October 2013 - mathfun@mailman.xmission.com

Re: [math-fun] math novels?
by Keith F. Lynch 12 Oct '13

12 Oct '13

Carl Sagan's novel _Contact_ is mostly about communications with extraterrestrials. But a far more interesting subplot is about a very different sort of communication. The aliens point out to mankind that if you calculate pi in base eleven, near the beginning there's a square number of consecutive digits that are all 0 or 1, and when plotted as a square they give a picture of a circle. A little further into the number there are reams of meaningful binary data. Before you complain hat nobody has write access to pi, note that there's no claim that pi ever *changed* value. Has anyone actually made a search in various bases that would find such things? Of course they might be in 2*pi or 1/pi or even (pi^2)/6. This subplot was left out of the movie, unfortunately. I can think of lots of interesting math short stories. Most interesting to me are "Luminous" and its sequel "Dark Integers" by Greg Egan. The profoundly anti-Platonic premise of these stories is that no mathematical theorem is true or false until it makes a difference to some physical system. (In some cases, that system may be the brain of a mathematician.) Fermat's Last Theorem may have been neither true nor false until Wiles proved it. This sounds unfalsifiable until a character points out that information can't travel faster than light, so if a theorem is proven true in one place it can still be proven false in another if no signal could have reached the latter from the former. This is by analogy with the mainstream physics idea that the values of the fundamental physical constants may be different in different places, with defects in space-time between these incompatible regions. So there could be inconsistencies is math, locked in since the first few nanoseconds after the Big Bang. The characters carry out a multi-year distributed search for inconsistencies is math, something like SETI@Home only even more controversial. They find some. And when they explore the boundary, they find they can move it by repeatedly proving "nearby" theorems on one side or the other of the border. All this is backstory. As the story starts, one of our two protagonists has been caught by bad guys who want to know where the defects are so they can move them to make lots of money. The bad guys don't care that the side effects of such tampering could be catastrophic. The story becomes like a James Bond movie in that the bad guys are using very advanced technology to try to extract the hidden information, but the good guy is using even more advanced technology to protect it, plus there's plenty of one-on-one violence, though nobody is actually killed. The good guys decide the only way to keep math safe is to destroy the alternate math. To do this they rent time on a Chinese supercomputer that's more powerful than all other computers in the world put together. This destruction goes well at first, but soon it starts to push back, as if someone was using an even more powerful computer to oppose their efforts. But who, and where? I can't say more -- or anything at all about the sequel -- without spoiling the story. For humorous math fiction, I recommend the old collections, _Fantasia Mathematica_ and _The Mathematical Magpie_, both edited by Clifton Fadiman. They contain math-related short stories, cartoons, poems, etc., including fiction by Martin Gardner.

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[math-fun] Are we simple, yet?
by Bill Gosper 11 Oct '13

11 Oct '13

I mentioned f(nth convergent(√3)) = 1/56 (-40 + 2^(3 n/2) (13 + 12 Sqrt[2] + (-1)^n (13 - 12 Sqrt[2]))) where f is Newman's rational counter. Why didn't I say f(1 + (-2 (1 - Sqrt[3])^(-1 + n) + 2 (1 + Sqrt[3])^(-1 + n))/(-(1 - Sqrt[3])^n + (1 + Sqrt[3])^n)) ? Because instead of this formula, I got the puzzling expression In[83]:= FullSimplify[ FindSequenceFunction[Convergents[Sqrt[3], 22]]@n, n \[Element] Integers] Out[83]= (2 Sqrt[3] (-2 - Sqrt[3])^n QPochhammer[2 - Sqrt[3], -2 + Sqrt[3], n])/ ((-1 + (-2 - Sqrt[3])^n) QPochhammer[-1, -2 + Sqrt[3], n]) (Strangely, the supposedly equivalent FullSimplify[FindSequenceFunction[Convergents[Sqrt[3], 22], n], n \[Element] Integers] simply gives up.) OK, let's cut it a break and do the numerators and denoms separately: In[84]:= FullSimplify[ FindSequenceFunction[Numerator /@ #, n]/ FindSequenceFunction[Denominator /@ #, n] &@ Convergents[Sqrt[3], 22], n \[Element] Integers] Out[84]= (3 (2 + Sqrt[ 3]) ((2 - Sqrt[3])^( n/2) (724 Sqrt[2 + Sqrt[3]] + 418 Sqrt[ 3 (2 + Sqrt[3])] + (989 + 571 Sqrt[3]) (-2 + (-1)^n (2 + Sqrt[2]))) + (2 + Sqrt[3])^( n/2) (1978 + 1142 Sqrt[3] + 724 Sqrt[2 + Sqrt[3]] + 418 Sqrt[3 (2 + Sqrt[3])] - (-1)^ n Root[16 + 31648 #1 + 7824944 #1^2 - 7912 #1^3 + #1^4 &, 3])))/((2 + Sqrt[3])^( n/2) ((2 + Sqrt[2]) (6393 + 3691 Sqrt[3]) + (-1)^ n Root[144 + 613728 #1 + 326963664 #1^2 + 51144 #1^3 + #1^4 &, 2]) - 4 I^n (2 - Sqrt[3])^( n/2) (Cos[(n \[Pi])/2] Root[9 - 81740892 #1^2 + #1^4 &, 4] + I (6393 + 3691 Sqrt[3]) Sin[(n \[Pi])/2])) Ingratitude. BtW, In[87]:= Convergents[1/(Sqrt[3] - 1), 9] Out[87]= {1, 3/2, 4/3, 11/8, 15/11, 41/30, 56/41, 153/112, 209/153} I didn't know that In[88]:= NestList[1 + 1/2/# &, 1, 8] Out[88]= {1, 3/2, 4/3, 11/8, 15/11, 41/30, 56/41, 153/112, 209/153} I'm a little surprised that neither this formula nor the first one above requires any odd-even machinery, since the continued fraction of √3 has period 2: 1 1 2 1 2 1 2 ... --rwg

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Re: [math-fun] Fractals are now recondite?
by Bill Gosper 11 Oct '13

11 Oct '13

Mike Stay>That's great! May I suggest that when the area is smallest, you should give those three regions the labels 1, 2, and 3? Thanks! How's http://gosper.org/flaky.gif ? --rwg On Fri, Oct 4, 2013 at 2:41 AM, Bill Gosper <billgosper(a)gmail.com> wrote: Funster Gary Antonick (NYT NumberPlay) fears that the Franceflake fractal boundary paradox might be too hairy for his readers, even though Martin Gardner explained it in his original Flowsnake piece. So I made an mgif (gosper.org/flake.gif) illustrating the problem, and offering a URL for an (indirect) explanation. --rwg _______________________________________________ -- Mike Stay - metaweta@gmail.comhttp://www.cs.auckland.ac.nz/~mikehttp://reperiendi.wordpress.com _____________________________________________

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[math-fun] what is: supersymmetry?
by Warren D Smith 08 Oct '13

08 Oct '13

Here's some notes by me attempting to figure this out... Supersymmetry (SUSY) is an additional hypothetical symmetry (or more than one such) of space-time which converts bosons into fermions (and vice versa). For example it would convert the photon, a boson, into a "photino" (hypothetical fermion) and the electron (a fermion) into a "selectron" (hypothetical boson). It is probably the biggest open question in fundamental physics whether SUSY exists, and its existence would be a prerequisite for supergravity (SUGRA) and superstring theory, as well as a possible explanation for "dark matter" (which would be composed of one or more of the superpartner particles which must exist if SUSY is true). So far there is zero convincing evidence for SUSY -- and indeed it seems clear from evidence that "unbroken SUSY" cannot be correct in our universe, only "broken SUSY"s could be correct, which are much more mysterious. But there are some highly unconvincing but tantalizing clues which perhaps constitute some kind of pro-SUSY "evidence" if you are an incredible optimist. Far as I can tell physicists like it just because it seems amazing and cool, not since there is any significant evidence. SUSY solves or partly solves some fundamental problems in physics provided you are enough of an optimist (i.e. are willing to accept highly speculative heuristic-aided reasoning in place of proofs, to get the "solves"). It was initially proved to be impossible for such a symmetry to exist -- before it was proved to be possible. Fortunately the inventors had not known about the impossibility proof. The reason is that the construction required setting everything not in conventional space-time at all (that cannot work), but instead in a higher-dimensional space in which the extra dimensions were not real numbers, but rather "grassman numbers." The simplest superspace R^(1|1) has 1 real coordinate and one non-real coordinate, got by adjoining an additional magic symbol eps to the reals, obeying eps*eps=0 and otherwise acting normally, i.e. commutative, associative, distributive. (Analogous to getting the complex numbers by adjoining i with i*i=-1.) [Do not confuse this superspace with the word "superspace" which has also been used in a better way by J.Wheeler to denote something entirely unrelated -- the space of metrics. Wheeler as usual makes good name choices while the other physicists make bad ones... that's just an inexplicable observed fact about Wheeler versus most other physicists.] "Grassman numbers," also called "fermionic dimensions," act like real numbers except multiplication within them anticommutes, causing all squares of pure-grassman quantities to be zero. The whole notion is plainly logically self-consistent because real matrices can implement it. For example, the general element of R^(1|1) is a+b*eps with a,b, real, which can be regarded as the 2x2 matrix a b 0 a then just matrix multiply and add as usual. Grassman numbers are simpler than real numbers, e.g. polynomials (and power series!) with grassman argument always have degree=1. Trying to do integral and differential calculus, delta functions, partial differential equations, etc, inside superspaces is a bit more challenging or anyhow head-hurting. Trying to do serious physics work in spacetime manifolds with grassman dimensions, appears to be hellish. Tensor notation was fairly nasty, but spinor notation is nastier, and then throw in grassman dimensions on top of spinor notation and voila -- welcome to utter notation hell; plus say goodbye to whatever phsyical intuitions you used to have to help you. My personal threshold is tensors. Due to my mental limitations I've never been able to achieve any sort of happiness dealing with (the numerous) spinor notations and their even-hairier descendants. I might be able to push further were I motivated, but the trouble is to motivate me you'd probably have to convince me that SUSY were true (or some equivalent motivation) which means I'm basically useless for the present-day's task of determining that. When you set up a superspace that includes our usual spacetime as an all-real subspace... it includes within it the usual Poincare-Lorentz relativistic group which all physical laws are invariant under... as a subgroup. Now within superspace, the magic boson<-->fermion exchanging symmetry operation is: to interchange the grassman and real coordinates. Well, no. Actually, it's hairier than that, an operator involving differentials acting on quantum fields. Anyhow, it is definable. Do I want SUSY to be true? No. I want it to be false, for the purely greedy reason that dealing with the hellish complexities of it is something I'd prefer humans not to have to do. (SUGRA? String theory? Those are even harder.) Despite the messiness, quite a lot has been actually worked out in the MSSM ("minimal supersymmetric extension of the standard model" of particle physics) that actually is reasonably close to stuff that experimentalists can try to observe. This contrasts with SUGRA and string theory where essentially nothing has ever been worked out that is anywhere near to being experimentally verifiable. The main obstacle is the whole business of "broken" supersymmetry, which must happen if SUSY happens, but is basically total guesswork and fantasized hopes, and gives those guys ENORMOUS leeway for dodging inconvenient experimental results and still trying to assert SUSY exists no matter what the experiments say. Far as I can see, absolutely nothing stops that from continuing forever. On the bright side, if they experimentally do see supersymmetry, then wow, big progress. So SUSY is kind of 1-sided: You can prove it, but you cannot disprove it. At least that's the way I perceive it. As such, it seems somewhat problematic whether we should consider it "science." Here's two "easy introductions" to this subject: http://arxiv.org/abs/hep-th/0108200 http://www.kitasato-u.ac.jp/sci/resea/buturi/hisenkei/sasaki/Kiryuu.pdf which I find very unsatisfactory. I have not yet found any introduction I like. -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)

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[math-fun] Comet hit Libya 28.5 Myr ago
by Warren D Smith 08 Oct '13

08 Oct '13

http://www.sciencedirect.com/science/article/pii/S0012821X13004998 seems convincing evidence they actually found a piece of the comet that did it. -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)

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[math-fun] Inverse Newman iteration
by Bill Gosper 07 Oct '13

07 Oct '13

In[1106]:= NestList[2*Ceiling[1/#] - 1 - 1/# &, 3/8, 22] Out[1106]= {3/8, 7/3, 4/7, 5/4, 1/5, 4, 3/4, 5/3, 2/5, 5/2, 3/5, 4/3, 1/4, 3, 2/3, 3/2, 1/3, 2, 1/2, 1, 0, Indeterminate, Indeterminate} So 3/8 is the "20th rational". The 68th iterate on √2: In[1127]:= ContinuedFraction[ Nest[Simplify[2*Ceiling[1/#] - 1 - 1/#] &, Sqrt[2], 68]] Out[1127]= {1, 1, 1, 1, 1, 1, 1, 1, 1, {2}} I.e., it tries to disguise √2 as the golden ratio by sticking nine 1s on the front. Many other iterates of this process produce CFs with only 1s and 2s. --rwg

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Re: [math-fun] Fractals are now recondite?
by Adam P. Goucher 06 Oct '13

06 Oct '13

I maintain that Cl(flowsnake) \ Int(Cl(flowsnake)) sounds catchier... Sincerely, Adam P. Goucher > ----- Original Message ----- > From: Bill Gosper > Sent: 10/07/13 01:22 AM > To: math-fun(a)mailman.xmission.com > Subject: Re: [math-fun] Fractals are now recondite? > > [...] > > In order to be Certifiable for Classroom Use, along with the link, the text > might say something like "tripling the boundary *while preserving the shape* > of a 2-dimensional figure". And label the diagrams "Triangle", "Square" and > "Franceflake" (except you can't seem to goog on the later, tsk tsk). > > > No problem. All the little kiddies have to do is type > https://www.google.com/search?q=flowsnake+boundary&source=lnms&tbm=isch&sa=… > --rwg > _______________________________________________ > math-fun mailing list > math-fun(a)mailman.xmission.com > http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun

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Re: [math-fun] Fractals are now recondite?
by Bill Gosper 06 Oct '13

06 Oct '13

[...] In order to be Certifiable for Classroom Use, along with the link, the text might say something like "tripling the boundary *while preserving the shape* of a 2-dimensional figure". And label the diagrams "Triangle", "Square" and "Franceflake" (except you can't seem to goog on the later, tsk tsk). No problem. All the little kiddies have to do is type https://www.google.com/search?q=flowsnake+boundary&source=lnms&tbm=isch&sa=… --rwg

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Re: [math-fun] Three flake frames
by Bill Gosper 06 Oct '13

06 Oct '13

On 2013-10-06 00:50, rwg wrote: > On 2013-10-05 23:40, Adam P. Goucher wrote: >>> Bill wrote: >>> >>> OMG, there really ARE 13 of them little goobers? >>> gosper.org/flake13.png >>> >>> That's not so boring. >> >> I don't think so, unfortunately, if you're just rotating that portion of >> the Cl(flowsnake)\Int(Cl(flowsnake)) -- according to my calculations, the >> complex number solution has a modulus slightly different from 1 (so you'll >> need to rotate and enlarge). >> >> Can you zoom further into the centre of the design to check whether the >> correspondence is exact? >> Will do. Meanwhile, note http://gosper.org/flake13a.png . I didn't compute > that arctan(7 √3/11). I just stumbled on it and it looked right. But > not exactly, I agree. > --rwg On zooming, 7/11 is definitely wrong. If there is a solution, it's closer to 16/25. But as I slide past the optimum, the line segments never quite superpose. When they're collinear, they slightly overhang. When not overhung, they're off by about a line thickness. Aargh, my initial doubt seems vindicated. How cruel and unusual. --rwg > >> Nevertheless, I can confirm that the second frame does work (but the angle >> is not 7pi/33). >> >> >> Sincerely, >> >> >> Adam P. Goucher >> >> >>> On Fri, Oct 4, 2013 at 10:06 PM, Bill Gosper <billgosper(a)gmail.com> wrote: >>> >>> > On Fri, Oct 4, 2013 at 9:52 PM, Bill Gosper <billgosper(a)gmail.com> wrote: >>> > >>> >> (Easy for YOU to say.) gosper.org/flakes.png >>> >> First frame is swing angle π/3, confirming DanA's guess about trisected >>> >> areas. >>> >> 2nd frame is implausible swing angle 7π/33, or VERY nearly, showing >>> >> subflake >>> >> at a peculiar angle. >>> >> 3rd frame is even weirder angle that seems to make a triskelion out of >>> >> ten tiny >>> >> flakes. Doubt this one. >>> >> >>> > >>> > Especially since they number thirteen. >>> > >>> >> --rwg

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Re: [math-fun] Three flake frames
by Adam P. Goucher 06 Oct '13

06 Oct '13

> Bill wrote: > > OMG, there really ARE 13 of them little goobers? > gosper.org/flake13.png > > That's not so boring. I don't think so, unfortunately, if you're just rotating that portion of the Cl(flowsnake)\Int(Cl(flowsnake)) -- according to my calculations, the complex number solution has a modulus slightly different from 1 (so you'll need to rotate and enlarge). Can you zoom further into the centre of the design to check whether the correspondence is exact? Nevertheless, I can confirm that the second frame does work (but the angle is not 7pi/33). Sincerely, Adam P. Goucher > On Fri, Oct 4, 2013 at 10:06 PM, Bill Gosper <billgosper(a)gmail.com> wrote: > > > On Fri, Oct 4, 2013 at 9:52 PM, Bill Gosper <billgosper(a)gmail.com> wrote: > > > >> (Easy for YOU to say.) gosper.org/flakes.png > >> First frame is swing angle π/3, confirming DanA's guess about trisected > >> areas. > >> 2nd frame is implausible swing angle 7π/33, or VERY nearly, showing > >> subflake > >> at a peculiar angle. > >> 3rd frame is even weirder angle that seems to make a triskelion out of > >> ten tiny > >> flakes. Doubt this one. > >> > > > > Especially since they number thirteen. > > > >> --rwg > >> > > > > > >> [...] > >> > > > _______________________________________________ > math-fun mailing list > math-fun(a)mailman.xmission.com > http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun

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