October 2007 - mathfun@mailman.xmission.com

[math-fun] radio math
by James Propp 26 Oct '07

26 Oct '07

I've been invited to speak on a college morning radio program next week, on the topic of mathematical proof and infinity. It'll be a conversation with two interviewers (no call-ins). I have no experience with this kind of public speaking, but when the opportunity came my way, it seemed like it might be a fun thing to try. I asked the producer what sort of people listen to the show, and he replied >Audience is primarily adults both UML staff and faculty as well as >residents of the Merrimack Valley. > >The show seems to appeal to people with an interest in a variety of >subjects and in "bright" conversations. Anyone out there have any suggestions for interesting analogies, points worth making, etc.? When I try to walk through a conversation about infinity in my mind with non-mathematicians, it usually doesn't go very well. There are a lot of ways an intelligent and well-educated person is likely to misunderstand the mathematical enterprise of making up rules about infinity and seeing where they lead us. In fact, the more intelligent a layperson is, the more objections he or she is likely to have to the very first steps of trying to talk about infinity as a well-defined mathematical notion! (To give just one example of how my inner conversations go awry: If I try to prove that the number of whole numbers has the same size as the number of perfect squares, people are apt to notice and fixate on the fact that one of the sets is a subset of the other, and so "must" be smaller. And even if I can convey the idea that we're using a new notion of measuring size, based on pairing elements, and that we have to relinquish all our intuitions that are based on finite sets until we can justify them in the new setting as consequences of our definition, the fact that the perfect squares have dual citizenship as both whole numbers and perfect squares makes the idea of the pairing confusing.) Note that you can't draw pictures over the radio, so you can't make a table showing a one-to-one correspondence between two infinite sets. Part of what I'm missing is a kit of good strategy for evading common pedagogical problems by cleverly choosing a plan of approach that prevents the issue from arising in the first place. For instance, if I use the Hilbert hotel picture, and talk about moving the person in room n to room n^2, then I can argue that there are just as many *rooms* of one kind as the other because the two sets can accomodate the same set of *people*, and the whole "dual-citizenship" thing doesn't arise. (I'm pretty good at solving pedagogical problems like this, but usually not in real time! Maybe the only way to get good at talking about math on the radio is to get lots of practice and make lots of mistakes; kind of like the way to get good at doing math...) I also feel that part of what I'm missing is a sales-pitch for a whole style of thought, namely "The Mind at Play", and good, friendly ways of encouraging people to relax, try out ideas, and not be afraid of being wrong. Also: Are there any books in particular that you think I should plug ("If you enjoyed listening to this conversation, then you should read X")? Jim

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Re: [math-fun] radio math
by Dan Asimov 25 Oct '07

25 Oct '07

I'd recommend that anyone going on the radio to ad lib a mini-lecture on math for lay listeners first practice in private on non-mathematician friends to see what works, and to develop confidence in their ability to give a successful ad lib mini lecture. --Dan P.S. I also second Joshua's recommendation of "A Certain Ambiguity", which is quite a fascinating book, though somewhat dense (for the lay reader) in some parts. (It also contains the usual quota of (generally non-crucial) mathematical errors, but Princeton University Press has told me that at least the ones I reported would be fixed in the next edition. I would *not* recommend "1,2,3,...Infinity" by George Gamow, as it contains crucial errors in its discussion of infinities. And Dover has refused to fix any of these, despite my reporting them over 10 years ago.

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Re: [math-fun] radio math
by Eugene Salamin 25 Oct '07

25 Oct '07

----- Original Message ---- From: James Propp <jpropp(a)cs.uml.edu> To: math-fun(a)mailman.xmission.com Sent: Thursday, October 25, 2007 1:20:28 PM Subject: [math-fun] radio math I've been invited to speak on a college morning radio program next week, on the topic of mathematical proof and infinity. It'll be a conversation with two interviewers (no call-ins). I have no experience with this kind of public speaking, but when the opportunity came my way, it seemed like it might be a fun thing to try. I asked the producer what sort of people listen to the show, and he replied >Audience is primarily adults both UML staff and faculty as well as >residents of the Merrimack Valley. > >The show seems to appeal to people with an interest in a variety of >subjects and in "bright" conversations. Anyone out there have any suggestions for interesting analogies, points worth making, etc.? When I try to walk through a conversation about infinity in my mind with non-mathematicians, it usually doesn't go very well. There are a lot of ways an intelligent and well-educated person is likely to misunderstand the mathematical enterprise of making up rules about infinity and seeing where they lead us. In fact, the more intelligent a layperson is, the more objections he or she is likely to have to the very first steps of trying to talk about infinity as a well-defined mathematical notion! (To give just one example of how my inner conversations go awry: If I try to prove that the number of whole numbers has the same size as the number of perfect squares, people are apt to notice and fixate on the fact that one of the sets is a subset of the other, and so "must" be smaller. And even if I can convey the idea that we're using a new notion of measuring size, based on pairing elements, and that we have to relinquish all our intuitions that are based on finite sets until we can justify them in the new setting as consequences of our definition, the fact that the perfect squares have dual citizenship as both whole numbers and perfect squares makes the idea of the pairing confusing.) Note that you can't draw pictures over the radio, so you can't make a table showing a one-to-one correspondence between two infinite sets. Part of what I'm missing is a kit of good strategy for evading common pedagogical problems by cleverly choosing a plan of approach that prevents the issue from arising in the first place. For instance, if I use the Hilbert hotel picture, and talk about moving the person in room n to room n^2, then I can argue that there are just as many *rooms* of one kind as the other because the two sets can accomodate the same set of *people*, and the whole "dual-citizenship" thing doesn't arise. (I'm pretty good at solving pedagogical problems like this, but usually not in real time! Maybe the only way to get good at talking about math on the radio is to get lots of practice and make lots of mistakes; kind of like the way to get good at doing math...) I also feel that part of what I'm missing is a sales-pitch for a whole style of thought, namely "The Mind at Play", and good, friendly ways of encouraging people to relax, try out ideas, and not be afraid of being wrong. Also: Are there any books in particular that you think I should plug ("If you enjoyed listening to this conversation, then you should read X")? Jim _______________________________________________ 1. You won't have time to cover two topics, i.e. both proofs and infinity, so arrange with the interviewers which one they will start with. My guess is that infinity is the easier of the two. There is so much that can be discussed: the countable and uncountable cardinals, Cantor diagonalization, the ordinals, the asymptotic infinity as in "let x approach infinity". 2. You can't practice within your own head; you must try it out in advance with several typical people. Try to explain the different infinities, see where they get hung up, and modify the explanation until it flows well. 3. I first learned, while still in elementary school, about the countable and uncountable from George Gamow's "One, Two, Three, Infinity". Gamow gave the diagonalization proof that the real numbers are uncountable, and I understood it readily. You could have a look at that book and see if it's helpful. Gamow calls the cardinality of the reals aleph-1; years later I learned that this is not correct, but that mistake is no big deal. That same book was also my first exposure to imaginary numbers, and that might also be a suitable topic for discussion. 4. If you're trying to demonstrate the countability of the rationals, the hard part is embedding the rationals into the integers. Here's a clever trick that I saw somewhere. Write a rational number as a character string as for example "123/456". Now think of "/" as a digit and the character string as a base-11 integer. The same character string trick can show the countability of polynomials with integer (or rational) coefficients, and even polynomials in several variables, and even countably many variables. Likewise, there are only countably many algebraic numbers, or computable numbers. Gene __________________________________________________ Do You Yahoo!? Tired of spam? Yahoo! Mail has the best spam protection around http://mail.yahoo.com

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Re: [math-fun] nice little sin(a+b+c+...) formula
by Henry Baker 24 Oct '07

24 Oct '07

I'm sorry, that posting was only intended for Bill, whom I hoped already knew what I was looking for. I'm looking for formulae only involving sinh, so the one with cosh doesn't work for me. Ditto for asinh in terms of asinh. At 02:50 PM 10/24/2007, Joshua Zucker wrote: >On 10/24/07, Henry Baker <hbaker1(a)pipeline.com> wrote: >> sinh(x*2^k) = ?? >> sinh(x+y) = > >These two are exactly the same as in the regular sine case, e.g. >sinh(2x) = 2 sinh(x) cosh(x) >sinh(x+y) = sinh(x) cosh(y) + sinh(y) cosh(x) >and so on. > >> asinh(x*2^k) = ?? >> asinh(x+y) = ?? > >I don't know. Either use the ln formulation of these or just use the fact that >asinh(x) = i * asin(-ix) >to change your favorite asin formulas into asinh formulas. I don't >have any favorite asin formulas to use to start with, though. > >--Joshua Zucker

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[math-fun] Wolfram's 2,3 Turing Machine Is Universal!
by Ray Tayek 24 Oct '07

24 Oct '07

http://science.slashdot.org/article.pl?sid=07/10/24/1429229 --- vice-chair http://ocjug.org/

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[math-fun] packing 7x11x13 box puzzle
by Richard Schroeppel 24 Oct '07

24 Oct '07

David Cantrell asks ... > How many boxes, each measuring 7 inches x 11 inches x 13 inches, can be packed into a crate measuring 4 feet x 5 feet x 6 feet? I'll start the bidding at 205. The trivial upper bound is 48x60x72/7x11x13 = 207 boxes, + 153 in^3 left over. I get 205 with a mundane packing: Four 11x13 rectangles can be packed into a 24x24 square, with a 2x2 hole in the middle. This tiles a 48x72 layer, 7" thick, with 24 boxes, and a wastage of 168 in^3 per layer. Regard the 60" dimension as vertical, and use the lower 49" to pack seven layers. This gives 168 boxes, wastage 1176 in^3, and a leftover slice 11" thick by 48x72. The slice can hold a maximum of 37 boxes (11x48x72/7x11x13 = 37 + 89/91). One way is to pack 7x13 rectangles into a 48x72 box. Cut out a 35x65 piece (= 25 rectangles), leaving a 13x72 stripe (= 10 ractangles) and a 35x7 stripe (= 2 rectangles). This gives 37 boxes in the top 11" slice, with a wastage of 979 in^3. Totalling up, 205 boxes, wastage 2155 in^3. Do I hear 206? Rich

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[math-fun] branchless programming
by Fred lunnon 23 Oct '07

23 Oct '07

I'm interested in algorithms which eliminate conditional branches: the following problem turns up in attempting to design a robust, straight-line, numerical quadratic equation solver, in the subcase of a double root. Find functions f = f(a,b,c), g = g(a,b,c) such that, given any 3 real numbers (a,b,c) /= (0,0,0) with b*b = a*c, it is guaranteed that (f*a + g*b, f*b + g*c) /= (0,0). One way to do this (harking back to a related discussion on math-fun a few years back) would be to set f = h(a a + b b), g = h(b b + c c), where define h(x) = ceiling(|x|) / max(1, ceiling(|x|)) = 0 when x is zero, 1 otherwise. [The absolute value is of course redundant in this application.] But I wonder whether it's possible to solve the problem more elegantly, using (say) polynomial f,g; alternatively, is there some obvious reason why this might be impossible? Fred Lunnon

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[math-fun] packing circles
by James Buddenhagen 22 Oct '07

22 Oct '07

Bill Gosper's packing circles in an oval puzzle reminded me of the following packing problem: Find the ellipse of smallest area which can contain n non-overlapping unit disks. This problem is interesting in that even for small n the answer can be non-intuitive. For example, for n = 3, I would have guessed that the best ellipse would be a circle. But it is not. For the best known results up to n = 24 see this page http://www.stetson.edu/~efriedma/cirinel/ at Erich Friedman's packing center. If you find any improvements let Erich know and he will update the page. Jim Buddenhagen

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Re: [math-fun] Re: Submarine problem
by Michael Kleber 22 Oct '07

22 Oct '07

Steve Witham wrote: > If I'm following, with that plan for n = 4 > you would shoot at 1, 1, 1, 1, 2, 2, 3, 4 -- 8 shots-- > but you only need 7, e.g.: 1, 1, 2, 2, 3, 3, 4. Gadzooks! I will now stop trying to prove my strategy was optimal. So can you generalize that to all p^3? --Michael Kleber -- It is very dark and after 2000. If you continue you are likely to be eaten by a bleen.

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[math-fun] that messy cot cot cot regular polygon sum
by Bill Gosper 21 Oct '07

21 Oct '07

I gave [...] Amazingly worse: n ==== \ pi j pi j pi j > cot(---- + f) cot(---- + g) cot(---- + h) = / n n n ==== j = 1 2 sin(h - g) cot(f n) sin(h n - g n) n (--------------------------------------- + ...)+ <hairy mess> sin(g - f) sin(h - f) sin(g n) sin(h n) from a harrowing derivation via 16-page intermediate results, including a quadrivariate generating fcn to circumvent a *nasty* cubic. Here's the stringout if you want to test or simplify it: 'SUM(COT(%PI*J/N+F)*COT(%PI*J/N+G)*COT(%PI*J/N+H),J,1,N) = N^2*(SIN(H-G)*COT(F*N)*SIN(... I found a neat way to drastically simplify it: Lim(n->oo, Sum(cotcotcot)/n) = Integral(cotcotcot,0,1) is finite (f#g#h#f), so the coeff of n^2 is zero! When the smoke clears, sum(cot(%pi*j/n+f)*cot(%pi*j/n+g)*cot(%pi*j/n+h),j,1,n) = -n*((sin(2*g-2*f)*cot(h*n)+sin(2*f-2*h)*cot(g*n)+sin(2*h-2*g)*cot(f*n)) /(2*sin(g-f)*sin(f-h)*sin(h-g))+cot(h*n)+cot(g*n)+cot(f*n))); n ==== \ %pi j %pi j %pi j > cot(----- + f) cot(----- + g) cot(----- + h) = / n n n ==== j = 1 - n ((sin(2 g - 2 f) cot(h n) + sin(2 f - 2 h) cot(g n) + sin(2 h - 2 g) cot(f n))/(2 sin(g - f) sin(f - h) sin(h - g)) + cot(h n) + cot(g n) + cot(f n)). Now what *is* that integral? This limit doesn't exist! Empirically, it's 0. Anyway, the sum answer is verifiably equal to the mess. --rwg __________________________________________________ Do You Yahoo!? Tired of spam? Yahoo! Mail has the best spam protection around http://mail.yahoo.com

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