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Re: [math-fun] Hexaflexagon paper
by John Golden 27 Mar '18

27 Mar '18

> > I?m sort of a newcomer to hexaflexagons. Any thoughts about what sort of > paper works best? > > (I might just go to a stationer?s and ask for samples of various weights of > paper, assuming they have samples sitting around. But that may not be an > option in the 21st century.) > > Jim Propp I’ve had good luck with K-8 learners with copier weight paper. A little less durable, a little more flexible for turning.

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[math-fun] Hexaflexagon paper
by James Propp 27 Mar '18

27 Mar '18

I’m sort of a newcomer to hexaflexagons. Any thoughts about what sort of paper works best? (I might just go to a stationer’s and ask for samples of various weights of paper, assuming they have samples sitting around. But that may not be an option in the 21st century.) Jim Propp

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[math-fun] Collatz thoughts
by Dan Asimov 25 Mar '18

25 Mar '18

Three observations: 1) The Collatz mapping can be thought of as just going from odd integers to odd integers, via f(x) = 2K+1 for any odd integer x, where 3x+1 = 2^n * (2K+1) for integers n >= 0 and K >= 0. This seems much more natural than dividing by 2 only once per "time step". Then the Collatz conjecture can be phrased as saying that for odd x > 0, iterating f to get f(x), f(f(x)), ... must end up in the (1 ) cycle. 2) No reason we can't look at this for odd x < 0. In that case the cycles I know of are (-1 ), (-5 -7 ), and (-17 -25 -37 -55 -41 -61 -91 ). 3) It seems even more natural to look, more generally, at Collatz on rational numbers of the form x = odd/odd. Let x be such a rational. Then much as in 1) set f(x) = p/q in lowest terms where 3x+1 = 2^n * p/q for n, p, q integers with p and q odd. There seem to be lots of cycles, but they don't seem easy to find. And it's not clear if studying this more general situation will shed any light on the original conjecture. —Dan

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[math-fun] For a dance...
by Bill Gosper 25 Mar '18

25 Mar '18

On 2018-03-24 14:43, françois mendzina essomba2 via math-fun wrote: > Hello... > > I noticed that : > > atan(x)=(205*x^7+2184*x^5+5760*x^3+4096*x)/(36*x^8+820*x^6+4368*x^4+7680*x^2+4096)+sum(S(n),n=1..infinity); > > with > > S(m)=2^(-m-2)*x*sum(1/((2*k-1)^2*2^(-2*m-4)*x^2+1)-1/(k^2*2^(-2*m-2)*x^2+1),k,1,2^(m+1)); > > we deduce from it: > > 1) > > Pi= 2449/850 + 4*sum(S(n),n=1..infinity); > > with > > S(m)=2^(-m-2)*sum(1/((2*k-1)^2*2^(-2*m-4)+1)-1/(k^2*2^(-2*m-2)+1),k,1,2^(m+1)) ; > [. . .] Dr. Essomba: While In[11]:= Pi==2449/850+4*Sum[S@n,{n,\[Infinity]}] In[13]:= S@m_- >2^(-m-2)Sum[1/((2*k-1)^2*2^(-2*m-4)+1)-1/(k^2*2^(-2*m-2)+1),{k,2^(m+1)}] Out[14]= S[m_]->1/8 (2^-m+2 I (2 I \[Pi] Csch[2^(2+m) \[Pi]]+HarmonicNumber[(1-I) 2^(1+m)]-HarmonicNumber[(1+I) 2^(1+m)]-PolyGamma[0,1/2+(1-I) 2^(1+m)]+PolyGamma[0,1/2+(1+I) 2^(1+m)])) In[15]:= %11/.% Out[15]= \[Pi]==2449/850+4 \!\( \*UnderoverscriptBox[\(\[Sum]\), \(n\), \(\[Infinity]\)]\( \*FractionBox[\(1\), \(8\)]\ \(( \*SuperscriptBox[\(2\), \(-n\)] + 2\ I\ \((2\ I\ \[Pi]\ Csch[ \*SuperscriptBox[\(2\), \(2 + n\)]\ \[Pi]] + HarmonicNumber[\((1 - I)\)\ \*SuperscriptBox[\(2\), \(1 + n\)]] - HarmonicNumber[\((1 + I)\)\ \*SuperscriptBox[\(2\), \(1 + n\)]] - PolyGamma[0, \*FractionBox[\(1\), \(2\)] + \((1 - I)\)\ \*SuperscriptBox[\(2\), \(1 + n\)]] + PolyGamma[0, \*FractionBox[\(1\), \(2\)] + \((1 + I)\)\ \*SuperscriptBox[\(2\), \(1 + n\)]])\))\)\)\) In[16]:= N@% Out[16]= True is numerically plausible, I'm afraid that under US law, this makes you a PolyGamist. --rwg

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[math-fun] Khinchin - Collatz connection??
by Bill Gosper 25 Mar '18

25 Mar '18

There are 356033 integers within 41 Collatz (#/2 | (3#+1)/2) steps of 1. 356033/(4/3)^41 = 2.685357138639965 . . . 843774/(4/3)^44 = 2.684860516180538 . . . 63149973/(4/3)^59 = 2.685268450566904 . . . 112268898/(4/3)^61 = 2.685322969193732 But N@Khinchin = 2.685452001065306 (Thank Julian for the 4/3.) Does anyone recall an objection that the usual definition isn't really the expected geometric mean because its derivation assumes consecutive partial quotients are independent? Is there a "CorrectedKhinchin"? https://en.wikipedia.org/wiki/Khinchin%27s_constant and http://mathworld.wolfram.com/KhinchinsConstant.html say flatly No. (My Finch is on loan.) --rwg (EWW: In (8), the ln k belongs in the exponent.) Happy 8th, Gabriel!

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[math-fun] Avoiding Collisions in Outer Space
by Bill Gosper 24 Mar '18

24 Mar '18

Fred, if you are reading the following for the first time, I sorely regret missing out on your eruption of Irish adjectives: https://en.wikipedia.org/wiki/Project_West_Ford https://www.wired.com/2013/08/project-west-ford/ --rwg On 3/25/18, Allan Wechsler <acwacw(a)gmail.com> wrote: I didn't say I was in favor of this plan. I was trying, merely, to interpret the intention. On Sat, Mar 24, 2018 at 8:10 PM, Fred Lunnon <fred.lunnon(a)gmail.com> wrote: You mean, having trashed the planet down here, we're going to start trashing it up there too? Brilliant plan ... WFL On 3/24/18, Allan Wechsler <acwacw(a)gmail.com> wrote: I'm not familiar with the proposal, but I think I can answer that question. The microsatellites will be very cheap, and mass-produced. Ambient conditions will certainly eventually cause them to fail. Using redundant circuits should be able to extend their usable lifetimes, but mostly, when they fail, send up another ten thousand.

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Re: [math-fun] Avoiding Collisions in Outer Space
by Keith F. Lynch 24 Mar '18

24 Mar '18

Henry Baker <hbaker1(a)pipeline.com> wrote: > The highway-in-the-sky approach works more-or-less for satellites in > exactly the same orbit, but keeping them equidistant may require > active maneuvers and hence fuel. Active maneuvers don't necessarily require fuel if the satellites are very light and in a fairly high orbit. Dump a can of Pringles (tm) in a circular orbit a few thousand miles up. Except they aren't potato chips, but computer chips. They're solar powered, and they have patches around their edges which can be turned dark or light so as to use sunlight to maneuver in both position and attitude. These satellite constellations are simultaneously communications satellites, directional antennas, computers, file servers, and cooling fins. They should be able to outcompete all ground-based computer and communications services except those that require low latency. See http://server-sky.com/ All credit goes to Keith Lofstrom, perhaps better known for his proposed launch loop. And now for some shameless name-dropping: http://KeithLynch.net/3keith.jpg is a photo of he and I sitting together on a couch in a hotel in Toronto. The third person is Keith Henson. I'm the only person on that couch who doesn't have a Wikipedia page about him.

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[math-fun] For a dance...
by françois mendzina essomba2 24 Mar '18

24 Mar '18

Hello... I noticed that : atan(x)=(205*x^7+2184*x^5+5760*x^3+4096*x)/(36*x^8+820*x^6+4368*x^4+7680*x^2+4096)+sum(S(n),n=1..infinity); with S(m)=2^(-m-2)*x*sum(1/((2*k-1)^2*2^(-2*m-4)*x^2+1)-1/(k^2*2^(-2*m-2)*x^2+1),k,1,2^(m+1)); we deduce from it: 1) Pi= 2449/850 + 4*sum(S(n),n=1..infinity); with S(m)=2^(-m-2)*sum(1/((2*k-1)^2*2^(-2*m-4)+1)-1/(k^2*2^(-2*m-2)+1),k,1,2^(m+1)) ; 2) Pi=139701*sqrt(3)/91504 + 3*sum(S(n),n=1..infinity); with S(m)=sqrt(3)*2^(-m-2)*sum(1/(3*(2*k-1)^2*2^(-2*m-4)+1)-1/(3*k^2*2^(-2*m-2)+1),k,1,2^(m+1)) ; 3) Pi=169189*sqrt(3)/96824 + 6*sum(S(n),n=1..infinity); with (2^(-m-2)*sum(1/(((2*k-1)^2*2^(-2*m-4))/3+1)-1/((k^2*2^(-2*m-2))/3+1),k,1,2^(m+1)))/sqrt(3);

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Re: [math-fun] Does anyone recall reading (here?) of TSA seriously grilling someone
by Henry Baker 24 Mar '18

24 Mar '18

Adders on a plane? (Of course, if you're in steerage, perhaps only half-adders?) At 10:10 AM 3/24/2018, Seb Perez-D wrote: >Apparently not a T-shirt but still about math in a plane (probably not a Euclidean one) > >Email from Warren Smith 8 May 2016 with content on: >Professor's airplane math leads to flight delay >AP 6:01 p.m. EDT May 7, 2016 > >Cheers > >Seb > >On Sat 24 Mar 2018 at 14:50, Bill Gosper <billgosper(a)gmail.com> wrote: > >> over a math T-shirt they didn't understand? --rwg

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[math-fun] Does anyone recall reading (here?) of TSA seriously grilling someone
by Bill Gosper 24 Mar '18

24 Mar '18

over a math T-shirt they didn't understand? --rwg

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