July 2018 - mathfun@mailman.xmission.com

[math-fun] 4-5-6
by Bill Gosper 14 Jul '18

14 Jul '18

This is probably old news, but triangles with two angles in ratio n seem always to have their shortest side a perfect nth power: Sides -> Angle ratios 4 | 5 | 6 -> {0.673407904146284, 2, 0.7424920273751127}, 40 | 39 | 25 -> {1.882031453653618, 1/2, 1.062681495634607}, 48 | 35 | 27 -> {1.363957995353242, 1/3, 2.199481223190493}, 70 | 51 | 49 -> {1.052656745727889, 1/2, 1.899954575047201}, 77 | 72 | 49 -> {1.710412883254859, 1/2, 1.169308311215516}, 117 | 88 | 81 -> {1.113181718197647, 1/2, 1.796651855941546}, 126 | 115 | 81 -> {1.62220636751334, 1/2, 1.232888761906277}, 132 | 119 | 64 -> {2.216537678860817, 1/3, 1.353462216596219}, 176 | 135 | 121 -> {1.153038717449732, 1/2, 1.734547131620664}, 187 | 168 | 121 -> {1.56846197132432, 1/2, 1.275134518123709}, 204 | 145 | 144 -> {1.008861547409698, 1/2, 1.982432579708385}, 228 | 217 | 144 -> {1.779802212058679, 1/2, 1.123720369853131}, 247 | 192 | 169 -> {1.181274044306326, 1/2, 1.693087230384759}, 260 | 231 | 169 -> {1.532276923765853, 1/2, 1.305247092728272}, 273 | 272 | 169 -> {1.982157307044563, 1/2, 1.009001653346091}, 280 | 279 | 125 -> {2.965423277339246, 1/3, 1.011659961977428}, 330 | 259 | 225 -> {1.202325336238882, 1/2, 1.663443279217926}, 345 | 304 | 225 -> {1.506251839762349, 1/2, 1.327799208076356}, 368 | 273 | 256 -> {1.086393099567274, 1/2, 1.840954255689428}, 400 | 369 | 256 -> {1.660214339091134, 1/2, 1.204663731006491}, 425 | 336 | 289 -> {1.21862553465025, 1/2, 1.641193248567542}, 442 | 387 | 289 -> {1.486633544904333, 1/2, 1.345321452522924}, 459 | 440 | 289 -> {1.809233016517494, 1/2, 1.105440803777561}, 476 | 305 | 256 -> {1.216537678860817, 1/4, 3.288019820105906} In[234]:= Times @@ {0.673407904146284`, 2, 0.7424920273751127`} Out[234]= 1. . . . In[233]:= Times @@ {1.2165376788608173`, 1/4, 3.288019820105906`} Out[233]= 1. These 2, 1/2, 1/3, etc, were Rationalize'd after FullSimplify failed utterly on all of them. --rwg

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[math-fun] How to make Bill Gosper fall off his chair
by Bill Gosper 14 Jul '18

14 Jul '18

Fortunately, I was already lying down. Shouldn't you be able to pack three copies of https://jjj.de/tmp-math-fun/ec3464-curve-tile-BBB-decomp.pdf around its lower left corner, erase the black components, and thereby get a Kochflake from six as well as three of those shapes? Wait, that doesn't add up. You need to put those black things back somehow. --rwg

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[math-fun] Rohan notices that integer 60º triangles come in pairs:
by Bill Gosper 13 Jul '18

13 Jul '18

On 2018-07-12 05:57, Fred Lunnon wrote: > "Two sides and a non-included angle" --- the one case when a triangle > is not uniquely specified by three of its six angles and sides. > > Knew that would prove useful some day ... > > WFL The pons ASSinorum. > > > > On 7/12/18, Allan Wechsler <acwacw(a)gmail.com> wrote: >> This makes perfect sense. Notice that if you take the sum of the varying >> values, you get the longest side: 3+5 = 8, 7+8 = 15, 5+16 = 21. >> >> In other words, the "average length" of the varying side is half the long >> side. >> >> This all becomes completely clear if you imagine putting a compass at the >> vertex between the two constant sides, and opening it to the length of the >> "hypotenuse" (which is, counterintuitively, the side of median length, not >> the longest side, which is a leg). When you sweep the compass, it cuts the >> opposite side in two places, producing the two lengths of the variable >> side. >> >> The Pythagoroid Formula for the sixty-degree case is A^2 + B^2 = C^2 + AB. >> So, 64 + 9 = 49 + 24 for the first triple, and 64 + 25 = 49 + 40 for the >> second. But no parametric solution, like with true Pythagorean? --rwg >> >> On Thu, Jul 12, 2018 at 7:40 AM, Bill Gosper <billgosper(a)gmail.com> wrote: >> >>> {7 | 8 | 3, 7 | 8 | 5, 13 | 15 | 7, 13 | 15 | 8, 19 | 21 | 5, 19 | 21 | >>> 16, >>> 31 | 35 | 11, 31 | 35 | 24, . . ., >>> 859 | 989 | 429, 859 | 989 | 560, 889 | 999 | 295, 889 | 999 | 704,. . . >>> Sometimes very close pairs: >>> 181 | 209 | 104, 181 | 209 | 105, . . ., 2521 | 2911 | 1455, 2521 | 2911 >>> | >>> 1456, . . . >>> --rwg

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[math-fun] "optimal control" theory question
by Henry Baker 13 Jul '18

13 Jul '18

I seem to recall someone from NASA telling me about some famous theorem of control theory that goes something like this: Suppose you have a system with multiple degrees of freedom, e.g., x,y,z in 3-space, and you have sensors capable of sensing the coordinate positions, but the *precisions of the different dimensions are different*. I.e., you might be able to get x,y positions to within one meter, but the z position might have a precision only to 5 meters. The classical optimal control algorithms will tend to operate in such a way that *most of the position uncertainty will be *forced* into the most imprecise coordinate. Example: the usual GPS system is more accurate in the lat&lon dimensions than it is in the *elevation* dimension. So you construct an optimal control system to fly an airplane using GPS. The system operates beautifully to get your plane to land on the correct runway, but unfortunately, it nearly always crashes the plane due to the altitude uncertainty. In other words, the optimal control strategy is indeed "tail-stuffing", whereby small uncertainties are moved into degrees of freedom which are less observable, and hence will be magnified to the point where they are. Thus, a banker who can manipulate the location of credit risk can manipulate it into degrees of freedom that are less observable, and furthermore, any optimal algorithm would do precisely the same.

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Re: [math-fun] Skripals' anagram
by Dan Asimov 13 Jul '18

13 Jul '18

SKRIPALS' anagram: Avian constants: LARKS' PIS Places Noah moors the boat: ARK SLIPS Just the things for photographing the small mammals: PIKA SLRS Very handy for those times you need to file down your necktie: SILK RASP. —And

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[math-fun] Rohan notices that integer 60º triangles come in pairs:
by Bill Gosper 12 Jul '18

12 Jul '18

{7 | 8 | 3, 7 | 8 | 5, 13 | 15 | 7, 13 | 15 | 8, 19 | 21 | 5, 19 | 21 | 16, 31 | 35 | 11, 31 | 35 | 24, . . ., 859 | 989 | 429, 859 | 989 | 560, 889 | 999 | 295, 889 | 999 | 704,. . . Sometimes very close pairs: 181 | 209 | 104, 181 | 209 | 105, . . ., 2521 | 2911 | 1455, 2521 | 2911 | 1456, . . . --rwg

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[math-fun] How to make Bill Gosper fall off his chair
by Joerg Arndt 12 Jul '18

12 Jul '18

Enjoy: https://jjj.de/tmp-math-fun/ This one is on the (3.4.6.4)-grid (aka, apparently: "Rhombitrihexagonal tiling", see https://en.wikipedia.org/wiki/Rhombitrihexagonal_tiling ) Best regards, jj

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Re: [math-fun] "Stochastic" integration (?)
by Dan Asimov 11 Jul '18

11 Jul '18

I had almost never thought about stochastic integrals, and the idea of a probabilistic vector field is cool. ----- the speed at one moment is NOT independent of the speed at the next moment ----- I'm not sure if I'm interpreting this right. One way might be (let's say for curves {A(t)} in the complex plane with A(0) = 0, A'(0) = 1: Then something like Brownian motion defines a measure mu_t, for each t > 0, on all possible continuous paths p : [0, t] —> C so that certain probabilistic consistency equations are satisfied. This could lead to really interesting possibilities. —Dan

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[math-fun] Skripals' anagram
by Bill Gosper 11 Jul '18

11 Jul '18

"COUNTERSPIES" ↔︎ "PERSECUTIONS" --rwg

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[math-fun] "Stochastic" integration (?)
by Henry Baker 11 Jul '18

11 Jul '18

I find myself driving quite a bit on freeways, and I'm curious about how my car GPS can calculate a reasonable ETA (Estimated Time of Arrival) for me. I learned calculus in high school & college, but I don't recall learning about how to integrate a *probabilistic* variable -- i.e., a variable which is subject to some probability distribution. So here's the thing: suppose I'm driving at a speed whose values come from some distribution, and I want to integrate this speed over some amount of time to compute another probability distribution about how much distance is being covered within that time, or to estimate a distribution of my arrival time at a particular destination. (Let's assume that time itself moves precisely, but the car speed varies.) Now an integral is merely a fancy *sum*, and we know how to estimate the sum of *independent* random variables. But in the case of my car, the speed at one moment is NOT independent of the speed at the next moment! The acceleration, maybe, but not the speed. More likely, even the acceleration isn't independent from moment to moment, but the "jerk" (dacceleration/dt) might very well be. So now we have an even bigger problem: we need to go back an integrate the jerk(t) for each moment of time t. But wait, there's more! If I'm using a program like Waze, which has a probability distribution for the speed on each segment of each highway, then I should be somehow computing an overall distribution for my trip time from some sort of integration of all of these little contributions. If these individual distributions were independent, then presumably my result would be some sort of convolution of all of the little distributions. But the distributions aren't independent. (Even worse: Waze knows that the speed distributions are time-dependent -- e.g., based upon time of day. But I don't even know how to do the simpler problem.) So what now?

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