Let’s say that you’re a geometric shape living in Edward Abbott’s Flatland. What methods could you use to determine that there was a third dimension? I know that if the “plane” that the shapes lived on was actually mapped onto a sphere the ability to make some straight lines and very accurate measurements would make it easy determine that the universe was curved, but are there other methods that could be used? What if you were on a Euclidian plane? The physics of such a world would be rather different from our own but for the sake of wrapping our heads around it everything works as close to our universe as possible only flat (with the possible exception of gravity which would either have to not exist, be so weak as to be unnoticiable, or pull us shapes onto the sphere). Any ideas?
Gravity in two dimensions is weird. It does produce detectable effects, but it does not actually manifest as a force, nor does it cause any curvature of space outside of masses. Mostly, the effect is deflections of geodesics, analagous in some ways to the Bohm-Aharonov effect.
More to the point, where could a place like “flatland” exist in our universe? Admittedly I have not traveled the entire extent, but it all pretty much appears to be infested with 3 dimensions everywhere you look. I don’t see how you could have a location that were strictly 2-dimensional; it seems like it would get some 3rd dimension rubbed off on it before too long.
An important question is, In what sense does a third (or fourth) dimension “exist” for those confined to two (or three)? There are different ways that extra dimensions might exist, and correspondingly different ways that they might be detected or inferred.
Curvature, such as the spherical geometry you mention, can be thought of as curvature “into a third dimension”; but it can also be thought of as an intrinsic property of the space and doesn’t strictly require the existence of extra dimensions. This sort of extra dimension is really just useful for intuition and can’t be measured in any real way. That is, you could deduce a “radius of curvature” for your space, but you couldn’t deduce that there exists a point that is the “center,” or anything apart from the spherical surface you see.
Abbott’s Flatlanders are apparently confined to a plane by some force which the three-dimensional creatures don’t feel; but they can also be removed from this plane by external forces (e.g., the Sphere). So matter in the plane, as Flatlanders know it, can apparently interact–electromagnetically, if the Flatland universe’s matter behaves like ours–with matter outside the plane. Flatlanders might be able to deduce the presence of unseen objects by measuring their effects within the plane. For example, a charged object close to the plane might cause deflection of trajectories of charged objects in the plane. A Flatlander could use a planar electronically-scanned radar array, for example, to map out the structure of “space” orthogonal to his plane, if he could somehow design the electronics. (Whether this electronics is feasible in Flatland depends on details of the physics that A. Square has not provided.)
String theories in modern physics typically require the existence of extra dimensions in order to work. One type of theory has all matter and nongravitational interactions confined to a lower-dimensional region–what we poor Flatlanders call “spacetime”–within this higher-dimensional space. The only thing that escapes this region is gravity. So this sort of extra dimension might be inferred indirectly (it can’t be “seen” directly with photons, of course) by measuring gravitational forces. Because the “extra dimensions” are usually assumed to be small and compact (“rolled up like a sock”) these measurements must be made at short distances. Thus at distances much larger than the size of these extra dimensions, gravity should behave “normally” with a 1/r[sup]2[/sup] force; but at distances smaller than these extra dimensions the force should fall off more quickly, since gravity now has more dimensions to expand into (think of Gauss’ Law). Some experiments have been performed along these lines; they’re difficult to do because gravity is so weak.
I imagine that the point would be that we might in fact live in a 3-d slice of a higher dimensional universe. And how we might go about discovering that.
Or did I get whooshed?
Some people think that you could tell because gravity would be a lot weaker than the other forces.
Professor Frink, in the two-dimensional world of The Simpsons, showed that there is indeed a third dimension.
Actually Walrus I’m working on some short fiction featuring two dimensional beings dealing with the Lovecraftian horror of three dimensional beings and I’m just trying to make sure I’ve got my head wrapped around it correctly. Yeah, some people think that we could be three dimensions mapped onto more (and the research into that is relavent to scaling it down to 2d looking for 3d), but I’m not trying to make a thinly veiled reference to that.
Thanks a lot everyone for your replies.
Please let us know how you work out the issue of their digestive tracts.
The “Treehouse of Horror” episodes are highly non-canon, and cannot be taken to mean anything about the greater Simpsons world.
Oh come on. That’s been dealt with in at least two ways already. I don’t know why everybody keeps bringing up the supposed impossibility.
Politely ignoring it. 
A.K. Dewdney’s The Planiverse would be great source material. 
Two ways I know of are a pouch-type digestive tract where you take in food, digest it, and spit the waste back out the same opening, and the “zipper” style tract where things cling together, but pull apart temporarily to allow passage of material.
Why is there a need for a digestive tract?
Why would any of the biological processes of a 2D creature resemble ours?
A 2D creature would require very different mechanisms for most if not all functions.
Note, incidentally, that no matter what he did, he’d be stuck with a twofold degeneracy, in that he’d have no way of knowing (or even defining) which side was which. Probably what he’d do would be to identify some “interesting” off-Plane object (perhaps the largest thing, or the closest, or whatnot), and then describe other objects as being on the same side or opposite side as that reference object.
Yep. This degeneracy is even a problem for Spaceland radar designers; no shielding is perfect.
I can think of two things, off hand. The first, is that if they’re living in 3-D, but are unequipped to realize it naively, maybe they can discover an unexpected anomoly; e.g., the explanation to the fact that light follows the inverse-square law doesn’t match what it should in two space, so therefore, a heterodox suggestion may be a third dimension.
The second thought is if they really exist in two space, and is basically stolen from Clifford Pickover. Suppose there are other spatial dimensions and when a creature passes through our three dimensions, we merely see a cross section. Suppose, there’s a creature with thousands of appendages, more or less identical, that intersect our three-space world. We could be seeing something like a coordinated hive and wondering how all those things get so organized. In other words, perhaps a bee colony is actually the manifestation of such a creature in the universe we know. That’s a goofy, but fun, what-if that Pickover mentioned in one of his books.
Now suppose that your two-space folk see some sort of coordinated hive. The coordinated hive is too coordinated. There’s no possible way a flatland bee can know of a flower when another bee, some distance away, just discovered it, but it does! Maybe it’s a new species of flatland bee, recently discovered, and the mystery thickens…
So, though they’re probably for the Cafe Society rather than GQ, those are the two things I can think of that may help.