August 2010 - mathfun@mailman.xmission.com

Re: [math-fun] Another(?) stick length illusion
by Robert Munafo 21 Aug '10

21 Aug '10

Here it is as a PDF file (1285 bytes, which is only 31% bigger :-) http://mrob.com/pub/math/speciner.pdf Mike, yes I did attach a pdf file, but apparently the mailing list server removed it (and without returning an error message) so thanks for letting me know. On Fri, Aug 20, 2010 at 12:48, Mike Speciner <ms(a)alum.mit.edu> wrote: > On Friday 20 August 2010 12:32:50 Robert Munafo wrote: > > Here it is as a PDF file (1285 bytes, which is only 31% bigger :-) > > > Did you forget to attach the pdf, or does the mailing list not allow > attachments (which is what I thought, and one of the reasons I didn't > bother trying to include a PDF; the other being that the PS is sufficiently > readable to "prove" that the two lines are the same length, although I > probably should have included some comments)? > > --ms > On Fri, Aug 20, 2010 at 10:10, Mike Speciner <ms(a)alum.mit.edu> wrote: > Try this: > > %!PS-Adobe-3.0 > %%BoundingBox: 0 0 612 792 > %%Orientagion: Portrait > [...] > > 1 0 moveto c1 s1 lineto -c2 s2 lineto -c2 -s2 lineto c1 -s1 lineto fill > 0 setgray > c1 s1 moveto c1 -s1 lineto stroke > > showpage > -- Robert Munafo -- mrob.com

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Re: [math-fun] The (_|_) and 6pents
by Bill Gosper 21 Aug '10

21 Aug '10

Hans H: > Joshua Zucker: > >> I connected them to form a black rectangle and then it looked like a >> rectangle (they looked the same length). It seemed like the "smaller" >> one stayed the same size when I took those perpendicular lines back >> off, while the "longer" one seemed to jump to a bigger size when they >> disappeared. > > I've played around with this a bit as well. The illusion seems > stronger to me when the lines are vertical: > > http://chesswanks.com/pot/TwoLines.jpg > > If my jpg conversion maintained my thinner lines' thickness, the > "longer" line also appears darker. The darkness business also seems backwards. But Hans's rendition makes me wonder if the length business is just a case of the classic \ / \ / /|\ | / | \ | | | | | | | \ | / | \|/ | / \ / \ --rwg

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[math-fun] The (_|_) and 6pents
by Bill Gosper 21 Aug '10

21 Aug '10

Whew, Paint on my Apple virtual XP is still sane. So I believe the two segments are now identical. Does anyone think they look identical? --rwg www.tweedledum.com/rwg/6pents.png On 8/20/10, Bill Gosper <billgosper(a)gmail.com> wrote: > Apologies. I had tried superposing the segments with Windows Paint, > which has a mode > where white <-> transparent. But for some bizarre reason, it enforced > opaque white > on this image, and I couldn't test it. (One *very* frustrating hour > later:) > I tried rewriting it at various pixel depths and as a gif. Paint not > only refused to > treat white as transparent, it even disabled the feature in the > toolbar, and left it disabled > when I loaded a file on which it formerly worked! > So I tried "physically" moving a rectangle containing the longer line > onto the shorter, > and the black part became transparent!! Paint has gone criminally > insane. If this were > something important, I would be, too. Anyway, I just manually fudged > the picture closer > to equality, and I still see the illusion. Feel free to repair it > with some sane software, > and see if it still works for you. > --rwg > On Thu, Aug 19, 2010 at 8:44 PM, Bill Thurston <wpt4(a)cornell.edu> wrote: >> What is your experimental method, guys? >> I copied the image into Adobe Illustrator and used the measuring tool to >> check. > The leftward segment measured 4.05", and the rightward segment > measured 4.17". A > 3% difference. Theoretical equality != actual equality. > > My experimental method was somewhat cruder: I held up a piece of paper > next to one line on my screen, made two marks, then slid the paper > over. Going back and forth a few times convinced me that it wasn't > just pencil mark error. > > --Joshua >

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Re: [math-fun] Another(?) stick length illusion
by Bill Gosper 20 Aug '10

20 Aug '10

Apologies. I had tried superposing the segments with Windows Paint, which has a mode where white <-> transparent. But for some bizarre reason, it enforced opaque white on this image, and I couldn't test it. (One *very* frustrating hour later:) I tried rewriting it at various pixel depths and as a gif. Paint not only refused to treat white as transparent, it even disabled the feature in the toolbar, and left it disabled when I loaded a file on which it formerly worked! So I tried "physically" moving a rectangle containing the longer line onto the shorter, and the black part became transparent!! Paint has gone criminally insane. If this were something important, I would be, too. Anyway, I just manually fudged the picture closer to equality, and I still see the illusion. Feel free to repair it with some sane software, and see if it still works for you. --rwg On Thu, Aug 19, 2010 at 8:44 PM, Bill Thurston <wpt4(a)cornell.edu> wrote: > What is your experimental method, guys? > I copied the image into Adobe Illustrator and used the measuring tool to check. The leftward segment measured 4.05", and the rightward segment measured 4.17". A 3% difference. Theoretical equality != actual equality. My experimental method was somewhat cruder: I held up a piece of paper next to one line on my screen, made two marks, then slid the paper over. Going back and forth a few times convinced me that it wasn't just pencil mark error. --Joshua

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[math-fun] generalized sphericons
by Bill Gosper 20 Aug '10

20 Aug '10

Yow, there are totient(n)/2 of these for every star polygon(n/d), i.e. (totient(n)/2)^2 for every n. The nine for n=7: www.tweedledum.com/rwg/sphericons7.PNG . --rwg

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[math-fun] That can't be right!
by Hans Havermann 20 Aug '10

20 Aug '10

Approximately six days ago, news organizations began reporting that, according to the United Nations, "one-fifth" of Pakistan was under water. Since then, the one-fifth has appeared again and again with assorted modifiers: up to, about, at least, etc. Looking at the Earth Observatory pictures... http://earthobservatory.nasa.gov/IOTD/view.php?id=45200 ... I am having a hard time seeing the reality behind that fraction. I've been unable to find the original UN source of the stories and I don't see a lot of skepticism out there. Perhaps by the "country" of Pakistan is meant something other than geographical shape but that is never explicitly stated. Anybody care to wade in?

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[math-fun] Another(?) stick length illusion
by Bill Gosper 20 Aug '10

20 Aug '10

While hacking the pentagon fill, tutor Julian found www.tweedledum.com/rwg/6pents.png, which seems to work backwards: The stick larger than its context appears smaller than the stick smaller than its context. --rwg

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[math-fun] semisesquisphericon
by Bill Gosper 19 Aug '10

19 Aug '10

www.tweedledum.com/rwg/semisesquisphericon.png, www.tweedledum.com/rwg/quinquangularsphericon.png A single (nonplanar) face is divided by a single edge into--a single face. www.tweedledum.com/rwg/pentagrammaticsphericon.png A single (nonplanar) face is divided by two edges into--a single face. I may have seen the first one as an oriental snack package. Anyway, you should be able to curl (not fold) them from single paper cutouts. I wonder if they're patented as packaging or something. --rwg

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[math-fun] New Fields Medallists
by Dan Asimov 19 Aug '10

19 Aug '10

The new Fields Medallists have been announced today at the International Congress of Mathematicians in Hyderabad, India: < http://www.icm2010.in/imu-prizes/prize-winners-2010 >. --Dan _____________________________________________________________________ "It don't mean a thing if it ain't got that certain je ne sais quoi." --Peter Schickele

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[math-fun] Solution Status of Monopoly Game ?
by Henry Baker 16 Aug '10

16 Aug '10

I heard a recent NPR Planet Money episode on the game of Monopoly & wondered whether Mpnopoly has been "solved", in the sense of finding the optimal algorithm for play, given the statistics of the dice. Of course, there are many variations on the game, so one would presumably have to fix the set of rules that would govern. http://en.wikipedia.org/wiki/Monopoly_%28game%29 Given the availability of modern super-computers, I would have thought that optimal/semi-optimal strategies would have been found by now.

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