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 :-)
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.)