August 2015 - mathfun@mailman.xmission.com

[math-fun] Computational effort
by James Propp 01 Mar '16

01 Mar '16

I don't know what the relevant units of computation are --- floating-point operations are the ones I've heard about most, but what if you're not using floating point? --- but I'm wondering if anyone has tried to quantify "how much computing" took place in (a) code-breaking at Bletchley Park, (b) the Manhattan Project, and (c) the Apollo space program. It seems likely to me that my laptop, running some silly number-crunching problem in Mathematica and returning an answer after six hours that I could probably have figured out in my head if I'd thought a little bit more before jumping in to write some code, has them all beat. What do you think? (And while we're on the subject: What's the canonical example of wasted computational effort? Isn't there some well-known old story of someone who had an early computer calculate the determinant of a singular matrix?) Jim Propp

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[math-fun] San Jose Math Decagon
by Bill Gosper 17 Feb '16

17 Feb '16

A couple of years ago Julian and I showed some animations to the San Jose Math Circle. gosper.org/decazoom.gif suggests why the area of a circle is the radius times half the circumference, and displays the continued fraction of a very lame approximation to π. --rwg

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[math-fun] xkcd
by rcs＠xmission.com 17 Dec '15

17 Dec '15

Today's XKCD cartoon has a nice collection of approximations. http://xkcd.com/1047/ There's the ancient expression 6 pi^5 for the mass ratio of the proton to the electron, and the more accurate (e^8 - 10)/phi. Robert Munafo & REIS are at least partially responsible. Rich

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[math-fun] Julian pimped my pump
by Bill Gosper 17 Dec '15

17 Dec '15

To run it <http://gosper.org/pumper_2.mp4> continuously, download with Miro, QuickTime-View-Loop. The pumped shapes split lengthwise resemble boats, so a grand entrance to a math museum would be a chain of these boats in an annular canal, passing through four rotors acting as an (approximate) air|water lock, with another four for the exit. Oh, the insurance. --rwg No manual entry for overboard

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[math-fun] A gallery of math art from Bridges 2015
by Alex Bellos 16 Nov '15

16 Nov '15

http://www.theguardian.com/science/alexs-adventures-in-numberland/gallery/2… <http://www.theguardian.com/science/alexs-adventures-in-numberland/gallery/2…>

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[math-fun] Folding a band saw blade
by James Propp 27 Oct '15

27 Oct '15

I took a look at https://www.youtube.com/watch?v=f-7x8X6I3RA because I needed to fold a kid's camping tent --- a task that poses similar differential-geometric challenges (but without the sharp teeth). The solution Paul Sellers advocates uses a table or other external object. Is there an easy-to-describe hands-only two-person solution? Also: Is there a write-up anywhere of the mathematics behind folding a band? I'm guessing that it's the kind of situation where you can get a (2n+1)-to-1 fold for any n. But I don't even see how the 3-to-1 fold works, nor do I see how one can rigorously prove that a 2-to-1 fold is impossible. Come to think of it, I don't even know how to state the problem as a mathematical one. Jim Propp

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[math-fun] Terminology: "Characteristic Generating Function"?
by Bill Gosper 23 Oct '15

23 Oct '15

"Indicator Generating Function"? "Predicate Generating Function"? What is the name for the G.F. that encodes an integer sequence in the exponents with coefficients in {0,1} ? E.g., the OGF for the squares is Sum[n^2*x^n, {n, ∞}] == -((x*(1 + x))/(-1 + x)^3), but the thing I'm looking for is Sum[x^n^2,{n,0,∞}]==(1/2)*(1 + EllipticTheta[3, 0, x]). (Years ago I sent some of you the GF for expressions for k as the sum of n (optionally distinct) members of such a set, in terms of the GF predicating membership in the set. It turned out to be a superspecial case of a formula in Generatingfunctionology that was so general I didn't even recognize it as the same problem.) The reason I ask is that Neil just came up with 1/2 (-1+EllipticTheta[2,0,x]/x^(1/4)+EllipticTheta[3,0,x]) predicating the quartersquares<http://en.wikipedia.org/wiki/Multiplication_algorithm#Quarter_square_multip…> A002620 <https://oeis.org/A002620>, from which he got (1/(2 (-1 + x)^2))((-1 + x) x^(3/4) EllipticTheta[2, 0, x] + x (3 - x + (-1 + x) EllipticTheta[3, 0, x])) generating what we think is the number of volumes you can pump with n of the recently discussed rotors, arranged optimally. --rwg

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[math-fun] Does anyone know if there are an infintude of these
by Bill Gosper 02 Oct '15

02 Oct '15

2F1[rational,rational,1 or 1/2,x] ==algebraic1[x] EllipticK[algebraic2[x]] identities? Or at least any more than a,b,c = 1/2,1/2,1 1/4,1/2,1 1/4,3/4,1 1/4/1/4,1 1/4,1/4,1/2 1/4,3/4,1/2 (sort of) 1/8,5/8,1 1/12,7/12,1 ? This last one features seriously unwieldy cubics. Tnx, --rwg

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Re: [math-fun] Dropbox still a pain in the ass...
by Henry Baker 29 Aug '15

29 Aug '15

There are 2 problems: 1. Getting the .gif file itself downloaded, rather than some .jpg still image approximation. Hopefully, appending "?dl=1" to the URL will fix this problem. 2. Once you have the .gif file itself downloaded, displaying it in animated form might require a tiny bit of work. On Windows, .gif files might be claimed by one or more web browsers (Explorer, Firefox, Opera, etc.), as well as "Windows Picture & Fax Viewer", Paint, Quicktime Player, etc. I find that "Windows Picture & Fax Viewer" is a bit slow to load, but it provides acceptable results. Most browsers also provide adequate display. Note that different browsers animate .gif files with different frame rates; I'm not aware that .gif files have a particular standard frame rate. I haven't searched for info on how to change the .gif frame rate in a browser, but I assume that it is one of the myriad changeable parameters (in Firefox check the "page" about:config). At 01:25 PM 8/28/2015, Fred Lunnon wrote: > The problems begin once you have accessed the file --- if you get lucky, your >system does the right thing: in this case, run the animated .GIF in >your browser. >If not, you have to tinker in some unpreductable fashion, as Henry and I both >discovered. > >WFL > >On 8/28/15, Mike Speciner <ms(a)alum.mit.edu> wrote: >> As near as I can tell, if you place a file in your Public dropbox >> subdirectory and copy the public link, then it is directly accessible to >> anyone using that link. >> >> --ms >> >> On 28-Aug-15 10:34, Henry Baker wrote: >>> Downloading this gif still required: >>> >>> 1. Turning on javascript (aka p0wnscript) for dropbox.com >>> 2. Turning on javascript for dropboxstatic.com >>> 3. Turning on javascript for ajax.googleapis.com >>> >>> 4. And then going through a dialog to say that I didn't want to sign up >>> for Dropbox. >>> >>> 5. And then finally clicking on a "download" button. >>> >>> Remember, you need "?dl=1" instead of "?dl=0" at the end of the URL below >>> in order to get the damn thing to download immediately. >>> >>> At 05:31 PM 8/27/2015, Fred Lunnon wrote: >>>> And Fred Lunnon seizes the opportunity to wheel out his animation >>>> of the seven-link robot / rigid kaleidocycle at >>>> https://www.dropbox.com/s/aeo6rxtc5j4p291/sevenring.gif?dl=0

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[math-fun] sums of three triangular numbers etc
by Neil Sloane 29 Aug '15

29 Aug '15

It is known that every number is the sum of three triangular numbers (that was when Gauss said Eureka!), and https://oeis.org/A008443 counts the ways. It is believed that every number is the sum of three palindromes (see A261422 or A261132, which have remarkable graphs). Both triangular numbers and palindromes have about O(sqrt(N)) terms below N. My question is, what is the least dense sequence S that works? One might think that one only needs O(N^(1/3)) terms below N to get S+S+S = Z+ Are there specific examples known of sparser sequences that work, Richard? (I didn't find anything in UPINT) Also, is there a simple proof of Gauss's theorem that doesn't work backwards from the 4-squares theorem? Is there a constructive algorithm that takes a number n and produces three triangular numbers that add to n? Same questions for sums of r squares, for r=2,3,4,... Best regards Neil

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