August 2015 - mathfun@mailman.xmission.com

[math-fun] Dropbox still a pain in the ass...
by Henry Baker 28 Aug '15

28 Aug '15

Downloading this gif still required: 1. Turning on javascript (aka p0wnscript) for dropbox.com 2. Turning on javascript for dropboxstatic.com 3. Turning on javascript for ajax.googleapis.com 4. And then going through a dialog to say that I didn't want to sign up for Dropbox. 5. And then finally clicking on a "download" button. Remember, you need "?dl=1" instead of "?dl=0" at the end of the URL below in order to get the damn thing to download immediately. At 05:31 PM 8/27/2015, Fred Lunnon wrote: >And Fred Lunnon seizes the opportunity to wheel out his animation >of the seven-link robot / rigid kaleidocycle at > https://www.dropbox.com/s/aeo6rxtc5j4p291/sevenring.gif?dl=0

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Re: [math-fun] simple rocket engine?
by Henry Baker 28 Aug '15

28 Aug '15

This is very similar to a proposal to bounce ground-originated photons off a mirror on the spacecraft. This solves the problems: 1. You don't have to carry mass that has to be thrown overboard -- no matter how energetic you eject it. 2. You don't have to carry the energy, either. 3. You're ejecting stuff at the speed of light, so you maximize the momentum transfer. Downside: you have to send a LOT of photons in any normal energy range. If you try to send photons with too much energy, they simply go right through the "mirror". Look at the effort required to "focus" an X-ray telescope: https://en.wikipedia.org/wiki/NuStar At 04:41 PM 8/27/2015, Warren D Smith wrote: >Say you're a spacecraft in space. You've got a source of lots of energy (solar panels). >You have limited mass, and even more limited mass that you are willing to throw away. >Also, you care mucho about reliability and simplicity. How do you go places? > >Here's a very simple idea. >Focus a laser beam pulse onto a surface. Blam. Get an extremely hot spot on >the surface which vaporizes, ejecting gas, thrusting spacecraft the other way. >The surface can then be melted to re-form it (or it perhaps was already a liquid); >then do it again. > >Variants: >1.the "surface" actually is a liquid drop detached from any solid and >blasted just as it moves into the focal spot; this lies inside a >rocket-type chamber+nozzle. > >2.The focal spot is situated inside a magnetic field pointed in the thrust direction. >Hot plasma then will fly off in field-line directions, but not in other directions. > >3. Rotating interruptor/shields in front of the laser could protect it from being hit by >hot ejecta coming back at it a little after it pulses. > >What is the point of all this? Well, >A. Simple. >B. Can be done on a very small scale. >C. The exhaust velocity, governed by the temperature of the laser-heated stuff, >it seems to me could be made extremely high, perhaps exceeding any currently-used >spacedrive.

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[math-fun] Fun, and argubly math
by David Wilson 28 Aug '15

28 Aug '15

https://www.youtube.com/watch?v=umlBrQoG6xk

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[math-fun] Baez on kaleidocycles
by Fred Lunnon 28 Aug '15

28 Aug '15

John Baez on Google+ discusses kaleidocycles at https://plus.google.com/u/0/117663015413546257905/posts/Y5nroEHTVvy?cfem=1 And Fred Lunnon seizes the opportunity to wheel out his animation of the seven-link robot / rigid kaleidocycle at https://www.dropbox.com/s/aeo6rxtc5j4p291/sevenring.gif?dl=0 WFL

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[math-fun] simple rocket engine?
by Warren D Smith 27 Aug '15

27 Aug '15

Say you're a spacecraft in space. You've got a source of lots of energy (solar panels). You have limited mass, and even more limited mass that you are willing to throw away. Also, you care mucho about reliability and simplicity. How do you go places? Here's a very simple idea. Focus a laser beam pulse onto a surface. Blam. Get an extremely hot spot on the surface which vaporizes, ejecting gas, thrusting spacecraft the other way. The surface can then be melted to re-form it (or it perhaps was already a liquid); then do it again. Variants: 1.the "surface" actually is a liquid drop detached from any solid and blasted just as it moves into the focal spot; this lies inside a rocket-type chamber+nozzle. 2.The focal spot is situated inside a magnetic field pointed in the thrust direction. Hot plasma then will fly off in field-line directions, but not in other directions. 3. Rotating interruptor/shields in front of the laser could protect it from being hit by hot ejecta coming back at it a little after it pulses. What is the point of all this? Well, A. Simple. B. Can be done on a very small scale. C. The exhaust velocity, governed by the temperature of the laser-heated stuff, it seems to me could be made extremely high, perhaps exceeding any currently-used spacedrive.

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[math-fun] The wandering sun --- gear-train polynomials
by Fred Lunnon 27 Aug '15

27 Aug '15

I have been developing a procedure for computing the polynomial associated with a given eccentric planetary gear train. In outline, it generates sunsets by solving inverse trigonometric equations of form belt-length = integer , numerically to high precision; then deduces an integer relation between successive powers of some chosen sunset, via LLL lattice basis reduction. The process is not currently fully automated: the user remains obliged to estimate suitable precision and polynomial degree, and to select a sunset (often but not always the minimum) not subsumed by some smaller train. As an example, consider the (symmetric) train [r, s; p, t, q] = [28, -12; 12, 8, 4] = 4*[7, -3; 3, 2, 1] . Computing possible sunsets h , then processing h -> h^2 * 2/(r-s) = h^2/20 (to reduce eventual polynomial degree and coefficient size) yields values to 6 decimal digits [0.164185, 0.177674, 0.203751, 0.250000, 0.334135, 0.504112, 0.937348, 4.00000] . (Maximum value 4 represents sun and ring touching externally: an overlapping, numerically pathological case best excluded!) Because this train is (concentric with a multiple of) 4x a smaller train, alternate values merely yield polynomials associated with 2x or 1x multiples. But applying LLL to powers of (say) the first value yields polynomial 400*X^6 + 768*X^5 + 22425*X^4 - 38960*X^3 + 19680*X^2 - 3840*X + 256 , with just 4 positive real roots, corresponding to the 4 remaining values. Its coefficients are highly composite: [2^4 5^2, 2^8 3, 3 5^2 13 23, -2^4 5 487, 2^5 3 5 41, -2^8 3 5, 2^8] . Substituting all 8 values into the polynomial yields syndrome [2.5 E-179, -1.0, -3.6 E-179, 5.7, -1.3 E-179, -1.9 E2, 2.5 E-177, 6.0 E6] . Working single precision was 180 digits, double that for solving equations and accumulating inner products; degree assumed was m = 12 ; polynomial factors of degrees 1,5 were dropped from the raw result. A remarkable feature: the smaller multiple [21, -9; 9, 6, 3] = 3*[7, -3; 3, 2, 1] yields processed values h^2/15 = [0.167544, 0.193318, 0.250000, 0.377104, 0.736875, 4.00000] (4 relevant), and polynomial 400*X^6 + 3684*X^5 + 22425*X^4 - 38960*X^3 + 19680*X^2 - 3840*X + 256 --- identical to that above except for a single coefficient! QUERY: Given dimension (ie. presumed polynomial degree) m , what precision is necessary to ensure successful lattice basis reduction? Fred Lunnon

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[math-fun] Space flight simulation program/game
by Henry Baker 27 Aug '15

27 Aug '15

FYI -- I suspect that the NASA/Musk and popular interest will force KSP to implement a full n-body simulation for orbital mechanics for a 2.0 product. 3.0 should include GR/"Interstellar" - type physics engine; perhaps PC's in a few years will have the horsepower to run such simulations in real time. >From what I can tell, this already looks to be a very cool game -- much more fun than *farming* or *building cities*. https://en.wikipedia.org/wiki/Kerbal_Space_Program Kerbal Space Program (commonly abbreviated as KSP) is a space flight simulator developed by Squad for Linux, OS X, Windows, Wii U, PlayStation 4, and Xbox One. In the game, players control a space program, build and fly spacecraft under physics simulation, and explore celestial bodies with characters called kerbals. The game was officially released out of beta on April 27, 2015. ... Notable members of the space industry have taken an interest in the game, including *NASA* and *Elon Musk of SpaceX*. While the game is not a perfect simulation of reality, *it has been praised for its representation of orbital mechanics*. Every object in the game except the celestial bodies themselves are under the control of a Newtonian dynamics simulation. Rocket thrust is applied accurately to a vehicle's frame based on the positions in which the force-generating elements are mounted. The strength of the joints connecting parts together is finite and vehicles can be torn apart by excessive or inappropriately directed forces. The game simulates trajectories and orbits using patched conic approximation instead of a full n-body simulation, and thus does not support Lagrange points, perturbations, Lissajous orbits, halo orbits and tidal forces. According to the developers, full n-body physics would require the entire physics engine to be rewritten.[22]

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[math-fun] Stephen Hawking: Enter, but leave your info at the door
by Warren D Smith 27 Aug '15

27 Aug '15

Is there actually a paper one can read explaining the math?

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[math-fun] The UUDDUUD sequence, or the 3/7th sequence
by Ed Pegg Jr 27 Aug '15

27 Aug '15

Start with 4. Progressive add or subtract the integers, repeating the sequence + + - - + + -. 4, 5, 7, 4, 0, 5, 11, 4, 12, 21, 11, 0, 12, 25, 11, 26, 42, 25, 7, 26, 46, 25, 47, 70, 46, 21, 47, 74, 46, 75, 105, 74, 42, 75, 109, 74, 110, 147, 109, 70, 110, 151, 109, 152, 196, 151, 105, 152, 200, 151, 201, 252, 200, 147, 201, 256, 200, 257, 315, 256, 196, 257, 319, 256, 320, 385, 319, 252, 320, 389, 319, 390, 462, 389, 315, 390, 466, 389, 467, 546, 466, 385, 467, 550, 466, 551, 637, 550, 462, 551, 641, 550, 642, 735, 641, 546, 642, 739, 641, 740, 840, 739, 637, 740, 844, 739, 845, 952, 844, 735, 845, 956, 844, 957, 1071, 956, 840, 957, 1075, 956, 1076, 1197, 1075, 952, 1076, 1201, 1075, 1202, 1330, 1201, 1071, 1202, 1334, 1201, 1335, 1470, 1334, 1197, 1335, 1474, 1334 FoldList[Plus,4,Flatten[Table[{1,1,-1,-1,1,1,-1}, {20}]] Range[140]] The differences are 1, 2, -3, -4, 5, 6, -7, 8, 9, -10, -11, 12, 13, -14, 15, 16, -17, -18, 19, 20, -21, 22, 23, -24, -25, 26, 27, -28, 29, 30, -31, -32, 33, 34, -35 Notice how all the numbers in the sequence repeat themselves two or three times. Only 3/7th of the integers appear this sequence, namely 7 times the triangular numbers plus (0,4,5) or (5,4,0). The numbers can be placed in a grid as follows. The start is 4, 5, 7, 4, 0, 5 -- that last move wouldn't be a king move, but from then on all moves are king moves. 0 . 12 . 21 . 47 . 70 . 110 . 147 . 201 . 252 . 320 4 . 11 . 25 . 46 . 74 . 109 . 151 . 200 . 256 . 319 5 . 07 . 26 . 42 . 75 . 105 . 152 . 196 . 257 . 315 Grid[Transpose[Table[7 n (n - 1)/2 + If[OddQ[n], {0, 4, 5}, {5, 4, 0}], {n, 1, 10}]]] Ed Pegg Jr

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Re: [math-fun] If you're looking for milepost 420, you won't find it in Idaho.
by Dave Dyer 26 Aug '15

26 Aug '15

At 09:44 AM 8/21/2015, Tom Rokicki wrote: >> It's also Hitler's birthday (b. 1889). Happy birthday, Adolph. > >Dude! Don't harsh our mellow. Not so much, it's also my mom's birthday. Back in the day she used to get teased. BTW, I always thought the origin was CSNY "four and twenty years ago..."

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