October 2014 - mathfun@mailman.xmission.com

Re: [math-fun] analytic function for new Airbnb logo?
by Bill Gosper 14 Dec '14

14 Dec '14

Henry>I wanted an approximation that was entire over the complex plane, so the straight lines and discontinuities would have to be fudged. If I take the image, convert it to black-and-white, and erode the boundaries to a single pixel width, I will get a "continuous" curve that goes around the origin twice. If I consider this curve in (r,theta) coordinates, I can now consider the Fourier transform. I'd like to reset the origin in such a way that the "fundamental" is zero -- i.e., the lowest frequency (which takes 2 revolutions) has a coefficient of zero. (There are other possible locations for the origin: the balancing point, the center of gyration, etc.) That having been done, I could alternatively use (log(r),theta) coordinates, which have some nicer properties, because it is the actual complex log function. Fourier analysis can approximate the discontinuous changes in curvature; hopefully modifying these coefficients a little will remove these discontinuities with a minimum of ringing. The sides of your rounded edge squares are impressively flat. <HGB At 03:48 PM 7/30/2014, Bill Gosper wrote: HGB> My approach would be to take the Fourier transform of the polar plot, rwg>Defined as piecewise linear|circular? and then simplify to a small number of coefficients. Of course, it will take 2 circuits around the origin to complete the figure. The sides are perhaps a little too flat, which will blow up the number of coefficients required. I guess if you transform [0,4pi) to [0,1) you could use some sort of Cheby polynomial approximation. It would be nice to be able to transform it back to p(x,y)=0, where p is a polynomial in x,y. At 02:56 PM 7/29/2014, Whitfield Diffie wrote: HGB>Anyone game for an analytic function to approximate Airbnb's new logo? They should manufacture a paper clip and give it out at trade shows. Whithttp://www.underconsideration.com/brandnew/archives/airbnb_logo_detail.… rwg> You can get some impressively straight segments with theta fcns: gosper.org/halph.pdf (10MB?!) or see http://gosper.org/thetpak.html (scrolled 69%). (.3281... is âˆš(one ninth constant), which I claim should be named Halphen's. It's the largest q for which the curve is convex.)) As I mentioned here before, as 0<q<1, the curve continuously deforms from a tiny "circle" to a big "+" sign, sweeping out a fairly convincing spaceplane: http://gosper.org/sst.png --rwg But a problem with this logo is that it switches curvatures discontinuously. <rwg Another possibility: modulate the arcspeed (slow down for curves, speed up for straightaways) before taking the Fourier.

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Re: [math-fun] Ménage primes
by Bill Gosper 03 Nov '14

03 Nov '14

On 2014-10-30 19:24, Adam P. Goucher wrote: > Here's a problem to be added to Ed Pegg's list of prize puzzles: find > another ménage prime. > > https://cp4space.wordpress.com/2014/10/31/menage-primes/ > > So far, only five such primes are known (although I haven't searched > beyond n = 10000): > > M(4) = 5 > M(5) = 13 > M(13) = 775596313 > M(53) = 567525075138663383127158192994765939404930668817780601409606090331861 > M(8645) = > 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> > Happy Halloween! > > > Sincerely, > > > Adam P. Goucher > A000179: 1, 0, 0, 1, 2, 13, 80, 579, 4738, 43387, 439792,... In[319]:= Table[Round[2*n/E^2*BesselK[n, 2]], {n, 9}] Out[319]= {0, 0, 1, 2, 13, 80, 579, 4738, 43387} BesselK!?! --rwg Hallowe'en: Yesterday, Jake's Pizza in Saratoga in the window had a certificate of approval from the Santa Clara County Board of Health. Inside: wall to wall spiders on massive fake webs.

2 2

[math-fun] Kerr metric
by Warren D Smith 31 Oct '14

31 Oct '14

>> Brent Meeker: >> It's probably pretty hard because you can't just have a rigid rotating hunk >> of matter (no >> rigid bodies in relativity). So you'd probably choose a perfect fluid (no >> viscosity) to >> model the matter with an appropriate equation of state to model the >> compressibility. No >> doubt it can be done numerically. >> Brent Meeker > WDS: Just because rigid bodies do not exist, > does not stop there from being a rigidly rotating hunk of matter (all > interpoint-distances > preserved as measured by the metric). > BM: But then the problem is what stress-energy tensor is implied by this hunk - nothing realistic. --it could well be realistic. Consider a rigidly rotating baseball. Or asteroid Ceres. Nothing unrealistic about it. > WDS: Further, if you produced a > solution with inviscid fluid and not preserving interpoint distances, > then I would dispute your solution because viscosity presumably would > cause some kind of energy loss. > BM: Loss to where? The energy can't get out. The effect of viscosity would just > be to make the stress-energy tensor more complicated and possibly even turbulent. ----the energy can and will get out. Viscous losses ==> heat ==> radiated photons ==> out. Almost all real fluids have viscosity, so therefore an inviscid fluid solution would not be realistic as a steady-state solution. A genuinely steady-state solution should exist with a Kerr exterior and realistic interior. One possible class of "realistic interiors" -- perhaps even the only class -- would be "a rigidly rotating mass distribution." Prove or disprove: there exists an interior in this class, and a way to glue it onto a Kerr metric exterior. If does exist, then start finding examples. If it doesn't exist, that sounds like a great theorem. You've now almost gotten me to talk myself into trying to do exactly this...

2 1

[math-fun] Hex that pumpkin!
by Alex Bellos 31 Oct '14

31 Oct '14

One for Halloween: Stereographic projections as light sculpture...amazing 3D-printed jack-o-lanterns by Henry Segerman and Saul Schleimer http://www.theguardian.com/science/alexs-adventures-in-numberland/2014/oct/…

4 4

[math-fun] "many interacting worlds" new quantum mechanics foundation?
by Warren D Smith 31 Oct '14

31 Oct '14

Michael J. W. Hall, D.-A. Deckert, Howard M. Wiseman: Quantum phenomena modelled by interactions between many classical worlds http://journals.aps.org/prx/abstract/10.1103/PhysRevX.4.041013 http://arxiv.org/abs/1402.6144 Postulates there are a large, but finite, number of interacting "worlds." There is a "universal interworld potential function" which accomplishes the interaction. The word "unitary" never occurs in their paper, which strikes me as one big red sign saying "caveat emptor.". But I don't understand it. -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)

1 0

[math-fun] Silly vortex model of atoms
by Warren D Smith 31 Oct '14

31 Oct '14

Adam Goucher's post about the old forgotten, and vastly discredited "vortex model of atoms" I guess shows this was a kind of "string theory," i.e. atoms supposedly were different kinds of knots on a closed string? Thus proving, you just cannot kill bad ideas :)

2 1

[math-fun] Ménage primes
by Neil Sloane 31 Oct '14

31 Oct '14

Adam, Very nice - and two excellent new sequences! Happy Halloween! Best regards Neil

1 0

[math-fun] Ménage primes
by Adam P. Goucher 31 Oct '14

31 Oct '14

Here's a problem to be added to Ed Pegg's list of prize puzzles: find another ménage prime. https://cp4space.wordpress.com/2014/10/31/menage-primes/ So far, only five such primes are known (although I haven't searched beyond n = 10000): M(4) = 5 M(5) = 13 M(13) = 775596313 M(53) = 567525075138663383127158192994765939404930668817780601409606090331861 M(8645) = 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06420366341628054044029157394722465065041981586227909418642811657034120997442060385846152527092482905442159703363321573929021059403835628817850402335451720050339581656616043196654599891734636885345643279250812121155137949015045429834855921402801205642580901076767440711928404157613 Happy Halloween! Sincerely, Adam P. Goucher

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[math-fun] Schroeppel wins national contest, starting top journalist's career :)
by Warren D Smith 27 Oct '14

27 Oct '14

Seems one of the earliest stories by later-famous investigative reporter Seymour M. Hersh was about Richard C. Schroeppel winning a nationwide math exam contest at age 16 in 1964. Hersh moved on from this initial triumph to become probably the greatest (or certainly one of the few best) investigative reporters of the 20th century... -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)

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[math-fun] Kerr metric
by Warren D Smith 26 Oct '14

26 Oct '14

>> WDS: >> As far as I know, an important still-open problem is to find a way to >> glue the Kerr vacuum metric as the exterior, onto the metric of some >> realistic hunk of rotating matter as the interior, to get combined >> metric representing a rotating mass and its gravity field. >> Seems quite embarrassing that nobody has been able to do that. >> How hard can it be? >> >> (But there are many successful ways to do this in the non-rotating case.) > Brent Meeker: > It's probably pretty hard because you can't just have a rigid rotating hunk > of matter (no > rigid bodies in relativity). So you'd probably choose a perfect fluid (no > viscosity) to > model the matter with an appropriate equation of state to model the > compressibility. No > doubt it can be done numerically. > Brent Meeker --Just because rigid bodies do not exist, does not stop there from being a rigidly rotating hunk of matter (all interpoint-distances preserved as measured by the metric). Further, if you produced a solution with inviscid fluid and not preserving interpoint distances, then I would dispute your solution because viscosity presumably would cause some kind of energy loss. (Although, I suppose you could argue a solution of your sort genuinely was valid for describing a mass made of superfluid liquid helium... but helium is only superfluid in a rather small pressure & temperature set.)

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