January 2015 - mathfun@mailman.xmission.com

[math-fun] Length of a stalactite
by Henry Baker 02 Jan '15

02 Jan '15

The recent discussions of Gaudi prompted me to think about stalactites. I wanted to understand how long they could get. Suppose a stalactite is made out of structural steel. Suppose that the stalactite is a solid body of revolution, and that the only forces are the vertical tensile forces that oppose gravity, and that the weight of the stalactite is uniformly distributed over the circular face of the base of the stalactite. We also assume that the steel doesn't stretch at all (bad assumption, but it probably doesn't change the weights very much, even though the lengths will grow somewhat). Structural steel has a tensile strength of approx. 250MPa = 250x10^6 kg m/sec^2. Structural steel has a density of approx. 7800 kg/m^3, and g = 9.81 m/sec^2. The optimum shape (with these constraints) for a stalactite appears to be an exponential function R(h)=exp(-h/B)/A, for suitable constants A,B>0. Let's consider a stalactite whose base area is 1 m^2. Then A=sqrt(pi). A steel base with area 1 m^2 can support 250*10^6 Newtons. We need to compute the volume of steel which weighs 250*10^6 Newtons. A 1x1x1 m cube of steel weighs 7800x9.81 = 76518 Newtons. So our base can support hanging 3267 m^3 of steel. So 3267 = integrate(pi*(exp(-h/B)/sqrt(pi))^2,h,0,inf) = integrate(exp(-2h/B),0,inf) = B/2, so B=6534. But our stalactite is infinitely long, which can't be right. Suppose that it ends with a single iron atom of radius 126 picometers = 126*10^-12 m. Then h = 145201 m = 89 miles. And this is ordinary structural steel; specialty wire steels are 10x stronger for the same density. http://en.wikipedia.org/wiki/Ultimate_tensile_strength

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[math-fun] Double moves
by James Propp 01 Jan '15

01 Jan '15

Has anyone studied variants of standard combinatorial games (such as Nim) in which each player makes two standard moves in succession on each turn instead of just one? (This is the way my 8-year-old son likes to play tic-tac-toe against me: he puts down two X's, then I put down two O's, then he puts down two X's and wins. Fair, right?) Note that this form of two-player game can be seen as a four-player game in which players #1 and #2 collaborate and players #3 and #4 collaborate. Jim Propp

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Re: [math-fun] Is Algebra Necessary?
by Henry Baker 01 Jan '15

01 Jan '15

My real point in using Maxima in the example below is that Maxima isn't currently being taught in high school, so schools may be wasting valuable time (as usual) teaching the wrong things. The way math is taught today is analogous to teaching you how to drive by walking 50 miles to your destination and taking your car apart. Where's the part where you actually learn how to drive? As cool as determinants and eigenvalues are (to me), these topics aren't going to be interesting to most people until they've pushed the limits of what they can accomplish without understanding them. At 02:22 AM 1/1/2015, Adam P. Goucher wrote: >Yes, you've taken the problem and expressed it in a form such that it can be >mechanically solved by an automaton. In the words of Sakunthala Panditharatne, >`the parts of coding that can be automated aren't coding'; I guess this >generalises to any creative or intellectual pursuit: > >https://twitter.com/sknthla/status/534848751417171969 > >Sincerely, > >Adam P. Goucher > >> Sent: Thursday, January 01, 2015 at 1:46 AM >> From: "Henry Baker" <hbaker1(a)pipeline.com> >> To: math-fun <math-fun(a)mailman.xmission.com> >> Subject: Re: [math-fun] Is Algebra Necessary? >> >> Cute! >> >> But is what I'm doing below "algebra" ? >> >> Maxima 5.28.0-2 http://maxima.sourceforge.net >> using Lisp GNU Common Lisp (GCL) GCL 2.6.8 (a.k.a. GCL) >> Distributed under the GNU Public License. See the file COPYING. >> Dedicated to the memory of William Schelter. >> The function bug_report() provides bug reporting information. >> (%i1) x+y=2000000; >> (%o1) y + x = 2000000 >> (%i2) 6*x+15*y=15000000; >> (%o2) 15 y + 6 x = 15000000 >> (%i3) globalsolve:true; >> (%o3) true >> (%i4) solve([%o1,%o2],[x,y]); >> 5000000 1000000 >> (%o4) [[x : -------, y : -------]] >> 3 3 >> (%i5) x/y; >> (%o5) 5 >> (%i6) >> >> Rentals:Sales::5:1 >> >> At 05:14 PM 12/31/2014, Hans Havermann wrote: >> >Dan Meyer twittered yesterday: "So glad the @nytimes found an answer to their question from 2012, 'Is Algebra Necessary?'" pointing to this Sunday, Michael Ciepy, NY Times information piece: >> > >> >"'The Interview' generated roughly $15 million in online sales and rentals during its first four days of availability, Sony Pictures said on Sunday. Sony did not say how much of that total represented $6 digital rentals versus $15 sales. The studio said there were about two million transactions over all."

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[math-fun] Is Algebra Necessary?
by Hans Havermann 01 Jan '15

01 Jan '15

Dan Meyer twittered yesterday: "So glad the @nytimes found an answer to their question from 2012, 'Is Algebra Necessary?'" pointing to this Sunday, Michael Ciepy, NY Times information piece: "'The Interview' generated roughly $15 million in online sales and rentals during its first four days of availability, Sony Pictures said on Sunday. Sony did not say how much of that total represented $6 digital rentals versus $15 sales. The studio said there were about two million transactions over all."

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[math-fun] 2015 -> 11111011111 (binary)
by Stuart Anderson 01 Jan '15

01 Jan '15

The most interesting property of the number 2015 that I could find is that in binary it is the palindromic string with a single zero 11111011111. Considering such palindromic numbers starting from 101, 11011, 1110111, 111101111, 11111011111 gives integers 5, 27, 119, 495, 2015, 8127, 32639, 130815, 523775, 2096127, 33550335, .... Using the formula a(n)= 2^(2n+1) - 2^n - 1 So the next time this happens is in 8127. This would be a nice opportunity for the NYE firework operators on the Sydney Harbour Bridge to display 2015 in binary - 5 large jets on each side, along with a circular firework from the center of the arch. Stuart

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Re: [math-fun] "The Imitation Game" -- review
by Henry Baker 01 Jan '15

01 Jan '15

Well, "Amadeus" isn't _perfect_ ! You have a point. When Mozart was a child, he was a bit of a parlor trick; his father would take him around the countryside giving concerts in which the young Mozart "improvise" on tunes that someone in the audience would throw in. Of course improvisation was similar to writing an essay: you start by addressing the question that is asked, and you then morph it into something you can talk further about. So from that perspective the Mozart/Solieri 1-upmanship anecdote was pretty accurate. I have no idea if such an encounter with Solieri actually happened, though. At 02:48 PM 12/31/2014, James Propp wrote: >Henry Baker wrote: > >Perhaps the best analogy would be a movie about Mozart by someone who was >> born deaf. No matter what anyone told this deaf movie-maker about Mozart's >> music, it would always be completely foreign hearsay, and could never >> convey Mozart's musical genius to someone who does know & love Mozart's >> work. ("Amadeus", of course, is the complete opposite, as it was made by >> people who really, really enjoyed & understood Mozart's music.) > >I actually remember having the opposite impression at one point in the >movie, where Mozart's genius is supposedly demonstrated by the way he >improves a piece of Salieri's. The way he did this was by morphing one of >Salieri's melodies into one of his own more recognizable themes (I forget >which) --- as if musical genius were a matter of inventing catchy tunes. If >such an interaction had really taken place between Mozart and Salieri, I'm >guessing Mozart would have left Salieri's musical idea mostly intact and >made subtle modifications to enhance it. But then most movie-goers would >have been deprived of the impression that they were hearing the difference >between a Salieri and a Mozart. > >Jim Propp

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[math-fun] The Joy of Repeated Text Substitution
by Robert Baillie 01 Jan '15

01 Jan '15

I thought people would enjoy this! http://www.dailykos.com/story/2014/12/31/1354858/-Breaking-EU-to-Make-Engli…

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