August 2016 - mathfun@mailman.xmission.com

Re: [math-fun] Sonic screwdrivers for dentists?
by Henry Baker 05 Aug '16

05 Aug '16

There's several issues here. Let's deal with the narrow frequency range issue first. In order to respond to a precise tone -- e.g., B-flat -- you need a tuned resonator with a high "Q". Furthermore, since we're talking about a bulk material, you need a whole bunch of tuned resonators, all tuned to the same frequency, and relatively unaffected by the size & shape of the bulk material. In order to have a high Q, the energy dissipation per cycle has to be small relative to the energy stored per cycle. This means that it can't radiate away in the form of sound/heat/light/radio waves, etc. This is a tall order, since responding to a particular frequency (466Hz in the case of Bflat/Asharp) requires a resonator of a reasonable physical size, which might not fit into your mouth. So let's assume we're talking about an ultrasonic frequency, whose resonators could be made quite small. Let's now attack fabrication. It might be possible to 3D print a structure whose resonances are reasonably sharply tuned in such a way that you can get catastrophic failure by vibrating it at certain critical frequencies. I'm guessing, but this structure would have to have a large number of identical cells, each of whom has a high Q, and each is relatively isolated from its neighbors so that its energy doesn't dissipate too quickly. BTW, plaster-of-paris -- as used in casts for broken bones -- is NOT *cut* away, so much as *vibrated* away. When I broke my arm in the 4th grade (in the 1950's), I had a cast for 2 weeks, and then had a new cast for an additional 4 weeks. To remove the casts, the doctor used a vibrating disk that literally shook the plaster apart. It would be too dangerous to use something like a saw which could easily destroy the arm that the doctor is trying to fix! I can assure you that my cast was not tuned to a particular frequency! At 07:29 AM 8/5/2016, James Propp wrote: >Could there be a stronger adhesive that loses its adhesive virtue when you >make it vibrate it just so? (Might dentists if the future say "Don't sing >any loud B-flats for the next two weeks"?)

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[math-fun] Sonic screwdrivers for dentists?
by James Propp 05 Aug '16

05 Aug '16

Is it possible to design materials so that they'll fall apart (or change state in other useful ways) when stimulated by vibrations of a particular frequency? I just had a temporary crown put on one of my teeth. The dentist used a weak adhesive (so that he'd be able to remove it a couple of weeks from now when the real crown is available), and the crown just broke (I'm guessing that the weak adhesive that he used was at least as much a factor as the scone I crunched). Could there be a stronger adhesive that loses its adhesive virtue when you make it vibrate it just so? (Might dentists if the future say "Don't sing any loud B-flats for the next two weeks"?) Jim Propp

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Re: [math-fun] An easy August sequence riddle (answer)
by Eric Angelini 04 Aug '16

04 Aug '16

Spoiler: https://oeis.org/A275590 -------------- >Hello Math-Fun, this is a permutation of the integers from 1 to 35; what are the next two terms? one, FOUR, two, three, FIVE, NINE, ELEVEN, TWELVE, six, seven, eight, ten, THIRTEEN, FOURTEEN, fifteen, sixteen, seventeen, EIGHTEEN, NINETEEN, TWENTY, twenty-one, twenty-two, twenty-three, TWENTY-FOUR, TWENTY-FIVE, TWENTY-NINE, THIRTY, THIRTY-FOUR, twenty-six, twenty-seven, twenty-eight, thirty-one, thirty-two, thirty-three, THIRTY-FIVE,... Best, É.

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Re: [math-fun] "Net" of a dodecahedron
by Bill Gosper 04 Aug '16

04 Aug '16

On 2016-08-03 07:20, Allan Wechsler wrote: > I wonder how many different correct nets there are? Here's one: Graphics@PolyhedronData["Dodecahedron", "NetEdges"] Bringing to mind http://gosper.org/Muller-Liar.gif Speaking of equilateral dodecahedra, besides regular, endo, rhombic, and Julian's, I recently learned of Bilinsky's: https://www.youtube.com/watch?v=xVBOlbjiHGI --rwg > On Wed, Aug 3, 2016 at 10:17 AM, James Buddenhagen <jbuddenh(a)gmail.com> > wrote: > >> correct net png form >> >> https://upload.wikimedia.org/wikipedia/commons/thumb/1/1b/Dodecahedron_flat… >> >> On Wed, Aug 3, 2016 at 9:15 AM, James Buddenhagen <jbuddenh(a)gmail.com> >> wrote: >> > Correct net here: >> > https://goo.gl/UIxKyP >> > >> > On Tue, Aug 2, 2016 at 7:34 PM, Veit Elser <ve10(a)cornell.edu> wrote: >> >> Good eyes Tom! >> >> The two "hemispheres" have to be joined differently, and just along one >> edge (not two). >> >> -Veit >> >> >> >>> On Aug 2, 2016, at 2:30 PM, Tom Rokicki <rokicki(a)gmail.com> wrote: >> >>> >> >>> This look wrong to anyone else? >> >>> >> >>> >> https://commons.wikimedia.org/wiki/File:Geometric_Net_of_a_Regular_Dodecahe… >> >>> >> >>> -- >> >>> -- http://cube20.org/ -- [ <http://golly.sf.net/>Golly link >> suppressed; >> >>> ask me why] --

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Re: [math-fun] buying gas last night
by Dan Asimov 03 Aug '16

03 Aug '16

Which pairs of pairs of rationals have the property that the sum of the cubes of one pair equals the sum of the cubes of the other pair? —Dan ----- The pump stopped > $3 short of the $20 I gave the Desi clerk, so I squeezed it up to $17 and, remembering that a new employee at Empress of India had once accepted one, demanded a $3 bill with Bill Clinton's picture on it. The clerk said $2.71. In disbelief, I walked back to the pump. It said $17.29. The clerk denied that 1729 meant anything to him, but became very enthusiastic when I mentioned Ramanujan. I still don't understand where that 29¢ came from. There now was actually a little gas spilled on the ground. -----

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[math-fun] "Net" of a dodecahedron
by Tom Rokicki 03 Aug '16

03 Aug '16

This look wrong to anyone else? https://commons.wikimedia.org/wiki/File:Geometric_Net_of_a_Regular_Dodecahe… -- -- http://cube20.org/ -- [ <http://golly.sf.net/>Golly link suppressed; ask me why] --

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[math-fun] jonathan borwein
by Bill Gosper 03 Aug '16

03 Aug '16

On 2016-08-02 19:48, Thane Plambeck wrote: > http://experimentalmath.info/blog/2016/08/jonathan-borwein-dies-at-65/ Yikes. And not a mention of https://en.wikipedia.org/wiki/Peter_Borwein (who I thought was the delicate one). --rwg

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[math-fun] Surprises of the Faraday cage
by Stuart Anderson 03 Aug '16

03 Aug '16

https://sinews.siam.org/Default.aspx?TabId=900&ArtMID=2243&ArticleID=757

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[math-fun] jonathan borwein
by Thane Plambeck 03 Aug '16

03 Aug '16

http://experimentalmath.info/blog/2016/08/jonathan-borwein-dies-at-65/ -- Thane Plambeck tplambeck(a)gmail.com http://counterwave.com/

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Re: [math-fun] Interactive virtual planimeter
by Bill Gosper 03 Aug '16

03 Aug '16

Try this: areaPolygon[L_List] := Total[MapThread[Im[#1\[Conjugate]*#2] &, {L, RotateLeft[L]}]]/2 pts = {{1, 1}, {9, 6}, {6, 9}}; LocatorPane[Dynamic@pts, Dynamic@Graphics[{Gray, Point[Join @@ Outer[List, Range@9, Range@9]], Red, Polygon@pts}], {{0, 0}, {9, 9}, {1, 1}}, LocatorAutoCreate -> True] Dynamic[areaPolygon[{1, I}.# & /@ pts]] Drag the points around. Create and destroy them with Cmd-left. Plain click seizes nearest. If a new point is out of sequence, permute by sliding things around. It is a bug in Mathematica that clicking a point floats its coordinates. --rwg On 2016-07-29 07:50, James Propp wrote: > Musing some more about the article I want to write, I think the best > accompanying interactive demo would be a virtual polar planimeter that only > handles polygonal regions with vertices lying on a grid. When you add a > vertex, a new triangle is added/subtracted. That would make it clear where > the "shoelace" formula comes from. It might also make certain kinds of > puzzles/challenges possible. > > Jim > > On Fri, Jul 29, 2016 at 9:06 AM, Bill Gosper <billgosper(a)gmail.com> wrote: > >> It seems to me that planimeters are winding-numberish, based on the formula >> >> areaPolygon[L_List] := Total[MapThread[Im[#1\[Conjugate]*#2] &, {L, >> RotateLeft[L]}]]/2 >> >> whereas Mike's is sort of Riemann-Stieltjesish. >> >> Here's <http://gosper.org/planimeter.pdf> a crude Mathematica attempt at >> the former. Drag-undrag-drag makes >> points-linesegment-points. >> >> OOPS, bug in Mathematica's pdf renderer! The top illustration should look >> like >> gosper.org/planimeter.png , i.e. winding number mod 2. I'm not sure if >> Wolfram has >> a site for running toy programs, but this one seems short enough to be >> fairly portable. >> --rwg >> On 2016-07-26 10:57, Mike Stay wrote: >> > Here's a cheap one I just whipped up: >> > >> >> https://www.khanacademy.org/computer-programming/planimeter/5456307495501824 >> > >> > >> > On Tue, Jul 26, 2016 at 10:33 AM, James Propp <jamespropp(a)gmail.com> >> wrote: >> >> Does anyone know of a good interactive virtual planimeter that will tell >> >> you the signed area of a region that you've specified by dragging around >> a >> >> cursor? Even a version that only works for polygons (where the user >> clicks >> >> successive vertices) would be better than nothing. >> >> >> >> I've looked on the web, but I couldn't find what I need. Maybe one of >> you >> >> is inclined to build one? I would feature it (with credit, of course) in >> my >> >> upcoming Mathematical Enchantments essay about signed area (later this >> >> year). >> >> >> >> I want something that runs from a browser, so readers of my blog can >> just >> >> click on a link and a new tab will open. I am not interested in >> explaining >> >> the mechanics of how physical planimeters work (though I suppose I >> wouldn't >> >> be opposed to a webpage that explained that too, as long as the visuals >> >> associated with the wheels and arms didn't submerge the main message). >> I'd >> >> much rather have extra visuals that reinforce the concept of signed >> area. >> >> For instance, if the demo does just polygons, then every time a new >> vertex >> >> gets added, one could see a new triangle joining the new vertex, the >> >> previous vertex, and the origin, with shading or color to indicate >> whether >> >> it's oriented clockwise or counterclockwise, and with appropriate >> >> cancellation of positive and negative area being represented visually. >> >> >> >> It'd be especially nice if the demo handled reentrant polygons >> sensitively, >> >> so that regions encircled m times had a different color than regions >> >> encircled n times, when the integers m and n (positive, negative, or >> zero) >> >> are unequal. >> >> >> >> Jim Propp

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