March 2008 - mathfun@mailman.xmission.com

[math-fun] quasi-icosahedral frustration
by wouter meeussen 23 Mar '08

23 Mar '08

hi all, when building a 3D set of points using a simple rule, I accidentally found a subset that seems to approach icosahedral symmetry. Why 'approach' ? because all points in my set are rationals, and those can never be icosahedral since they lack the necessary Sqrt[5] irrationality. The procedure is simple enough: start from the tetrahedron (zeroth generation) {0, 0, 0}, {1, 1, 0}, {1, 0, 1}, {0, 1, 1} and on each triangular face, add the point to form the fourth vertex of a new tetrahedron: {{4/3, -1/3, -1/3}, {-1/3, -1/3, 4/3}, {-1/3, 4/3, -1/3}, {4/3, 4/3, 4/3}} The second generation is: {{5/9, 4/9, -11/9}, {4/9, -11/9, 5/9}, {-11/9, 5/9, 4/9}, {5/9, 20/9, 5/9}, {20/9, 4/9, 4/9}, {4/9, 4/9, 20/9}, {4/9, 20/9, 4/9}, {5/9, 5/9, 20/9}, {5/9, -11/9, 4/9}, {-11/9, 4/9, 5/9}, {4/9, 5/9, -11/9}, {20/9, 5/9, 5/9}} and so on, with the n-th generation having denominators 3^n, and the number of points tripling each generation. The quasi-icosahedron becomes apparent when selecting all points at Sqrt[2] from the origin. The zeroth generation provides 3 such, the first generation also, the second provides six, and the k-th generation provides 3*2^k of them. Apart from plotting and admiring their inane beauty, I didn't succeed in understanding where the underlying connection to their pseudo-icosahedral grouping comes from. You got to plot them to believe it. squarely yours, Wouter.

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[math-fun] the road-coloring theorem
by James Propp 23 Mar '08

23 Mar '08

Has anyone seen this? Has anyone on this list gone through the proof? http://www.haaretz.com/hasen/spages/966679.html (See http://en.wikipedia.org/wiki/Road_Coloring_Conjecture for background.) Jim Propp

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[math-fun] Spoiler (hopefully) for Cute Puzzle: What Are The Two Numbers?
by Bill Gosper 22 Mar '08

22 Mar '08

>On 3/1/08, Fred lunnon <fred.lunnon(a)gmail.com> wrote: > >> I'm afraid the bound A,B < 101 is too high: >> an alternative solution overlooked by the setter is >> A = 4, B = 61 >> [and I think I also know the reason he missed it]. > >> WFL >Oops --- apologies in order. >In fact, the given bound is unnecessarily low: I find A,B < 219 >still gives a unique solution! WFL I get the pair 4&61 arising when A,B <123. 63*2, 62*3, 61*4, 60*5, 59*6, 58*7, 57*8,...,53*12,...,47*18,...,43*22,...,41*24,...,37*28,...,33*32 then all have ambiguous factorings. So S chirps on the 61+4. P chirps, knowing 2+122 would've stumped S, being absent from the chirp list [11, 17, 23, 27, 29, 35, 37, 41, 47, 51, 53, 57, 59, 65*, 67, 71, 77, ...] of pan-ambiguous sums. But then S knows P couldn't have resolved 63*2, e.g., because 6+21 is also on the list. Likewise 6+31, 15+20, etc . --rwg * but only if A=122 is legal. >Sumit: You don't know them. >Pradeep: I do now! >Sumit: Likewise! --------------------------------- Looking for last minute shopping deals? Find them fast with Yahoo! Search.

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[math-fun] Fwd: [purplecomet-l] Online Math Contest April 14-18, 2008
by rcs＠xmission.com 20 Mar '08

20 Mar '08

This might be interesting for middle & high-school students. --Rich ----- Forwarded message from no_reply(a)uww.edu ----- Date: Wed, 19 Mar 2008 16:32:04 -0600 From: Purple Comet Math Meet <no_reply(a)uww.edu> Reply-To: Purple Comet Math Meet <purplecomet-l(a)listmanager.uww.edu> Subject: [purplecomet-l] Online Math Contest April 14-18, 2008 To: rcs(a)CS.Arizona.EDU Take part in a growing INTERNATIONAL math meet for middle and high school students. The Purple Comet! Math Meet is a FREE, ONLINE team math competition. Purple Comet! Math Meet begins at noon GMT April 14, 2008 and ends at noon GMT April 18, 2008. So pick the time that works best for you and your students. For complete details, go to PURPLECOMET.ORG[1] and sign up as many teams as you want. Our new Web site is easier to use and has a poster you can print to advertise our contest to your students. Links: ------ [1] http://www.purplecomet.org ----- End forwarded message -----

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[math-fun] A Step Towards Proving the Riemann Hypothesis
by Ray Tayek 20 Mar '08

20 Mar '08

http://science.slashdot.org/science/08/03/20/1728236.shtml --- vice-chair http://ocjug.org/

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[math-fun] Amicable Math Art
by Eric Angelini 20 Mar '08

20 Mar '08

... by Hideki Nakazawa: http://www.aloalo.co.jp/nakazawa/portfolio/jpg/op104moneyamount29.html Best, É.

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[math-fun] functions behaving badly
by James Propp 19 Mar '08

19 Mar '08

Are there any theories of integration out there that would make the function {-(2 Cos[1/x^2])/x + 2 x Sin[1/x^2] for x not equal to 0, f(x) = { { 0 for x = 0 integrable on the interval [-1,1]? Note that the function f(x), although discontinuous at x=0, is equal to the derivative of the function {x^2 Sin[1/x^2] for x not equal to 0, F(x) = { { 0 for x = 0 for ALL x (including x = 0). So, if we were to blindly apply the Fundamental Theorem of the Calculus without attending to the satisfaction of its hypotheses, we would conclude that the integral of f from -1 to 1 equals F(1) - F(-1). Proceeding less blindly, it seems safe to say that any theory of integration that assigned a value to the integral would have to assign it the value F(1) - F(-1). But the unbounded behavior of f(x) in the vicinity of x=0 makes things difficult. Indeed, the function f(x) is neither Riemann integrable nor Lebesgue integrable on [-1,1]. See http://jamespropp.org/142/FunctionsBehavingWorse.nb for a sketch of the graph. (Can nonstandard analysis help here? For some reason I have a feeling that the Stone-Cech compactification might be of use.) Another question arising from this example is: Can things be even worse? Say a function is locally unbounded at a if it is unbounded on every neighborhood of a. If a function F is differentiable everywhere, what can be said about the points at which the derivative is locally unbounded? Can it be dense? Can it be all of R? Jim

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[math-fun] What are the values of A and B?
by Bill Gosper 19 Mar '08

19 Mar '08

>There are two integers, A and B, which are greater than 1 and less than 101. >Neither Sam nor Pete knows what they are, but Sam knows their sum, and Pete knows their product. >The following conversation takes place. >Pete: ``I don't know what the numbers are.'' >Sam: ``I knew that you did not know what the numbers are.'' >Pete: ``Now I know what the numbers are.'' >Sam: ``Then, so do I.'' >What are the values of A and B? I'm confused. Pete's first remark tells Sam nothing. Why can't this be shortened: Sumit: You don't know them. Pradeep: I do now! Sumit: Likewise! ? --rwg --------------------------------- Be a better friend, newshound, and know-it-all with Yahoo! Mobile. Try it now.

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Re: [math-fun] A very peculiar paper
by Mike Stay 18 Mar '08

18 Mar '08

On Mon, Mar 17, 2008 at 2:40 PM, Steve Witham <sw(a)tiac.net> wrote: > For all that, and the intractability, I think Solomonoff's is a nice > pinning down of Ockham's razor. He was this far in the...1950's? > 1960's? 60's. He, Chaitin and Kolmogorov were independently inventing this stuff at the same time. -- Mike Stay - metaweta(a)gmail.com http://math.ucr.edu/~mike http://reperiendi.wordpress.com

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[math-fun] Bad news about David Gale
by Sandra M. Gilbert 18 Mar '08

18 Mar '08

Dear friends, I'm sorry to have to tell you that on Friday, March 7, my beloved partner, David Gale, suffered a heart attack & cardiac arrest. He was maintained in Critical Care at Alta Bates Hospital for a few days, where the doctors undertook a 24-hour hypothermia treatment (now evidently the standard of care) to try to prevent neurological damage, but he never regained consciousness, and died today--Friday, March 7--exactly at noon. My children & I, together with David's three daughters & their partners were at the bedside so we had a chance to say goodbye. One of his grandsons & his wife was also able to join us. As most of you know, David was 86 & in reasonably good health, but he did have cardiac problems. Nevertheless, he was active & vital until the very end: on Tuesday, just before he got sick, he had been updating his prize-winning mathematics website! I can't begin to imagine, at this point, how I'll live without him, & I know his daughters feel the same way, especially because their mother died three weeks ago. If you would like to write to them, I think the best & most appropriate thing to do would be to address mail to all three--Katharine Gale, Karen Gale, & Kirsten Cutler--at their family home: 791 Hilldale, Berkeley, CA, 94708. If you'd like me to send you any email addresses for them, I can do that too. With my very best wishes & much gratitude for your friendship & support, Sandra Sandra M. Gilbert Distinguished Professor of English Emerita University of California, Davis phone: (510) 527-1260 fax: (510) 527-7130 www.SandraMGilbert.com

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