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Brakke and Weeks on visualizing 4D
Posted:
Mar 28, 1994 10:50 AM
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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?
GK
******************************************************************
From: weeks@sun.mcs.clarkson.edu (Jeff Weeks)
Dear Ken:
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.
So . . . would you like to make up a test?
Jeff
******************************************************************
From brakke@einstein.susqu.edu (Ken Brakke)
Jeff,
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.
Ken
******************************************************************
Dear Ken:
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!
Jeff
******************************************************************
From brakke@einstein.susqu.edu
Jeff,
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.
Ken
******************************************************************
Dear Ken:
> 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.
Jeff
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.
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