December 2010 - mathfun@mailman.xmission.com

[math-fun] A distant mirror
by Bill Gosper 19 Dec '10

19 Dec '10

Coincidentally, Neptune's orbit is about where things in geocentric coordinates violate the Speed Limit. --rwg _______________________________________________ ES>You will also need a large telescope mirror to view the Earth image 2000 ly distant. At 10^13 km/ly, this is 2 x 10^19 m. In order to see 1 m details, it will need to resolve an angle of 5 x 10^-20 rad. The standard diffraction limit for wavelength λ and aperture diameter a is λ/a. For λ = 500 nm, the required mirror diameter is 10^13 m = 60 AU. This is the size of Neptune's orbit. The distant mirror also needs to be at least this big. -- Gene

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[math-fun] Valuations of EllipticK
by Bill Gosper 19 Dec '10

19 Dec '10

For a horizontally swung pendulum (thetamax = pi/2), the Fourier series for am gives instead a fundamental of 4 sech(pi/2) = 1.594, exceeding pi/2 by < 2%. The "sechand" pi/2 fell out of some EllipticKs whose values were among the few currently known to FunctionExpand. But my more-or-less systematic collection of special values of eta provides an abundance of K values, because EllipticK[EllipticTheta[2, 0, q]^4/EllipticTheta[3, 0, q]^4] -> 1/2 Pi EllipticTheta[3, 0, q]^2 , i.e., EllipticK[(16 EllipticEta[q]^8 EllipticEta[q^4]^16)/EllipticEta[q^2]^24] -> Pi EllipticEta[q^2]^10/(2 EllipticEta[q]^4 EllipticEta[q^4]^4) , where EllipticEta[q_] -> EllipticTheta[1, \[Pi]/3, q^(1/6)]/Sqrt[3] , I.e., DedekindEta[tau]->EllipticEta[Exp[2*I*Pi*tau]] E.g., EllipticK[1/2 + Sqrt[2] 3^(1/4) (-2 + Sqrt[3])] -> Gamma[1/4]^2/(2 3^(3/4) (-1 + Sqrt[3]) Sqrt[2 Pi]), EllipticK[1/2 - Sqrt[2] 3^(1/4) (-2 + Sqrt[3])] -> 3^(1/4) Gamma[1/4]^2/(2 (-1 + Sqrt[3]) Sqrt[2 Pi]), EllipticK[1 - (-2 + Sqrt[2] - Sqrt[3] + Sqrt[6])^4] -> 3^(3/4) Gamma[1/3]^3)/ (2 2^(5/6) (1 + Sqrt[3])^(5/2) Sqrt[-3 - 4 Sqrt[2] + 5 Sqrt[3]] Pi) EllipticK[8 2^(1/4) (9 - 4 2^(1/4) - 3 Sqrt[2]))/(-1 + Sqrt[2])^6] -> (-1 + Sqrt[2])^(5/2) Gamma[1/4]^2/(4 Sqrt[2 (9 - 4 2^(1/4) - 3 Sqrt[2]) Pi]) EllipticK[-16 + 12 Sqrt[2]] -> 2 (2 + Sqrt[2]) Gamma[5/4]^2)/Sqrt[Pi] --rwg

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[math-fun] A distant mirror
by Stephen B. Gray 18 Dec '10

18 Dec '10

I suggest a project to see if there is a mirror in space facing us 1000 light years from Earth. If it's out there and if it's big enough, we could peer at it and see what happened here 2000 years ago. Maybe some extraterrestrial civilization has taken pity on our ignorance of ~0 CE history in Palestine and elsewhere. To uncover history we might need to see details as small as a few meters, which implies a mirror diameter of X. Maybe someone could compute X. If X is large enough, the builders would (?) ....... many interesting questions. This notion is not suggested as something practical, just something that is "logically possible," showing that the information still exists, except for that created indoors or under the sheets. Steve Gray

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Re: [math-fun] racing around a triangle
by Henry Baker 16 Dec '10

16 Dec '10

Ok, I've been playing with the "optimal" paths with acceleration vectors (At+B)/|At+B|, as described in the Corrozza, et al, paper. I've been trying to find an optimal "orbit" path around a square inscribed in the unit circle, where the acceleration vector a always has |a|=1. Here "orbit" means that the path goes around exactly once, hits every vertex, is continuous and its velocity vector is continuous. Although there is no requirement that the absolute value of the velocity vector at each vertex is the same, there doesn't appear to be any reason to doubt that a truly optimum path would have this property. If I take 4 segments of the type described in the paper, I can get very close to the circular path, both in space and in time, but I can't seem to be able to beat the circular path in terms of minimal time. The times are very close -- differing only by a percent or so, but they're still different; the path of the type given in the paper is always a bit slower than the circular path. It doesn't seem to help, whether the non-circular path goes outside the circular path or inside. Since the paper claims that its "optimal" paths are optimal, is the paper wrong? I don't know enough calculus of variations to be able to critique the paper. At 07:36 AM 12/8/2010, Henry Baker wrote: >Thanks very much, Andy, for the Mathematics Intelligencer reference. I found this description: > >http://blog.sciencenews.org/view/generic/id/64589/title/Winning_the_World_S… > >Here's the full paper (basically Carroza's 2009 undergraduate thesis at Williams College): > >Carroza, Davide; Johnson, Stewart; Morgan, Frank. "Baserunner's Optimal Path". Math. Intelligencer 32, 1 (2010), 10-15. > >http://www.springerlink.com/content/t7q081384758kl5l/fulltext.pdf > >-- > >The Carrozza, et al paper is quite relevant -- especially: >--- >Lemma 2. > >A fastest C(1,1) path from one point to another in the plane, given initial velocity, final velocity, and a bound sigma>0 on the magnitude of the acceleration a, > >a = sigma*(At+B)/|At+B| > >for some constant vectors A, B, is unique. >--- > >The cool thing about this lemma is that the optimal paths are _catenaries_ in _velocity space_ ! > >--- > >There are obviously several potential variations on the problem: > >1. A path that goes _through_ all the vertices versus a shortest path that goes completely around the triangle (it may be faster for some really slim triangles to not even attempt to touch one of the vertices, but you still don't cross the triangle boundary). My initial goal doesn't require going through all the vertices (so it isn't completely analogous to the "running bases" problem), so it is conceivable that a simple circular orbit with the largest triangle side as diameter may be the best solution (for some non-obtuse triangles). > >2. The path is a true cyclic "orbit": the positions & velocities match seamlessly at the beginning & end. It is conceivable that the fastest such "orbit" may actually go around the triangle twice (or k times), in which case we take the total time & divide by k. I.e., it may not be necessary to match accelerations & velocities on a single pass, so long as we match them at some fixed integer number of passes. For example, an optimal path that starts with initial velocity 0 might get faster & faster on each trip around the triangle, but converge to an optimal cyclic path. > >I guess that it is unlikely, but perhaps possible to have a optimal chaotic solution which _never_ matches, in which case we may have to look for a limit of the running average time. My model for this sort of chaotic behavior is the Minsky circle-drawing method from Hakmem. > >3. Obvious extension to (convex) polygons -- e.g., base-running. > >At 09:35 PM 12/7/2010, Andy Latto wrote: >>On Tue, Dec 7, 2010 at 5:58 PM, Henry Baker <hbaker1(a)pipeline.com> wrote: >>> I'm still trying to solve for the optimum path around a triangle, subject to the acceleration constraint |a|<=1. >> >>By "around a triangle", it seems you mean any path that passes through >>the three vertices of the triangle; is this correct? >> >>Also, I think you have an additional implied constraint that the route >>be cyclic, that is, the velocity at the end of the path is the same as >>the velocity at the beginning. Otherwise the best solution for the >>degenerate triangle would be to start at one end of the line segment >>with velocity 2, decelerate until you reach the other end with >>velocity zero, and then continue to accelerate in the same direction >>until you return to the original vertex, again with speed 2, but now >>in the direction away from, rather than towards, the other vertex. >> >>The Mathematical Intelligencer had an article a month or two back >>about the optimal way to run the bases in baseball. That is, the >>fastest path with bounded acceleration starting at one vertex of a >>square with velocity 0, passing through the other three vertices, and >>returning to the original vertex (though without the requirement that >>the final velocity be 0). I think much of this article is relevant to >>your triangle problem. >> >>Andy

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[math-fun] Constant-shadow solids
by Dan Asimov 16 Dec '10

16 Dec '10

Suppose that a closed bounded convex body X in 3-space, with nonempty interior, is such that the area of its orthogonal projection onto any plane is independent of the plane. Does this imply X is a round 3-ball, or are the other examples? What about for higher dimensions? My hunch is that beginning at some dimension, round balls will be the only examples. --Dan Those who sleep faster get more rest.

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Re: [math-fun] q. re Farey-ish fractions with weighted mediants
by James Propp 16 Dec '10

16 Dec '10

>I don't see where the restriction to odd denominators comes from. --Rich Maybe I missed the drift of the conversation, but I thought the topic was my weighted-mediants version of the Stern-Brocot tree, starting with 0/1 and 1/1. If a/b and c/d have odd denominators, so must (2a+c)/(2b+d) and (a+2c)/(b+2d). (This doesn't apply to the numerators, since we start with 0/1 which has even numerator.) Jim

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[math-fun] Math collaboration over distances
by Dan Asimov 16 Dec '10

16 Dec '10

The biggest thing I miss when trying to collaborate on math research with someone who's elsewhere is the ability for us to see what each other is writing, such as on a blackboard or pad. No doubt this has long since been technnologically possible, but such technology is certainly not widely distributed. Can people suggest reliable low-cost solutions? --Dan Those who sleep faster get more rest.

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Re: [math-fun] Constant-shadow solids
by Adam P. Goucher 16 Dec '10

16 Dec '10

> I don't understand this point in the argument. Why can't the shadow > change shape, in a way other than rotating, that keeps the area > constant? Yes, I misunderstood the original problem, and thought that he was referring to solids where the shadow is a constant *shape*, rather than area. Sincerely, Adam P. Goucher

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Re: [math-fun] q. re Farey-ish fractions with weighted mediants
by James Propp 15 Dec '10

15 Dec '10

>I haven't looked at "Farey-depth". Maybe someone on this list >could work out the initial terms of the two sequences in 2.? Earliest occurrences of odd denominators 3,5,7,...,201: 1,2,2,2,3,3,4,3,3,4,4,4,3,4,4,5,5,5,5,4,4,5,4,5,6, 4,4,6,5,6,5,5,5,5,5,5,5,5,5,4,5,6,7,6,6,6,6,6,5,6, 5,6,6,6,6,5,6,7,6,6,6,6,6,6,6,5,6,5,7,7,6,6,7,7,6, 7,6,6,6,5,5,7,6,6,6,7,7,7,6,6,6,7,7,6,7,7,7,6,7,7 Latest occurrences of odd denominators 3,5,7,...,29: 1,3,3,4,5,6,7,8,9,10,11,12,13,14,15 (The glitch in the second term of the second of these sequences reflects the fact that 2/5 and 3/5 don't show up until the 3rd iteration; whereas for n>2, it appears that the last fraction with denominator 2n+1 to show up is 1/(2n+1), and that this fraction shows up after exactly n iterations.) Here's the Mathematica code I used: Denom[L_, k_] := Module[{M, i}, M = {}; For[i = 1, i <= Length[L], i++, If[Denominator[L[[i]]] == k, M = Append[M, L[[i]]]]]; Return[M]] Earliest[k_] := Module[{i}, For[i = 1, Length[Denom[WF[i], k]] == 0, i++]; Return[i]] Latest[k_] := Module[{i}, For[i = 1, Length[Denom[WF[i], k]] < EulerPhi[k], i++]; Return[i]] Table[Earliest[2 n + 1], {n, 1, 100}] Jim

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[math-fun] BBC4: A Brief History of Mathematics--Podcasts
by Henry Baker 15 Dec '10

15 Dec '10

FYI -- Put this link into iTunes (or other podcast program) itpc://downloads.bbc.co.uk/podcasts/radio4/maths/rss.xml A Brief History of Mathematics Professor Marcus du Sautoy argues that mathematics is the driving force behind modern science. Ten fifteen minute podcasts that reveal the personalities behind the calculations from Newton to the present day. How do these masters of abstraction find a role in the real world? 10 Nicolas Bourbaki Fri, 1 Oct 10 Duration: 15 mins The mathematician that never was. Professor Marcus du Sautoy describes the life and mathematics of an elusive hero. The collected works of Bourbaki represents one of the most ambitious enterprises in mathematical history: an attempt to unify shapes and numbers into single discipline. 09 Hardy and Ramanujan Fri, 1 Oct 10 Duration: 15 mins A mathematical romance. Professor Marcus du Sautoy describes how a passion for prime numbers united a Cambridge professor and an unknown Indian clerk. 08 Henri Poincare Thu, 30 Sep 10 Duration: 15 mins An embarassing error and the mathematics of chaos. Professor Marcus du Sautoy describes how a mistake in Poincare's working led him to an astonishing conclusion: some mathematical problems don't have a reliable solution. 07 Georg Cantor Thu, 30 Sep 10 Duration: 15 mins Infinity. Professor Marcus du Sautoy describes the troubled life of this radical mathematician who shocked his colleagues by proving there's more than one infinity. With contributions from Emeritus Professor of Mathematics Roger Penrose. 06 The Mathematicians who helped Einstein Wed, 29 Sep 10 Duration: 14 mins Seeing in four dimensions. Professor Marcus du Sautoy on the pioneers who pushed mathematics into new dimensions and the strange new geometries they created. Emeritus Professor Roger Penrose confirms that even Einstein sometimes struggled with his maths. 05 Carl Friedrich Gauss Wed, 29 Sep 10 Duration: 14 mins The 19th century mathematical celebrity. Professor Marcus du Sautoy describes how a study of asteroids led Gauss to describe the normal distribution. With contributions from Chairman for the Commission for Racial Equality Trevor Phillips, who believes statistics are the most powerful weapon we have for fighting prejudice. 04 Evariste Galois Tue, 28 Sep 10 Duration: 14 mins Professor Marcus du Sautoy on his favourite mathematician, an angry, young genius who did his best maths in prison and died in a duel, aged 20. 03 Joseph Fourier Tue, 28 Sep 10 Duration: 15 mins Professor Marcus du Sautoy describes the life and mathematics of this Napoleonic soldier, with contributions from musician Brian Eno who loves Fourier's analysis and uses it to create sounds that have never been heard before. 02 Leonard Euler Mon, 27 Sep 10 Duration: 14 mins The man who calculated as other men breathe. Professor Marcus du Sautoy on the mathematical omnivore without whom no history of mathematics is complete. 01 Newton and Leibniz Mon, 27 Sep 10 Duration: 15 mins The battle over the calculus. Professor Marcus du Sautoy reveals how the great hero of British science is rather less gentlemanly than his German rival. An astronaut and investment analyst pay homage to the enormous power of the calculus.

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