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Re: [math-fun] Random slice of a cube
by James Propp 16 Oct '17

16 Oct '17

I googled for five minutes and looked at half a dozen articles; I saw various models of random polytopes, but none of them appeared to be of the form "choose a random plane that intersects a polytope nontrivially and then take that intersection". I also didn't find the resilt about random polygons that Warren mentioned (the one that says that a random polygon has 4 sides on average). So if Warren or anyone else can provide more leads, I'd appreciate it! Jim Propp On Sunday, October 15, 2017, Warren D Smith <warren.wds(a)gmail.com> wrote: > I believe there is a known published model of "random polytopes" > in which the exact expected value is 4. >

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[math-fun] Gosper's identities
by François Mendzina Essomba 16 Oct '17

16 Oct '17

Hello, I found in the page http://www.tweedledum.com/rwg/idents.htm strange identities, as incredible as each other ... I have observed at length some of the identities ranging from C105 to C115. And this is what I deduced from it: (9) D106 product(2*cos(theta/3^(n/m))-1,n=1..infinity)=product(cos(3^(i/m)*theta/2),i=0...m-1) ; (10) D105 product(2*cos(theta/2^(n/m))-1,n=1..infinity)=product((1+2*cos(2^(i/m)*theta))/3,i=0..m-1); (11) D110 product((2*cos(theta/3^(n/m))+1)/3,n=1..infinity)=(2^m/(theta)^m)*product(sin(3^(i/m)*theta/2)/3^(i/m),i=0..m-1); However I am a little perplexed as to the last formula that differs from Gosper's D111 identity by a factor of 4. The result I get is this: (12) D111 product((2*cos(theta/3^(n/2))+1)/3,n=1..infinity)=(2^2/(theta)^2)*product(sin(3^(i/2)*theta/2)/3^(i/2),i=0..2-1); I suppose it must be the same for this small series, I have yet to check. Best regards

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Re: [math-fun] Random slice of a cube
by Dan Asimov 15 Oct '17

15 Oct '17

Keith Lynch wrote: ----- ... I would ask, not what the average number of sides is, but the proportion of each number of sides ... ----- And then for each number of sides, its proportion is averaged over all planes (obtained by the 3-points-on-circumsphere method)? Or if not, what? —Dan

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[math-fun] Da Vinci "orb" & optics
by Henry Baker 15 Oct '17

15 Oct '17

You're going to be hearing a lot about the recently "authenticated" painting by Leonardo da Vinci called "Salvator Mundi", because it will be auctioned at Christie's in November 2017 for hundreds of millions of dollars. https://en.wikipedia.org/wiki/Salvator_Mundi_(Leonardo_da_Vinci) The Wikipedia link to a jpeg of the picture itself: https://upload.wikimedia.org/wikipedia/commons/1/1f/Leonardo_da_Vinci_or_Bo… Here's a major problem IMHO with this painting: The (supposedly rock crystal) "orb" should distort the image behind it far more than it does in this painting, and da Vinci *would have known this*, as he did a lot of experiments with lenses and mirrors, designed lens and mirror grinding apparati, knew about the camera obscura, and supposedly invented the first *contact lens*! (I myself own a solid glass sphere much like the one in the painting, and I can vouch for the substantial distortion it makes when you view objects through it.) Da Vinci was famous for constantly updating his paintings (many of which were never finished) when he learned something new that would change how the painting should look. E.g., Isaacson's new biography of da Vinci mentions the fact that da Vinci updated the neck area of a painting after his human dissections showed that the neck muscle was different from what he had previously assumed. There are several possible explanations for the lack of orb distortion in the painting being auctioned by Christie's: * the particular area of the painting has undergone *bad conservation* by conservators who demonstrated their profound ignorance of optics when "repairing" the painting; * the orb isn't a sphere at all, but a thinner *lens*; but why in the world would da Vinci replace the standard spherical orb symbol with a lens? Possibly as an inside joke re his use of lenses to get his images right (see artist David Hockney's theories about these Renaissance painters and their optical devices)? https://en.wikipedia.org/wiki/Hockney%E2%80%93Falco_thesis * the orb isn't solid at all, but more like a transparent glass Christmas tree ornament, so it wouldn't distort the image underneath nearly as much as a solid rock crystal. Glass blowers could have made such a hollow orb for da Vinci, and da Vinci was known to have experimented with hollow glass "lenses" which he filled with water & other liquids for experimentation. Isaacson and the "experts" at Christie's claim that da Vinci went to a lot of trouble to paint "inclusions" (small bubbles) in the supposedly rock crystal material. They even claim that the crystal could be doubly refracting, and that da Vinci captured that. Then why in the world would da Vinci screw up the even more obvious feature -- the distortion caused by this supposedly solid crystal object? Perhaps there is a simpler explanation: the little dots aren't *inclusions* at all, but *specular reflections* of da Vinci's lights used to illuminate the orb model while it was being painted? Specular highlights would also be consistent with a hollow glass sphere or a lens. BTW, the modern computer graphics technique of "ray tracing", which is the computer implementation of "geometric optics", is required to properly handle refractive objects -- e.g., lenses, orbs. Ray tracing is quite computationally intensive, requiring large supercomputers to implement; most of the latest Hollywood animated features incorporate the sophisticated effects obtained from ray tracing, although they don't want to go so far as to make the animations look "too good", else people will forget that they *are* animations. Some of the later Dutch painters loved to show off their incredible skill by including such distorted images from reflections in curved objects and through lenses. There was a Scientific American article from 30+ years ago that discussed some of these effects. https://en.wikipedia.org/wiki/Ray_tracing_(graphics) https://en.wikipedia.org/wiki/Geometrical_optics If one has a good enough model of the image behind such an orb, it should be possible to *invert* the ray tracing computation to reveal the actual shape of the lens/orb which is providing the distortion.

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[math-fun] Looking for a recurrence related to Stern's diatomic array
by Neil Sloane 14 Oct '17

14 Oct '17

Stern's diatomic array (A049456) is a triangle in which row n lists the denominators in the n-th row of the table of Farey fractions. Another version of A049456 is the Stern-Brocot sequence (A002487) which has the recurrence a(2n)=a(n), a(2n+1)=a(n)+a(n+1), with a(0)=0, a(1)=1. There are a huge number of references. Someone recently asked me about a sequence which I was able to show was equal to the number of distinct terms in the n-th row of Stern's A049456. Surprisingly, this was not in the OEIS, although it is now - see A293160: 1, 2, 3, 5, 7, 13, 20, 31, 48, 78, 118, 191, 300, 465, 734, 1175, 1850, 2926, 4597, 7296, 11552, 18278, 28863, 45832, 72356, 114742, ... Can someone find a recurrence? (The smallest positive missing number from row n is A135510, which has an 1850 reference to Eisenstein.)

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[math-fun] Winning number
by James Propp 13 Oct '17

13 Oct '17

The number 2 (\[Pi] - ArcTan[2 \[Pi]]) turned up in a problem I'm working on, and I asked Mathematica to compute it numerically; it reported the answer as 3.45726 My instinctive reaction was that 3,4,5,7,2,6 would be a really good hand in some version of poker. Jim Propp

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[math-fun] Golden ratio and Pi
by François Mendzina Essomba 13 Oct '17

13 Oct '17

Bonjour à tous : Two small formulas for the day. formula 7 Golden ratio in an infinite nested root (Phi)^(-1)=(exp(-Pi)*sqrt(5*exp(2*Pi)-4)-1)/2+exp(-Pi)/sqrt(1+sqrt((5*exp(2*Pi)-4)+(5*exp(2*Pi)-4)*sqrt((5*exp(2*Pi)-4)+(5*exp(2*Pi)-4)*sqrt((5*exp(2*Pi)-4)+(5*exp(2*Pi)-4)*sqrt((5*exp(2*Pi)-4)+(5*exp(2*Pi)-4)*sqrt((5*exp(2*Pi)-4)+(5*exp(2*Pi)-4)*sqrt((5*exp(2*Pi)-4)+...))))))); formula 8 a small algorithm for Pi ^ 4 u_(0)=-1; u_(n+1)=(1/2)*sqrt(2+2*u_(n)); a_(n)=2^(4*n+5)*(3-2*sqrt(2+2*u_(n))+u_(n)); n → infinity ; a_(n) → Pi^4;

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[math-fun] Largest sum-free subset of {0,1,2,...,2n}
by James Propp 13 Oct '17

13 Oct '17

Prove that if S is a subset of {0,1,2,...,2n} containing more than n elements, then there must exist x,y,z in S such that x+y=z. (This smells like an Olympiad problem to me. In fact, I may well have solved this very problem forty years ago, but I don't see how to solve it now. Some sort of pigeonhole argument, sure, but how to set it up?) The bound is tight; both 1,3,5,...,2n-1 and n+1,...,2n are sum-free. Jim Propp

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[math-fun] Unsolved puzzle from Le Monde
by Don Reble 12 Oct '17

12 Oct '17

> First problem: in Alice's small city, 10 council members had to be > elected. Each voter could vote for 10 candidates, out of a list of 25. > What is the largest possible number of voters... > [such that] every voter has at least one of the candidates he voted > for elected in the council, no matter what... > First problem: 1358 One can show that 1358 voters can be satisfied, using a fairly obvious algorithm to choose the elected from the ballots. The problem is to rig 1359 ballots so that (at least) one voter must be left unsatisfied. Here's an easier (?) derived problem: there are 17 candidates; 2 will be chosen; and 8 voters, each voting for 10. (7 voters can be satisfied.) Can you rig 8 ballots? -- Don Reble djr(a)nk.ca

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[math-fun] Formulas (5, 6) ... number pi
by françois mendzina essomba2 12 Oct '17

12 Oct '17

Hello,Two smallapproximation formulas of pi. Formula 5:A formula that weighs,the number of terms explodes with m. Pi=Limit(1/2^(m-1)*(1+2*Sum(sqrt(1-((n^2)*2^(-2*m))),n=1..2^(m)-1)),m=infinity); Formula 6:if Lambda istaken in the interval ] 1; 1 [ and different from 0, -1 / 2,1 / 2, -1 / 4,1 / 4. Pi=(1/lambda)*Sum(((-1)^(n+1)/(2*n-1))*((tan(lambda*Pi))^(2*n-1))*cos((4*n-2)*lambda*Pi),n=1..infinity)+(1/lambda)*Sum(((-1)^(n+1)/(2*n))*((tan(lambda*Pi))^(2*n))*sin((4*n)*lambda*Pi),n=1..infinity);

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