November 2013 - mathfun@mailman.xmission.com

[math-fun] French Hens: open problem on space-filling curves / IFS
by Michael Kleber 03 Nov '13

03 Nov '13

Since Joerg mentioned Ventrella's book _Brain-Filling Curves: A Fractal Bestiary_, let me take this opportunity to re-mention an open problem I'm fond of. (At least, I think it's still open.) Recall that 1^3 + 2^3 + 3^3 + ... + n^3 = (1+2+...+n)^2. That means that one 1x1 square, two 2x2 squares, ..., n nxn squares together all have the same total area as one square of side length 1+2+...+n. Robert Wainright's "Partridge problem" asks whether you can realize this numerical identity as a dissection of a big square into the requisite set of smaller squares. It turns out to be possible for all n >= 8; you can see some pictures of solutions on pages by Funsters at http://www2.stetson.edu/~efriedma/mathmagic/0802.html or http://www.mathpuzzle.com/partridge.html. (The "all n >=" part is very not obvious even once you have an n=8 solution.) The name "partridge problem" comes from "four calling birds, three french hens, two turtledoves, and a partridge in a square tree", of course. This identity can manifest itself as a tiling problem not just for squares, but for an arbitrary planar figure P. If kP means a copy of P scaled linearly by a factor of k, then the question is whether you can use one P, two 2P's, ..., n nP's to cover a (1+2+...+n)P. If so then we say P has "partridge number" n (or maybe only for the minimal such n, depending on who you ask). For example, the square has partridge number 8, the equilateral triangle has partridge number 7, the 30-60-90 right triangle has a partridge number of 4, and so on. The 30-60-90 find is by Patrick Hamlyn, and is a truly beautiful puzzle; I've had a physical copy sitting on my desk for years. Anyway, the open problem part of all this! Patrick Hamlyn's 30-60-90 is the polygon with the smallest known partridge number. As far as I know, it is open whether there are polygons OR fractals with partridge numbers 2 or 3. The fact that 2 is open, even for polygons, really bugs me. I really want to know whether there is a polygon P such that a P and two 2P's fit together to make a 3P. (For a while I thought I had a proof of impossibility, and I was all set to write a paper called "On the Necessity of French Hens", and that my proof fell apart is a great disappointment, though I have to admit more for the title of the non-paper than for its theorem.) Anyway, the fractal version of the partridge-2 or partridge-3 problem is very much along the lines of Ventrella's book. In particular, I think that if a solution exists then then must be a space-filling curve that realizes it. --Michael On Sun, Nov 3, 2013 at 9:19 AM, Joerg Arndt <arndt(a)jjj.de> wrote: > A pointer to a neat picture book featuring > very many "space filling" curves: > > Jeffrey J. Ventrella: > Brain-Filling Curves: A Fractal Bestiary, > LuLu.com, (2012). > > The book can be downloaded for free at > > http://archive.org/download/BrainfillingCurves-AFractalBestiary/BrainFillin… > Warning: it is a hefty 235 Megabytes! > > Two pertinent URLs are > http://www.fractalcurves.com/HorrorVacui.html (book as html) > http://www.brainfillingcurves.com/ (needs scripting activated) > > On page 107 (top) there is a curve corresponding > to the L-system (and turns of 120 degrees) > Start: F > Rules: > F --> F-F+F+FF-FF > + --> + > - --> - > > Here is a rendering > http://jjj.de/3frac/R7-x1-curve.pdf > > What Ventrella does not show is how neatly 3 of > these combine into either > http://jjj.de/3frac/R7-x1-closed-trihook.pdf > or > http://jjj.de/3frac/R7-x1-closed-gosper-island.pdf > > The better known splittings of Gosper's island are > http://jjj.de/3frac/gosper-split7.pdf > and possibly (3 renderings) > http://jjj.de/3frac/R7-2-gosper.pdf > http://jjj.de/3frac/R7-2-sty1-gosper.pdf > http://jjj.de/3frac/R7-alt-hex-gosper.pdf > Here is a bubbly one: > http://jjj.de/3frac/R7-bubble-gosper.pdf > > (cf. http://www.fractalcurves.com/Root7.html ) > > > For the truly bored there are more images under > http://jjj.de/3frac/ > > Best, jj > > > _______________________________________________ > math-fun mailing list > math-fun(a)mailman.xmission.com > http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun > -- Forewarned is worth an octopus in the bush.

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[math-fun] can some with access to The Symmetries of Things
by Bill Gosper 03 Nov '13

03 Nov '13

please tell me if Conway et al have a more standard terminology for these things I've been calling alternately concave pentiprisms and overtruncated trapezohedra? Do they recognize the more logical terms, e.g., pentatrapezohedron vs the more standard but illogical "pentagonal trapezohedron", which has no pentagonal faces? Tnx, --rwg

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Re: [math-fun] Root of math Genius Sought
by Stuart Anderson 03 Nov '13

03 Nov '13

> > 4. Root of Maths Genius Sought (Ray Tayek) > > Work harder and get smarter http://qz.com/139453/theres-one-key-difference-between-kids-who-excel-at-ma… Terry Tao http://terrytao.wordpress.com/career-advice/does-one-have-to-be-a-genius-to…

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Re: [math-fun] Combining results of scientific experiments -- CODATA claimed to be wrong
by Warren D Smith 02 Nov '13

02 Nov '13

> From: Fred Lunnon <fred.lunnon(a)gmail.com> > To: math-fun <math-fun(a)mailman.xmission.com> > Subject: Re: [math-fun] Combining results of scientific experiments -- > CODATA claimed to be wrong > Message-ID: > <CAN57YqveH-4rHSe+UkdOmBi7dA4ZjRy5THkSU2XeVDFMU=tj6g(a)mail.gmail.com> > Content-Type: text/plain; charset=ISO-8859-1 > > How much difference would your suggestion make to the currently agreed > values? WFL --I do not currently know. Finding out may not be so easy. There are two issues (a) the values and (b) the error bars and their interpretation. Either one changing, is important. A ton of physics papers are based on the claimed error bar of something or another, e.g. muon magnetic moment "disagrees" with theory, hence there "must be new physics such as supersymmetry" and on and on. A ton of that kind of stuff is presumably now bullshit. As a start to make progress: must get ahold of the database of measurements that CODATA must have that underlies all their stuff.

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[math-fun] Root of Maths Genius Sought
by Ray Tayek 02 Nov '13

02 Nov '13

http://science.slashdot.org/story/13/10/30/1423236/root-of-maths-genius-sou… Posted by <http://unknownlamer.org/>Unknown Lamer on Wednesday October 30, 2013 @10:51AM from the army-of-cloned-math-nerds-not-very-terrifying dept. ananyo writes "He founded two genetic-sequencing companies and sold them for hundreds of millions of dollars. He helped to sequence the genomes of a Neanderthal man and James Watson, who co-discovered DNA's double helix. Now, entrepreneur Jonathan Rothberg has set his sights on another milestone: <http://www.nature.com/news/root-of-maths-genius-sought-1.14050>finding the genes that underlie mathematical genius. Rothberg and physicist Max Tegmark, who is based at the Massachusetts Institute of Technology in Cambridge, have enrolled about 400 mathematicians and theoretical physicists from top-ranked US universities in a study dubbed 'Project Einstein'. They plan to sequence the participants' genomes using the Ion Torrent machine that Rothberg developed. Critics say that the sizes of these studies are too small to yield meaningful results for such complex traits. But Rothberg is pushing ahead. 'I'm not at all concerned about the critics,' he says, adding that he does not think such rare genetic traits could be useful in selecting for smarter babies. Some mathematicians, however, argue that maths aptitude is not born so much as made. 'I feel that the notion of "talent" may be overrated,' says Michael Hutchings, a mathematician also at Berkeley." --- co-chair http://ocjug.org/

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[math-fun] Combining results of scientific experiments -- CODATA claimed to be wrong
by Warren D Smith 01 Nov '13

01 Nov '13

Combining results of scientific experiments CODATA tells us the values of physical constants, such as proton radius, based on combining results of experiments by different people. I will here argue they are doing it wrong and consequently all scientific "fundamental constants" from CODATA are screwed up. 1. The way they do it (naive description) is: each experiment gives a value V and an error bar +-B. They then weighted-average all the values, using weights=1/B^2, to obtain final value Vcombined. And they sum the squares of the B's and compute the square root to get Bcombined. 2. The way they do it (less naive) is that many of the physical constants and measured quantities are theoretically related, namely the logs of these obey certain known linear relations. So there is a high-dimensional space of all these log-quantities, which thanks to the linear relations is really a lower-dimensional subspace (which is still pretty high dimensional). And they work in this space using "error bars" which really are ellipsoids. Also there are additional issues related to handling "systematic" versus "statistical" errors. 3. The less-naive version happens to be the same as the naive description if no linear relations involved. 4. What is wrong with that? The underlying justification for the value-combining via weighted-average method in (1), is that each measurement is assumed to be normally distributed with the claimed mean and std.deviation. Then this produces a max-likelihood estimate. Which also for normals happens to be the Bayesian estimate. But in fact, that is JUST FALSE. Example, 137.035 999 679 (94) Previous CODATA 2006 value of 1/alpha... oops. 137.035 999 049 (90) CODATA 2010, mostly from Rb spectroscopy which disagree at 4.84 sigma, which is to say, >99.99995% confidence that those were NOT normal errors. There are a ton of such examples. Experimenters are constantly getting their error estimates wrong, or the statistics are nongaussian -- which amount to the same thing. The number of times experimenters have been off over 10sigma is large, massively refuting the gaussianity assumption. 5. So the correct thing to do, is to combine results NOT using an underlying assumption of gaussianity, but rather using the CORRECT assumption, based on historical record, of the true non-gaussian error distribution. (And frankly the correct distribution is more like Cauchy than Gaussian...) I think a decent model would be more like power-law tails where the correct power would need to be determined by a fit to historical data. I would suggest a density that is something like the gaussian density out to 1.5sigma, with power law tails surgically grafted on beyond that. The class of "stable densities" seems theoretically motivated. 6. CODATA implicitly already admits this -- they say they discard data as "outliers that they do not believe" based on their inner prejudices and judgments. If however they were using the right nongaussian assumption they would not need to do that discarding. 7. So... how should data be combined if we instead assume power law tails? The density we are recommending is log-concave and hence the max-log-likelihood estimate can be got by a concave-down function maximization problem (or in multidimensions) for which there is a polynomial time algorithm. This will still produce a weighted average as the combination, but the weights will be different. 8. And how should errors be combined? One answer is via whatever weighted average you used to combine the data -- use those to combine the variances. A different answer is, if we are using "stable densities" then when these combine you still get a stable density and just state its parameters. There would with stables actually be no variance (it would be infinite) and the meaning of the "error bar" would be something else. This is kind of a whole new way to do statistics I'm suggesting, but it also clearly seems needed. -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)

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Re: [math-fun] Julian's dodecahedron
by Bill Gosper 01 Nov '13

01 Nov '13

DaA> RWG, it's a lovely and informative animation! Thanks! The reason I asked about the rotation is that someone disliked it, even though I added it to remedy some unrevealing poses during pauses. Just one thing: I wish it were a bit slower, and that the words remained visible a bit longer. (I never took an Evelyn Wood course.) --Dan We can't risk small children reading them aloud and getting soap stuffed in their mouth. On 2013-10-31, at 10:57 PM, Bill Gosper wrote: http://gosper.org/pyrominia.gif . Please let me know if you strongly like or dislike the slight rotation. --rwg Here's another one, perhaps showing why Julian thinks his result is too trivial to publish: http://gosper.org/pentatrapezohedra.gif --rwg

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