The following correspondence between Ken Brakke and Jeff Weeks was evoked by an article by Evelyn Sander which appeared on the Forum last June. (Evelyn Sander: "4D Part 3: Minimal Surfaces", available by ftp from the Forum archives, monthly.digests geometry.college archive.June.93). Anyone else want to take the Brakke test?
I just read the following in a Forum article by Evelyn Sander.
> Brakke says that even after his work in 4D, he is unable to > really visualize it. His responded with skepticism to > the idea that anyone can visualize 4D. He said, "Give them > a test. Show them a bunch of projections of similar 4D > objects and ask them which are pictures of the same > object. Show them one 'side' of a 4D object, and have them > describe the other 'side.' I think you'll find that > nobody can really visualize 4D. The best they can do is try > to understand a few theorems and see that the objects > obeys these theorems."
I certainly feel subjectively that I can visualize 4D, so I'd like to try the Brakke Test. Is this something that you can send in written form, or does it require the physical presence of the person administering it?
Just to lay the groundwork for my potential alibi in advance, I should point out that asking the subject to produce projections of a 4D object may be too hard a test. For example, consider the following variation of the Brakke Test, which tests whether the subject can visualize 3D. You show the subject orthogonal projections of an octahedron viewed "vertex-on", then ask the subject to draw projections of the octahedron "edge-on" and "face-on". My guess is that few non-mathematicians could pass this test, even though virtually all humans can visual 3D space fairly well. Even if you made the test easier by letting them hold a physical octahedron for a minute, then hiding the octahedron and asking them to draw the projections, I think most people would fail. So the Brakke Test shouldn't require the subject to understand 4D any better than your average 3-year-old understands 3D.
I didn't have any specific test all set up, but I'll make one up for you. What I want to test is if someone can really handle all four dimensions at once, and not just solve things by analogy to three dimension. I am aiming at Wizard Geometers, so objections about man-in-the-street incompetence in 3D are irrelevant.
Snake Test: Define an n-snake to be a string of n hypercubes with cube k+1 glued to a face of cube k. One can define a snake by starting with cube 1 centered at the origin and aligned with the axes, then giving the axis directions for adjoining successive cubes. For example, +w-x-y+z would be a 5-snake. Test: take two 5-snakes and tell whether they are the same up to rotation purely by visualizing them. If they are the same, visualize the rotation that takes one to the other, and give the invariant planes and rotation angles for the rotation. Also give the other snakes that can be reached by continuing that rotation.
Opaque Hypercube Test: Given some hypersurfaces inside a hypercube, is there any line of sight through the hypercube not blocked by the hypersurfaces? (A little vague, but I hope you get the idea)
Illusion Test: I presume you are familiar with the Impossible Triangle (and other such Escher-like illusions) which can be easily drawn on paper, and constructed in 3D if you fix the viewpoint. Test is to construct an Impossible Tetrahedron in 4D.
I delighted that you're willing to make up a test! I'm pretty tired now, but I'll think about your Illustion Test tomorrow (I'm a morning person by nature). The Snake Test seems easy enough, but lends itself to "cheating": one could easily work out equivalent snakes based on one's knowledge of how the coordinate axes behave under rotation.
> What I want to test is if someone can really handle > all four dimensions at once, and not just solve things by analogy > to three dimension.
This certainly seems to be the case, at least subjectively. My personal impression is that I "feel" 4-d space more than I "see" it. I am aware of the 3-d slice that I'm in, but this slice feels infinitely thin in the 4-d space. It's kind of fun feeling 3-d space (and my own body) as being infinitely thin.
> I am aiming at Wizard Geometers, so objections > about man-in-the-street incompetence in 3D are irrelevant.
I disagree. I certainly don't visualize (or, more to the point, manipulate) 4-d objects at the Wizard Geometer level. My only claim is to be able to visualize it at the toddler level.
But speaking of Wizard Geometers, have you proposed this test to, say, Thurston or Conway?
Along these same lines, I suspect the Opaque Hypercube Test could be too hard. Indeed, even in 3-d trying to deduce opaqueness from an abstract description of the surface(s) inside could be completely baffling. The most relevent test would be something that's completely trivial to do visually in 3-d, but very hard to do in 4-d if one relies on analogy.
Anyhow, I'll think about the Illusion Test tomorrow. Thanks!
The snake test can be figured out by cheating (since I can't visualize 4D as well as I'm trying to test for, I have to cheat myself). But you're honor-bound not to cheat. The part about finding the invariant rotation planes is a little tougher, though.
If you can feel yourself in 4D, can you move your body out of a hyperplane? Can you imagine yourself in a fully 4D body? Interesting suggestion not to use vision; our visual system may be too hardwired to 3D. Our body sense may be more flexible.
> If you can feel yourself in 4D, can you move your body out of > a hyperplane?
Yes, but only via parallel translation. I can't do sommersaults (sp?), but I wish I could because I'm sure SO(4) is an even better environment for sommersaults than SO(3)!
Usually I just stay put, and move the objects I'm looking at.
> Can you imagine yourself in a fully 4D body?
No, not at all. When I try to, I end up with a sort of 4-d body, but it has a preferred 3-d slice defined by its intersection with the hyperplane I imagine myself to be in, which means that really I'm still imagining myself in a 3-d body, but with a lot of 4-d gunk attached to either side.
> Interesting suggestion not to use vision; our visual system > may be too hardwired to 3D.
Actually I use the hardwired visual system to keep track of what would be the projection into (x,y,z) space. The "depth" that corresponds to what would be the w coordinate is the part that's "felt". So there is (unfortunately) always a preferred direction present in the experience. But at least the feeling of 4-d space is there.
I haven't really sat down to attack the Illustion Test (too much real work to get done), but to the extent that I've thought about it I find myself starting to "cheat", that is, carefully analyze how the 3-d (actually 2-d) version is drawn, with the thought of doing the same thing for a thickened tetrahedron. But clearly this is not what is intended.
So long for now.
P.S. Come to think of it, it's more correct to say that the 3-d part of the experience (the (x,y,z) projection) is really seen as 2-d, with the depth simply "felt".
In fact, isn't this how everyone (mathematician or not) sees the world? Shut your eyes and imagine a wire frame model of a cube. What you see is actually a 2-dimensional image; you "feel" (but don't actually "see") the depth. Imagining 4-d is similar. You see a 2-d imagine, feel the depth using your everyday spatial abilities, and feel the w-coordinate using something similar.
The difference between feeling a third dimension and feeling a fourth is that we've been living for decades in a 3-d space, so we're pretty good at imagining rotations. In 4-d our experience is limited to however many hours we've devoted to mental practice, so (for me at least) a rotation is easily visualized iff its axis (which is a 2-plane) lies in the (x,y,z) hyperplane.