February 2013 - mathfun@mailman.xmission.com

[math-fun] telescoping triggish products
by Bill Gosper 02 Mar '13

02 Mar '13

Decades ago in an MIT talk, I gave a method to enumerate a family of telescoping products akin to Product[(1/2)*(1 + x^2^(-n)), {n, Infinity}] == (-1 + x)/Log[x] . (Which telescopes because you can shift the index by changing the x variable.) Recently, Mourad Ismail sent a paper reminding me to carry out the enumeration, on which Neil and I started yesterday. Neil found Product[(1/2)*(z^(-2)^(-k) + 1), {k, 1, Infinity}] == (z - 1)/(z^(2/3)*Log[z]) and fears it to be centuries old, after getting severely scooped on a continued fraction identity last week. I found Product[-(-1)^(2/3) + (-1)^(1/3)*z^(-2)^(-k), {k, Infinity}] == (-1 + (-1)^(2/3)*z)/((-1 + (-1)^(2/3))*z^(2/3)) which looks oscillatory and nonconvergent, but is really just equivalent to Product[2*Sin[Pi/6 + (-1/2)^k*t], {k, Infinity}]==(2*Cos[Pi/6 + t])/Sqrt[3] from http://www.tweedledum.com/rwg/idents.htm . (Nice(?) exercise.) Two of the above identities come from the scheme (a*x^4 + b*x^2 + c)/(c*x^2 + b*x + a) reducing to a polynomial, which Reduce promises contains three more. And there are other schemes... --rwg

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[math-fun] this HAS to be known
by Bill Gosper 01 Mar '13

01 Mar '13

Sum[2^k/(1 + z^2^k), {k, -Infinity, Infinity}] == 1/Log[z] Can somebody tell me where? --rwg And, as Neil empiricizes, Sum[2^k/(1 + z^2^k), {k, 0, Infinity}] == 1/(z-1)

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[math-fun] Wolfram Alpha large factorial bug
by Bill Gosper 01 Mar '13

01 Mar '13

MRob> I discovered a calculation error problem in Wolfram Alpha. When given "20000!" (the factorial of 20000) it gives the answer in two forms: * a "Decimal approximation", 1.8192063202303451348276... x 10^77337 * a "Power of 10 representation", 10^(10^4.888364627124846). My own calculations (using the Stirling series for the log Gamma function) indicate that the first answer is correct. One can see that the second answer disagrees with the first, by computing log(77337+log(1.81920632023)) using base-10 logarithms. The exponent should be 4.888388... not 4.888364... For "LogGamma[20001]", Wolfram Alpha gives 178075.621737198700312867928177..., which agrees with its first 20000! answer and with my calculations. I also checked this in GP/PARI, which of course gives all the digits of 20000!, but you can multiply by 1.0 to put it in the more familiar scientific notation, and it agrees with Wolfram Alpha's first answer. I'd be curious to find out what Mathematica gives (I suspect it will be the correct first answer). I sent feedback to WoldramAlpha to report the problem. -- Robert Munafo -- mrob.com Follow me at: gplus.to/mrob - fb.com/mrob27 - twitter.com/mrob_27 -mrob27.wordpress.com - youtube.com/user/mrob143 - rilybot.blogspot.com In[487]:= N[20000!, 22] Out[487]= 1.819206320230345134828*10^77337 In[488]:= Log[10, Log[10, %]] Out[488]= 4.888388780625234145206724465 In[489]:= N[Log[10, Log[10, 20000!]], 23] Out[489]= 4.8883887806252341452067 --rwg

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[math-fun] "printing" 3D fractals
by Henry Baker 01 Mar '13

01 Mar '13

I've seen some examples of 3D fractals "printed" by these new 3D printers. Is anyone making CAD software for doing this sort of thing -- i.e., designing parts that incorporate 3D fractals as part of their structure?

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[math-fun] Turmite Busy Beavers
by ed pegg 28 Feb '13

28 Feb '13

I've been working on Turmites, or 2-D Turing machines. What Turmite runs longest before becoming predictable? Various people have been building up the following page: https://code.google.com/p/ruletablerepository/wiki/EdPeggsBusyBeaverTurmite… For RULS (Right, U-turn, Left, Straight), the Langton's Ant series can considered as RL -- 9,977 steps until highway RUL -- 2,669,000 steps until highway RUUL -- 1,589,104,000 steps until highway RUUUL -- 217,782,000,000 steps until highway I haven't yet figured out how order emerges out of chaos in this series. I could probably run RUUUUL in a month on my current machine, but I'm hoping someone will figure out something more clever. I have no idea how to approach making bounds for the number of distinct, unpredictable-for-awhile machines of type 2c3s, 3c2s, or other combinations. If anyone has insights, anecdotes, asides, ideas, or new records, feel free to make comments on the pages. --Ed Pegg Jr

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Re: [math-fun] Riddle
by Henry Baker 28 Feb '13

28 Feb '13

Very cool! Thanks! Of course, quantum theorists know that even "vacuum" isn't really empty... And even when it _is_ empty, weird things continue to happen: http://en.wikipedia.org/wiki/Aharonov%E2%80%93Bohm_effect At 05:57 AM 2/28/2013, Charles Greathouse wrote: >On the optimist vs. pessimist water glass, I like >http://what-if.xkcd.com/6/ > >Charles Greathouse >Analyst/Programmer >Case Western Reserve University > >On Wed, Feb 27, 2013 at 4:13 PM, David Makin <makinmagic(a)tiscali.co.uk>wrote: > >> To a realist - the glass is always full (of something). >> >> To an opportunist - the glass is a free drink while the rest argue. >> >> On 26 Feb 2013, at 23:06, Tom Karzes wrote: >> >> > I always liked this one: >> > >> > To a pessimist, the glass is half empty. >> > To an optimist, the glass is half full. >> > To an engineer, the glass is twice as big as it needs to be. >> > >> > Tom

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Re: [math-fun] nice little sliding block head exercise
by Bill Gosper 28 Feb '13

28 Feb '13

mrob>I simply rotated the whole board and got the S tetromino to the lower-right in 1 "tort". Heck no. "Torts" apply to individual pieces. You are guilty of a "crass action". --rwg On Fri, Feb 22, 2013 at 6:09 PM, Bill Gosper <billgosper(a)gmail.com> wrote: Scott H.> If moving 1 piece pushes/moves N-1 other pieces, does that count as 1 move or N moves? N. As Neil explained it to me, that's one "action". A "move" is one piece along any path. A "step" is one piece moving one square. --rwg Your sneaky moderator used rotation and claims six moves vs nine. This is so unthinkably illegal that I doubt there is even a term for such a maneuver. A "deviation"? A "tort"? "The solution requires five moves and a malfeasance."-- Robert Munafo -- mrob.com Follow me at: gplus.to/mrob - fb.com/mrob27 - twitter.com/mrob_27 -mrob27.wordpress.com - youtube.com/user/mrob143 - rilybot.blogspot.com

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Re: [math-fun] Riddle
by Guy Haworth 28 Feb '13

28 Feb '13

A mathematician, a physicist and an engineer take a train from London to Yorkshire. As they cross the county boundary, they see a field of black sheep. The engineer says "All the sheep in Yorkshire are black." The physicist says "You cannot say that. All you know is that all the sheep in this field are black." The mathematician says "No! All you can say is that at least one side of every sheep in that field is black." Guy

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[math-fun] Wolfram Alpha large factorial bug
by Robert Munafo 28 Feb '13

28 Feb '13

I discovered a calculation error problem in Wolfram Alpha. When given "20000!" (the factorial of 20000) it gives the answer in two forms: * a "Decimal approximation", 1.8192063202303451348276... x 10^77337 * a "Power of 10 representation", 10^(10^4.888364627124846). My own calculations (using the Stirling series for the log Gamma function) indicate that the first answer is correct. One can see that the second answer disagrees with the first, by computing log(77337+log(1.81920632023)) using base-10 logarithms. The exponent should be 4.888388... not 4.888364... For "LogGamma[20001]", Wolfram Alpha gives 178075.621737198700312867928177..., which agrees with its first 20000! answer and with my calculations. I also checked this in GP/PARI, which of course gives all the digits of 20000!, but you can multiply by 1.0 to put it in the more familiar scientific notation, and it agrees with Wolfram Alpha's first answer. I'd be curious to find out what Mathematica gives (I suspect it will be the correct first answer). I sent feedback to WoldramAlpha to report the problem. -- Robert Munafo -- mrob.com Follow me at: gplus.to/mrob - fb.com/mrob27 - twitter.com/mrob_27 - mrob27.wordpress.com - youtube.com/user/mrob143 - rilybot.blogspot.com

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[math-fun] junk DNA is junk
by Warren Smith 26 Feb '13

26 Feb '13

Well, looks like math-fun beat the scientific community to it, but now they are coming out in the popular press like infuriated hornets attacking the claims (and claimers) that junk DNA is not junk, see http://www.guardian.co.uk/science/2013/feb/24/scientists-attacked-over-junk…

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