimensions into 1 spatial dimension that way. (As mentioned in the comments, another way is to treat the 2d as a complex number and then keep the magnitude, perhaps representing the argument as a rotation about the new axis, or using color (hue works well). Anyone know any others?)
Yeah, that’s a good method. And then a 4-sphere is a dynamic movie of these 3d filmstrips. I seem to default to picturing a 3-sphere as spheres that overlap (and I just know that they don’t actually intersect because they’re separated in the 4th dimension). I’m idly curious whether your 2-spheres intersect, or if they’re lined up side-by-side but separate.
Actually now that I think about it more it seems like the fimstrip visualization is better for the things I was using “groundhog day” for. E.g. the 1,3 saddle point level curves make good sense as a movie of 3d filmstrips, and likewise the 2,2 saddle point. That I came up with groundhog day instead of filmstrip is a bit of evidence that there’s something easier about it in some cases, but next time I’ll try starting with filmstrip.
They’re separate, and equally spaced (like actual film). That means that the difference in radius between the first and second 2-spheres has to be much larger than the difference between the middle and next-to-middle ones. I don’t visualize more “frames” than I need for whatever I’m doing, though fewer than 5 doesn’t really work, so I think most often I use 5. You can still get an “all on top of each other” (2d) “view” by making the 2d spheres semi-transparent and looking at the filmstrip from one end.
It actually extends okay into a planar grid of 3d frames for 5d; less well to 6d (things start “occluding” others too much) but maybe still sometimes useful. You can even add meta-film and sort of get it up to 9d. Anything beyond that I don’t find it possible to actually see any variations in all the dimensions at once (I’d REALLY like to have an intuitively meaningful visualization of the Leech lattice, but 24d just doesn’t seem possible with any technique I can think of...)
In my experience / opinion, the biggest problem with these techniques is that rotations that are partly in one “level” of the visualization and partly in another really aren’t natural… of course, for the special case of a sphere, rotational invariance means that doesn’t matter :-)
Interesting. I visualize a 3-sphere as a “filmstrip” of 3d film, i.e. a line of 2-spheres of increasing and then decreasing size. This gives up continuity, but something has to give to use the human visual system for higher-D spaces. I find mapping a space dimension to time gives me little idea about the simultaneous spatial structure (even the “visualize a 3d object by viewing its 2d cross-sections one after another” version seems quite hard to me).
Lexicographic ordering is one useful way to collapse 2 dimensions into one, I just prefer to collapse 2 spatial dimensions into 1 spatial dimension that way. (As mentioned in the comments, another way is to treat the 2d as a complex number and then keep the magnitude, perhaps representing the argument as a rotation about the new axis, or using color (hue works well). Anyone know any others?)
Yeah, that’s a good method. And then a 4-sphere is a dynamic movie of these 3d filmstrips. I seem to default to picturing a 3-sphere as spheres that overlap (and I just know that they don’t actually intersect because they’re separated in the 4th dimension). I’m idly curious whether your 2-spheres intersect, or if they’re lined up side-by-side but separate.
Actually now that I think about it more it seems like the fimstrip visualization is better for the things I was using “groundhog day” for. E.g. the 1,3 saddle point level curves make good sense as a movie of 3d filmstrips, and likewise the 2,2 saddle point. That I came up with groundhog day instead of filmstrip is a bit of evidence that there’s something easier about it in some cases, but next time I’ll try starting with filmstrip.
They’re separate, and equally spaced (like actual film). That means that the difference in radius between the first and second 2-spheres has to be much larger than the difference between the middle and next-to-middle ones. I don’t visualize more “frames” than I need for whatever I’m doing, though fewer than 5 doesn’t really work, so I think most often I use 5. You can still get an “all on top of each other” (2d) “view” by making the 2d spheres semi-transparent and looking at the filmstrip from one end.
It actually extends okay into a planar grid of 3d frames for 5d; less well to 6d (things start “occluding” others too much) but maybe still sometimes useful. You can even add meta-film and sort of get it up to 9d. Anything beyond that I don’t find it possible to actually see any variations in all the dimensions at once (I’d REALLY like to have an intuitively meaningful visualization of the Leech lattice, but 24d just doesn’t seem possible with any technique I can think of...)
In my experience / opinion, the biggest problem with these techniques is that rotations that are partly in one “level” of the visualization and partly in another really aren’t natural… of course, for the special case of a sphere, rotational invariance means that doesn’t matter :-)
Look at the Connection Machine CM-1 and CM-2 (http://tamikothiel.com/cm/cm-design.html) for a really cool physical realization of this, btw.
What’s meta-film?
A filmstrip (or filmgrid, etc.) each frame of which is itself a filmstrip (filmgrid, etc.)