July 2015 - mathfun@mailman.xmission.com

[math-fun] Did speeding American manhole cover beat Sputnik into space?
by Henry Baker 16 Jul '15

16 Jul '15

FYI -- http://www.theregister.co.uk/2015/07/16/america_soviets_space_race/ "in a test codenamed Pascal B" "The camera was set up to record one frame every millisecond. When the nuke blew, the lid was caught in the first frame and then disappeared from view. Judging from the yield and the pressure, Dr Brownlee estimated that it left the ground at more than 60 kilometres per second, or more than five times the escape velocity of our planet."

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[math-fun] Math museum error?
by James Propp 16 Jul '15

16 Jul '15

Does anyone know anything about this? http://www.dailymail.co.uk/news/article-3152858/Teen-catches-math-error-gol… The press release (sounding more political than mathematical) asserts that the way the Museum presents the Golden Ratio in its exhibit "is in fact the less common - but no less accurate - way to present it". So, what was the "mistake" in the Eames exhibit, and was it in fact a mistake? The newspaper story doesn't give details. Jim Propp

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[math-fun] the 19th singular value
by Bill Gosper 16 Jul '15

16 Jul '15

ETA(%E^-(2*SQRT(19)*%PI)) = ((3*SQRT(57)+1)^(1/3)-8/(3*SQRT(57)+1)^(1/3))^(1/8)*(GAMMA(1/38)*GAMMA(5/38)*GAMMA(9/38)*GAMMA(11/38)*GAMMA(7/19)*GAMMA(17/38)*GAMMA(23/38)*GAMMA(25/38)/(38*GAMMA(13/38)*GAMMA(8/19)))^(1/4)/(2*2^(53/152)*%PI) where Corey's denester neatly whacks the 8th root: (-(8/(1 + 3*Sqrt[57])^(1/3)) + (1 + 3*Sqrt[57])^(1/3))^(1/8) == -(2^(3/8)/3) + ((-1 + 3*Sqrt[57])^(1/3)*(4*(-5 + Sqrt[57]) + (11 + Sqrt[57])*(-1 + 3*Sqrt[57])^(1/3)))/(96*2^(5/8)) . It also denests LambdaStar[19], but the result is badly undersimplified. Fullsimplify thuggishly reverts to the original Root object, even when given a ComplexityFunction that enormously penalizes _Root ! But it is possible to guess the general form from the denester mess and determine coefficients with PSLQ: LambdaStar[19]==(((-Sqrt[57] + Sqrt[19] + 7*Sqrt[3] + 11)*(3*Sqrt[57] + 1)^(1/3)/ 16) - ((Sqrt[57] + Sqrt[19] - 7*Sqrt[3] + 11)/(2*(3*Sqrt[57] + 1)^(1/3))) - ((Sqrt[19] - 1)/2))/(3*Sqrt[2]) I was unable to Google whether this denesting is known. Is there a table of singular values more extensive than MathWorld's? --rwg Joerg>Let's not forget Broadhurst's very neat expressions: Message-ID: <200803140250.m2E2oHi4002315(a)philter.princeton.idaccr.org> Date: Thu, 13 Mar 2008 22:50:17 -0400 From: David Broadhurst <D.Broadhurst(a)open.ac.uk> To: NMBRTHRY(a)LISTSERV.NODAK.EDU Reply-To: David Broadhurst <D.Broadhurst(a)open.ac.uk> Subject: Singular value for Euler's numerus idoneus N=1365 default(realprecision,1050); a_1 = 550*sqrt(13) + 318*sqrt(39) + 750*sqrt(7) + 433*sqrt(21); a_2 = 1986 + 1145*sqrt(3) + 208*sqrt(91) + 120*sqrt(273); b = ((a_1-a_2)^2 - 2)/2; k_1365 = sqrt(1/2 - b^2/4*sqrt(4-b^2) - (2-b^2)/4*sqrt(1-b^2)); --------------- The denester also reduced the depth of k_1365, but made a proverbial Very Large Expression, more than Mathematica could further chew.

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surprise that Plutos surface unmarred by craters
by Henry Baker 16 Jul '15

16 Jul '15

FYI -- Plutos Portrait From New Horizons: Ice Mountains and No Craters 'A second surprise was that the dwarf planets surface was unmarred by craters. We have not yet found a single impact crater in this image, Dr. Spencer said.' http://www.nytimes.com/2015/07/16/science/pluto-flyby-photos-reveal-mountai… I don't see the problem. Most of the chaotic behavior during planet formation occurs in the inner Solar system -- in particular, inside of the large planets -- e.g., Jupiter, Saturn, Uranus, Neptune. Outside of these planets, the density is far, far smaller, and the opportunity for collisions is also far, far smaller. Time -- at least measured by number of impact craters -- runs far, far slower in these outer regions. Not only is the "year" for Pluto much longer, but the density of material to run into is much smaller -- even if you integrated it all the way around its orbit.

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[math-fun] surprise that Pluto’s surface unmarred by craters
by Warren D Smith 16 Jul '15

16 Jul '15

From: Henry Baker <hbaker1(a)pipeline.com> 'A second surprise was that the dwarf planet’s surface was unmarred by craters. “We have not yet found a single impact crater in this image,” Dr. Spencer said.' http://www.nytimes.com/2015/07/16/science/pluto-flyby-photos-reveal-mountai… I don't see the problem. Most of the chaotic behavior during planet formation occurs in the inner Solar system -- in particular, inside of the large planets -- e.g., Jupiter, Saturn, Uranus, Neptune. Outside of these planets, the density is far, far smaller, and the opportunity for collisions is also far, far smaller. Time -- at least measured by number of impact craters -- runs far, far slower in these outer regions. --I disagree with HB, this is a surprise. Obviously, whenever Pluto formed, stuff was falling in. So there must have been many craters. Why are there none now? There are only 2 options (a) when Pluto formed it was liquid (e.g. "magma ocean") and bombardment stopped before it solidified. But if HB is correct about "time running slower" then Pluto would have had time to solidify before bombardment ended, in which case (a) refuted. (b) Pluto had geological activity such as volcanism and plate tectonics, which erased the craters and built the mountains, this all happening much later. This is surprising. Also, Pluto has many moons, guessed to have formed in a collision between two big objects yielding Pluto and its moons. If so, then there should have been bombardment at that time. Charon appears to have craters. I therefore conclude the correct answer is probably (b), that is Pluto had later geo-activity, and/or melted, while Charon did not. It makes sense that the biggest one (Pluto bigger than all its moons) should be the one to melt (more gravitational energy) -- or to exhibit geo-activity: more radioactive heat per surface for it to leak out of, if the activity is driven by radioactivity.

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[math-fun] Mathematica 10
by James Propp 16 Jul '15

16 Jul '15

This question may be premature, but I'm curious what people's reactions are to Mathematica 10. I actually stuck with Mathematica 8 even after Mathematica 9 came out, so if version 10 is to version 9 as version 9 was to version 8, I'll probably just ignore the release of version 10 and stick to version 8 until Wolfram pries it from my cold dead hands, or until version 8 falls to pieces. Or does version 10 have so many nifty features (or boring-but-important bug-fixes) that a serious computer-assisted math researcher simply has to have it, even if the interface is annoying? (Come to think of it, is there a way to run the version 9 or version 10 kernel using the version 8 front end?) Jim Propp

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[math-fun] rational-angled triangles with rational side ratios. classification.
by Warren D Smith 15 Jul '15

15 Jul '15

https://dl.dropboxusercontent.com/u/3507527/RatRatTriangles.txt gives a complete classification in Euclidean plane, and a conjecturally-complete classification in nonEuclidean planes. -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)

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Re: [math-fun] New challenge: Offset Sonsky Gears
by Fred Lunnon 14 Jul '15

14 Jul '15

My thanks Bill and Tom for gear simulations. WFL

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[math-fun] Why Roman arches circular?
by Warren D Smith 14 Jul '15

14 Jul '15

Circular is the best arch, IF it is deep inside a giant mass of other stuff. And the Roman arches often were used in precisely that situation. Freestanding arches were less common. Also, circular is simplest.

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Re: [math-fun] Draft of "The Lessons of a Square-Wheeled Trike"
by Henry Baker 14 Jul '15

14 Jul '15

http://www.egyptorigins.org/catenarycurve.htm "An Ancient Egyptian Catenary Construction Curve" "When I first examined the curve displayed by the Saqqara Ostracon I recognized that it was not circular in form. My first attempt to define the curve also showed it was not elliptical. Refer to remarks by Gunn, who confirms both assessments. I also could not believe it was a hyperbolic." "The following solution should be regarded as incorrect, but curious. I offer it as an example of how solutions can reach far afield in attempts to understand the ancient Egyptian geometricians." "I was haunted by the possibility that it might be a catenary." ... http://www.egyptorigins.org/saqqaraostracon.htm "(Refer to my proposed mathematical solution in the accompanying paper, An Ancient Egyptian Catenary Construction Curve, where I explain the difference in "sharpness" of the drop. EPM)" At 02:02 PM 7/14/2015, W. Edwin Clark wrote: >Try Google Images Egyptian catenary arch ><https://www.google.com/search?q=Egyptian+catenary+arch&safe=active&biw=1280…> > >On Tue, Jul 14, 2015 at 4:54 PM, James Propp <jamespropp(a)gmail.com> wrote: > >> Interesting! >> >> If someone can steer me towards a picture of an Egyptian catenary(-ish) >> arch, I can probably use it in my essay, which will be going live on Friday. >> >> Thanks, >> >> Jim >> >> On Tue, Jul 14, 2015 at 3:00 PM, Henry Baker <hbaker1(a)pipeline.com> wrote: >> >> > The Egyptians experimented with catenaries for St Louis-type arches >> > perhaps 3000-4000 years ago. >> > >> > I've seen a picture in a book somewhere showing a quite good approximation >> > to a catenary arch from ancient Egypt. >> > >> > Of course, they didn't know it as a "catenary", and obviously knew nothing >> > about cosines -- hyperbolic or otherwise. It is conceivable that the >> > ancient Egyptians discovered the Gaudi's principle, wherein hanging >> > catenaries become catenary arches when flipped over and compression & >> > tension exchange places. If some clever Egyptian had discovered this >> > principle, it was quickly lost, as the ancient world dnd't seem to be >> > filled with catenary curves (excepting ship rigging, which the ancient >> > Egyptians excelled at). >> > >> > It is extremely interesting to me that once the Greeks & Babylonians >> > discovered semicircular arches & parabolic arches, no more experimentation >> > seems to have been done with other shapes. >> > >> > In particular, the ancient Romans (to my knowledge) used ONLY circular >> > arches, even though the amount of material required was substantially >> > greater than what would be required for other types of arches. >> > >> > The Romans were outstanding engineers, and while waiting for their slave >> > labor to carry the excess material required to build a circular arch, these >> > smart people had plenty of time to ponder whether there was a better way. >> > >> > There are several possible explanations why they stuck with circular arches: >> > >> > 1. The Greeks became enamored of the elegance of circles, and -- whether >> > or not another shape would have been more "efficient" -- they preferred >> > circles for aethetic/moral/religious reasons. Analogous problems limited >> > astronomers until Galileo risked death/excommunication to argue for ellipses. >> > >> > 2. The ancient Romans may have depended upon a method of building arches >> > that depended critically upon their being arcs with a consistent radius. I >> > have conjectured that the ancient Romans devised a system of "centering" >> > that did NOT require a huge amount of scaffolding. This is because the >> > Romans built so many arches, in so many locations -- including locations >> > without any lumber for scaffolding -- that they required a modular method >> > using a minimal amount of lumber that had to be carried along with the >> > troops for other purposes -- including building siege towers and >> > catapults. In this case, the Romans weren't worrying about minimizing >> > material within the arch itself, but minimizing the material required for >> > the "centering" of the arch. >> > >> > https://en.wikipedia.org/wiki/Centering >> > >> > 3. Strict liability. The engineers who designed & built structures in >> > the ancient world were required to stand under them when load was first >> > applied. This strict liability principle guaranteed significant >> > overbuilding -- which is why some Roman roads & arches are still in use >> > today after 2000 years -- but so severely penalized experimentation that no >> > one was brave enough to consider alternatives. >> > >> > http://www.commonlaw.com/Hammurabi.html >> > >> > 229. If a builder has built a house for a man, and has not made his work >> > sound, and the house he built has fallen, and caused the death of its >> > owner, that builder shall be put to death. >> > >> > 230. If it is the owner's son that is killed, the builder's son shall be >> > put to death. >> > >> > 231. If it is the slave of the owner that is killed, the builder shall >> > give slave for slave to the owner of the house. >> > >> > At 12:35 PM 6/30/2015, James Propp wrote: >> > >I've finished a draft of installment #1 (as opposed to #0) of >> > "Mathematical Enchantments", and I'd welcome comments.

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