From: baez@math.ucr.edu (john baez) Newsgroups: sci.physics.research [1] Re: Bell Ringing in D=26? Date: 13 Dec 1996 15:00:49 GMT Organization: University of California, Riverside Lines: 51 Approved: bunn@leporello.berkeley.edu (sci.physics.research) Message-ID: <58rr31$cn1@agate.berkeley.edu> References: <58ncfr$ebi@agate.berkeley.edu> NNTP-Posting-Host: leporello.berkeley.edu Content-Type: text Originator: bunn@leporello In article <58ncfr$ebi@agate.berkeley.edu>, Craig Pryor wrote: >I recall once hearing something along the lines of... >If you "ring a bell" (i.e. you excite some system which has a >set of normal modes) the frequencies uniquely determine the >shape of the bell only in D=26. That's a bit vague, but it seems false as stated, and I don't know any way to mess with it a bit to make it come out true. In Kac's classic "can you hear the shape of drum?" puzzle you are trying to reconstruct the geometry of a n-dimensional manifold (possibly with boundary) from the eigenvalues of the Laplacian on that manifold. In lowbrow lingo, you're trying to figure out a shape knowing the resonant frequencies at which it vibrates. It's been known ever since Weyl that you can figure out a bunch of interesting stuff about the shape, which is why the puzzle is interesting. But recently two 2-dimensional regions in the plane with different shapes were discovered that have the same eigenvalues. (See Gorden, Webb, and Wolpert, Bull. AMS 27 (1992), 134.) One says they are "isospectral". So no, you can't hear the shape of a drum! What about higher dimensions? Well, actually some counterexamples in higher dimensions were known before the 2d counterexamples. But it seems clear that from the 2d counterexamples you can get counterexamples in *all* higher dimensions. Just take the 2d shapes and take their cross product with a fixed n-dimensional shape, like an n-dimensional torus. By separation of variables the eigenvalues of the Laplacian on a 2d shape determine the eigenvalues of the Laplacian on its cross product with an n-dimensional torus so we obtain two different (n+2)-dimensional shapes whose Laplacians have the same eigenvalues. I suspect it's possible you're thinking about the no-ghost theorem, which says roughly that bosonic string theory works nicest in 26 dimensions. That's pretty different! For various other charming features of 26 dimensions, try http://math.ucr.edu/home/baez/week95 and the references therein, including references to previous issues. There used to exist pictures and movies of isospectral 2d drums on the web, but I can't find them now. Perhaps some brave web surfer out there could find them? End of article 5625 (of 5709) -- what next? [npq] sci.physics.research (moderated) #5639 (2 + 123 more) ( )--(1)+-[1] From: cpryor@zariski.ftf.lth.se (Craig Pryor) \-[1]--[2] [1] Re: Bell Ringing in D=26? In article <58rr31$cn1@agate.berkeley.edu> baez@math.ucr.edu (john baez) writes: > In Kac's classic "can you hear the shape of drum?" puzzle you > are trying to reconstruct the geometry of a n-dimensional manifold > (possibly with boundary) from the eigenvalues of the Laplacian on that > manifold. This is exactly the problem I was thinking of. I believe I had been misinformed about the d=26 part. With the keyword "isospectral" altavista turned up a general discussion of the problem for 2d at http://math.ucsd.edu/~doyle/docs/drum/planar/planar.html and the movies you mentioned are at http://www.cs.umsl.edu/movies.html End of article 5639 (of 5709) -- what next? [npq] sci.physics.research (moderated) #5654 (1 + 123 more) ( )--(1)+-(1) From: bflanagn@sleepy.giant.net \-[1]--[2] [1] Re: Bell Ringing in D=26? Thanks very much for the following. Does anyone happen to know how the symmetries of the drum & its vibrations relate to the symmetries of the sound the drum makes? On 13 Dec 1996, john baez wrote: [Moderator's Note : Unnecessary quoted text deleted. -WGA] > In Kac's classic "can you hear the shape of drum?" puzzle you > are trying to reconstruct the geometry of a n-dimensional manifold > (possibly with boundary) from the eigenvalues of the Laplacian on that > manifold. In lowbrow lingo, you're trying to figure out a shape > knowing the resonant frequencies at which it vibrates. It's been known > ever since Weyl that you can figure out a bunch of interesting stuff > about the shape, which is why the puzzle is interesting. BJ: What other interesting stuff, please? End of article 5654 (of 5709) -- what next? [npq] sci.physics.research (moderated) #5665 (0 + 123 more) ( )--(1)+-(1) From: baez@math.ucr.edu (john baez) \-(1)--[2] [2] Listening to the shape of a drum In article , wrote: >John Baez wrote: >> In lowbrow lingo, you're trying to figure out a shape >> knowing the resonant frequencies at which it vibrates. It's been known >> ever since Weyl that you can figure out a bunch of interesting stuff >> about the shape, which is why the puzzle is interesting. >BJ: What other interesting stuff, please? There will be lots of eigenvalues of the Laplacian, or resonant frequencies, so let N(lambda) be the number of eigenvalues less than lambda. N(lambda) grows asymptotically like lambda^{n/2} where n is the dimension of the drumhead. So the easiest thing to spot is the *dimension* of the drumhead. Say that N(lambda) = C lambda^{n/2} + smaller error terms Then from the constant C you can figure out the *area* of the drumhead. (Well, the term "area" is appropriate if the dimension n is 2. If n = 3 we'd call it the volume, and so on. Anyway, we can figure out how big the drumhead is.) But then there are little correction terms to the above formula. If the boundary of the drum is reasonably smooth, a more accurate formula is N(lambda) = C lambda^{n/2} + D lambda^{(n-1)/2} + smaller error terms >From the constant D you can figure out the *length of the boundary* of the drumhead. (Well, the term "length" is appropriate if the dimension n is 2. If n = 3 we'd call it the area, and so on.) There is something called the Weyl-Berry conjecture about what happens when the boundary of the drumhead is a fractal! Suppose the drumhead has fractal dimension d. Then the Weyl-Berry conjecture is that we should replace the above formula with N(lambda) = C lambda^{n/2} + D lambda^{d/2} + smaller error terms My colleague here at UCR, Michel Lapidus, works on this sort of thing. One of the subtleties is that there are different concepts of "fractal dimension". The most well-known one, the Hausdorff dimensions, does *not* make the Weyl-Berry conjecture come out to be universally valid: there are counterexamples. The Minkowski dimension seems to work better, but as far as I know, the whole subject of drums with fractal boundary becomes more and more complicated, the closer and closer you look at it.... What else can you figure out? Well, if you generalize the problem a bit and look at the eigenvalues of the Laplacian not on functions but on p-forms, you can figure out a lot about the *topology* of the drum, using deRham theory. To be precise, you can figure out its Betti numbers. There is probably a lot more you can figure out, but I don't know exactly what it is! For example, what do the still smaller correction terms in the asymptotic formula above mean? Dunno, but I bet *someone* knows. And what about information obtained, not from the behavior N(lambda) as lambda -> infinity, but the from the first few eigenvalues. I'm pretty sure they study this a bit in the subject of "spectral geometry", but I don't know what the deal is. Can someone help out? End of article 5665 (of 5709) -- what next? [npq]