Wikipedia:Reference desk/Archives/Mathematics/2006 October 22

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October 22

calculus

Role of trignometic results in differenciation —The preceding unsigned comment was added by 61.17.223.27 (talk • contribs) .

Do you mean things like Trigonometric substitution in integration? x42bn6 Talk 13:03, 22 October 2006 (UTC)

Do you mean the discussion at Wikipedia:Reference desk/Archives/Mathematics/2006 October 20#How do I prove this??  --Lambiam^Talk 15:24, 23 October 2006 (UTC)

Powers of ten using first-order logic and simple arithmetic

I've been re-reading Gödel, Escher, Bach and I'm puzzled by one of the problems offered to exercise the reader's grasp of TNT (a number theory system). Basically, I'm trying to work out how to represent the statement, a is a power of ten in one statement using only first order logic, with operations only of addition and multiplication. I've managed to represent a is a power of two in the following way (using equivalent mainstream notation rather than that developed for TNT, which is isomorphic):
$\forall x:(((\exists y:y\times x=a)\land \neg (\exists y:(\exists z:z\times y=x)))\rightarrow (x=2))$
which can be interpreted as "for every x, if x divides a and no y divides x, then x is two," or, more succinctly, "the only prime that divides a is two." but I can't work out how to do it for powers of ten. Every time I try to do it, I either say something not powerful enough, or too powerful, or just wrong. I can say things like "the only primes dividing a are 2 and 5" and "The only numbers that divide a that are less than (some fixed limit) are (some powers of ten)" but neither is what I want. I still want to solve this problem for myself but I would like a gentle nudge in the right direction. Can anybody help me out? Maelin 13:55, 22 October 2006 (UTC)

I should probably add that the book does warn not to try doing the power of ten one unless you have several hours to spare for it and also know quite a bit of number theory. Maelin 14:34, 22 October 2006 (UTC)

This is not a direct answer, but have a look at Matiyasevich's theorem. It implies that the problem posed by GEB has a solution. The proof of this theorem is constructive, so it allows you to find a solution. Most likely there are much simpler solutions than the one you will obtain this way.  --Lambiam^Talk 18:09, 22 October 2006 (UTC)

sphere clusters

Yesterday someone mistook me for an expert on sphere packing ;) and asked about what I think is called clustering: the shapes formed when hard spheres seek to minimize the sum of distances between them (or their mutual gravitational potential energy), without constraint on the convex hull. I know I've seen something on this subject on the web, but couldn't find anything; perhaps I misremembered the keyword. —Tamfang 16:57, 22 October 2006 (UTC)

Try Kissing number, maybe? - Rainwarrior 17:36, 22 October 2006 (UTC)

No, because there's no requirement that some central sphere touch all the others. —Tamfang 18:52, 22 October 2006 (UTC)

Oh, there is a name for that? Cool. I remember, as a kid, realising that six circles (probably coins, glasses or boccia balls) seemed to fit around a seventh, but I didn't know much maths then, so I couldn't prove it. —Bromskloss 20:35, 22 October 2006 (UTC)

Though I am also reminded of image processing techniques opening and closing which (our articles on them aren't great) take an image and pass a shape around it (usually some sort of circle), using the area covered by that shape as the new image. It could be done in 3 or more dimensions too. I can't quite figure out what you're looking for by your description though. - Rainwarrior 17:47, 22 October 2006 (UTC)

What do you mean by "without constraint on the convex hull"? —Bromskloss 20:35, 22 October 2006 (UTC)

I think the problem can be formulated in one form as: find n points in Euclidean space maximizing $\sum _{1\leq i<j\leq n}d_{ij}^{-1}$ subject to the constraint $d_{ij}\geq 1\,$, where $d_{ij}$ is the distance between point i and point j. This is related to the question how many unit spheres you can pack inside a sphere of (large) radius R. For both problems the asymptotic density should be that given by Kepler's conjecture. I don't know a name for this version or similar versions. I'd call it huddling spheres.  --Lambiam^Talk 22:13, 22 October 2006 (UTC)

Yes, that's exactly the problem I had in mind (or, alternatively, minimize $\sum d_{ij}$). One important difference from packing is that in this problem you're less likely to get "rattlers", balls which have some freedom of movement even in the optimal solution. —Tamfang 01:55, 23 October 2006 (UTC)

In mail, one Henry Cohn pointed me to [1] which has lists of solutions and a few pix. —Tamfang 07:05, 24 October 2006 (UTC)

This might not actually answer your question, but it's an interesting read nonetheless. Rob Cockerham from Cockeyed.com has a page about filling a car with small plastic balls for a competition. You can read about it here: http://cockeyed.com/inside/trailblazer/trailblazer.html -Maelin 23:41, 22 October 2006 (UTC)

Crystal systems formed from identical atoms follow several spherical packing patterns (crystal structures). Among the most common are hexagonal close pack, body-centered cubic, and face-centered cubic. StuRat 00:51, 23 October 2006 (UTC)

Yeah, so? —Tamfang 07:05, 24 October 2006 (UTC)

This is an answer to your question. Look at those articles at the links for more info on sphere packing. StuRat 17:35, 25 October 2006 (UTC)

No, it isn't an answer to my question, because packing and clustering are distinct problems (though the solution of one is often a solution to the other). Why is it that the more I say "I'm interested in X rather than Y" the more people tell me about Y? —Tamfang 06:19, 30 October 2006 (UTC)