December 2012 - mathfun@mailman.xmission.com

[math-fun] Treefoil
by Adam P. Goucher 14 Dec '12

14 Dec '12

Here's a puzzle I pondered earlier today and eventually solved. Does there exist a cycle (self-avoiding cyclic walk) in Z^3 such that: (a) The three projections on to the xy, yz and zx planes are trees; (b) The cycle has order-3 rotational symmetry; (c) The cycle is a trefoil knot? Satisfying properties (b) and (c) is very easy, whereas satisfying (a) (even on its own, without (b) and (c)) is rather difficult (try it). Amazingly, it is possible for a cycle to satisfy all three properties simultaneously: http://cp4space.wordpress.com/2012/12/12/treefoil/ Can you find a smaller solution than 11 x 11 x 11? I know for certain that there isn't one in a 3 x 3 x 3 box. Sincerely, Adam P. Goucher

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[math-fun] Im(z!)=0
by Bill Gosper 13 Dec '12

13 Dec '12

Plot3D[Log[-((x + I*y)! - (x - I*y)!)^-2], {x, -5, 15}, {y, -5, 15}, PlotPoints -> 69] produces http://gosper.org/groovy.png , whose ridges indicate where z! is real. It's three families of "hyperbolas" plus the big ridge for the real axis, which is also the axis of symmetry. What are the asymptotes of the "hyperbola" just above the real axis in the first quadrant, heading off northwest in the second quadrant? Spoiler (Please ogle the png first): StringReverse[".)x-(gol ekil ylhguor gniworg trap yranigami eht ,etotpmysa tnerappa on htiw ,latnoziroh drawot kcab stcelfni *yllaudarg* cra tsewhtron ehT .noitseuq kcirt a saw siht tuB .x gol/1 ekil egrevnoc ot smees trap yranigami ehT ).foorp a sah nailuJ( .cra drawtsae eht fo etotpmysa eht si sixa laer eht taht raelc ytterp s'tI"] Experimentally, the ridges bunch gradually with increasing imaginary part, perhaps reminiscent of the bunching of the nontrivial roots of Ϛ. --rwg

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[math-fun] base 12 things and nothing happened today at 12:12:12
by Simon Plouffe 12 Dec '12

12 Dec '12

Hello, believe it or not, there are somepeople that are convinced that a system in base 12 would be better, for reading the hour and some trigonometric reasons, there is even a film aboutit, some aliens with 12 toes, that came out in the newspaper 'le monde' since today at 12:12:12 andon year 12, nothing happened, here is the film : http://youtu.be/B6415P4bh5M and here is a link to the Duodecimal Association, http://www.dozenal.org/index.html they call themselves Dozenalist, I presume they must like eggs a lot? best regards, Simon Plouffe

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Re: [math-fun] math-fun Digest, Vol 118, Issue 16
by Warren Smith 12 Dec '12

12 Dec '12

> Dan Asimov: > Anyway, as Ramanujan is famous for pointing out to Hardy, > > 9^3 + 10^3 = 12^3 + 1 > > . > > > This is a counterexample to Fermat's Last Theorem . . . almost. > It's off by one. > > QUESTION: Are there other examples of integers x, y, z for which > > x^3 + y^3 = z^3 + 1 --see http://www.math.psu.edu/vstein/preprints/Rachel11.ps for a beginning.

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Re: [math-fun] New lighting technology breakthrough?
by Christian G. Bower 11 Dec '12

11 Dec '12

I was biking home a few nights ago and did some observing. I noticed a car at a stoplight and saw that the red tail lights were slightly yellower than the stop light. Neither was as yellow as the "Don't Walk" sign which I would say was about half way between red and orange. Most of the tail lights I saw were indistinguishable from the first and yielded the same comparison to the stop light however I saw one or two that I could detect as being slightly off norm. The stoplight comparison I think rules out my inability to distinguish different wavelengths of red as the reason. I suspect it has more to do with the red plastic all being the same and there being little demand for designer versions of it. ------ Original Message ------ Received: Mon, 03 Dec 2012 11:22:51 PM PST From: Bill Gosper <billgosper(a)gmail.com> To: math-fun(a)mailman.xmission.com Subject: Re: [math-fun] New lighting technology breakthrough? > [Not math, but hopefully fun] > Automotive lights: Besides the traditional yellowish white headlights, > we're seeing cold white, and pale green, blue, pink, > and lavender. Plus those annoying prismatic things. But crawling down > I280 tonight, I was struck by the uniformity of > the red in the miles of taillights stretching out before me. Both LED and > conventional. Is there some strict spectral > standard? Or does my color discrimination roll off fast at the red end? > > hgb>To a first approximation, today's "white" LED and today's "white" > fluorescent light are the same: They both excite a fluorescent dye with a > blue/ultraviolet light. They differ in how the blue/ultraviolet light is > created. The visible "white" color (or color "temperature") is dependent > upon the type of dye. The dyes can be made from "quantum dots", which can > be engineered to have the appropriate spectrum. "White" light can also be > made using 3-4 colored LED's (green LED's are less efficient, so you tend > to need RGGB instead of just RGB); this is used in large LED billboards and > for LED TV's. Since you need 2 G's anyway, you can use G's with 2 different > wavelengths to get a wider "color gamut" -- RGG'B. There are some LED TV's > that utilize this idea. There are now LED stage/studio lights whose colors > can be computer-programmed for different colors using these combinations. I > would speculate that white light made from LED's of individual colors > _might_ be more efficient (in terms of electrical power) than the white > light from downconverted UV, but I haven't been able to confirm this. My > best guess is that downconverted UV is cheaper/simpler to use, because it > requires only one type of bulb, and because it draws upon a century of > fluorescent light technology. At 04:50 PM 12/3/2012, Warren Smith wrote: > > The news hype media is full of stuff about how David Carroll, prof. at Wake > Forest Univ. in NC, has invented new wondrous lighting technology, > environmentally friendly, efficient as LEDs and "at least twice as > efficient as fl.tubes," but simpler and looks better, wholy solid state, > simple, most any shape (sheets, rods, bent rods,...), glowing capacitor > using magic nanoparticles embedded in plastic, no toxic mercury like > fl.tube has, long lived (>10 years), will be commercialized with in 1 year. > I tracked it to this press release > http://news.wfu.edu/2012/12/03/taking-the-buzz-out-of-office-lights/ and > this pre-paper of some ilk: > http://www.sciencedirect.com/science/article/pii/S1566119912004831 What the > media hype machine never mentioned in about 10 articles I saw, but is > revealed in the latter document, is that they incorporate substantial > amounts of Iridium and Indium. That sounds like an immediate major lose > right there. -- Warren D. Smith > > Just today I daydreamed about making a platinum-iridium disk puzzle which > people > couldn't game thermally. > --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] colouring of R^3
by Christian G. Bower 11 Dec '12

11 Dec '12

It's not entirely correct to say that the statement of CH doesn't make sense w/o AC. In fact, in many ways it's more interesting that way. You say without AC we can't compare the sizes of infinite sets. More accurately, in ZFC, cardinality is a (class) linear or total order relation. In ZF it is a (class) partial order relation. Thus we can compare sizes, but sometimes the answer is "incomparable." Consider the following 2 statements: [Here c = 2^aleph_0 = continuum] (1) There is no cardinal k : aleph_0 < k < c (2) c = aleph_1 In ZFC, these statements are equivalent and correspond to CH. In ZF, they are not equivalent. (2) => (1) but (1) !=> (2). Also (2) => the reals can be well ordered. (1) does not. Another relation I found even more surprising related to the Generalized CH: (1) For every infinite cardinal k, there is no cardinal j: k < j < 2^k (2) For every ordinal a, 2^aleph_a = aleph_{a+1} In ZF, (1) => 2, (2) !=> (1), (1) => AC, (2) does not. ------ Original Message ------ Received: Sun, 09 Dec 2012 08:22:41 AM PST From: "Adam P. Goucher" <apgoucher(a)gmx.com> To: "Warren Smith" <warren.wds(a)gmail.com>,math-fun <math-fun(a)mailman.xmission.com> Subject: Re: [math-fun] colouring of R^3 ... > > NB: I've liberally assumed the Axiom of Choice everywhere, but > that's okay because the statement of the continuum hypothesis > doesn't even make sense without it (without AC, we can't compare > the sizes of infinite sets). ...

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Re: [math-fun] Antikythera device in volume production ?
by Henry Baker 11 Dec '12

11 Dec '12

Thanks, Robert, for posting this terrific paper! Mike G. Edmunds, Cardiff University, Tony Freeth, Antikythera Mechanism Research Project. "Using Computation to Decode the First Known Computer". IEEE Computer July 2011. Some comments in this paper caught my eye: "We believe the gears were laid out and cut by hand, with the teeth appearing to be *** simple equilateral triangles, *** slightly rounded at the tips." "The Antikythera mechanism is *** engineered at a tiny scale *** Â-the offset between the axis of the gear with the pin and that of the one with the slot is just over 1 millimeter." "For example, *** the random error in tooth-center placement varies between gears from about 0.1 to 1 degree. *** A few showed a more systematic variation characterized by a sinusoidal variation around the whole gear." "The moon position indicator, however, involves Âgearing up,Â amplifying angular errors so that *** the indicated position could differ by 20 degrees from the designerÂs intention. *** Such a large error is easily noticed when comparing predictions with the moonÂs actual position in the sky; *** it [the error] exceeds the amplitude of the Âfirst anomalyÂ variation that the mechanism was so cleverly designed to incorporate. *** A possible interpretation is that truly accurate prediction was not the deviceÂs primary purposeÂapproximate prediction of astronomical phenomena by mechanical means was achievement enough." My conclusions: 1. The Antikythera device was clearly a scaled-down version of previous devices, some of which were probably made over 100 years prior. 2. The Antikythera device was in _volume production_, since the triangular gears and "play"/"error" in the mechanism would have exceeded the designer's original specifications. Why bother designing & building a clever anomaly mechanism if it didn't work at the end of the day? Clearly, the high precision of this particular function was not essential for the bulk of the customers of the device. The triangular gears are likely the result of faster production, and smaller size, since triangular gears are not ideal in terms of strength, wear or "play". 3. The manufacturer of the device was disconnected enough from the designer, that the designer was not in a position to veto the production of a sloppy mechanism. Those of us in the software business are well aware of the "maintenance programmer" who -- with insufficiently complete understanding of the workings of the original program -- unwittingly makes modifications which destroy some of the features of the original design. At 01:49 PM 12/6/2012, Robert Munafo wrote: >The known gearing, along with the number of teeth in most of the known >gears, is described in detail here: > > http://cs.iupui.edu/~tuceryan/pdf-repository/Edmunds2011.pdf

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[math-fun] Antikythera device -- dismissive reply from expert(Prof. Mike Edmunds)
by Warren Smith 10 Dec '12

10 Dec '12

--Prof. Mike Edmunds: Dear Dr Smith, Thank you for your interesting message. Although the remnants of the Antikythera Mechanism were found on a ship, we are certain it was not a navigational instrument. All of the functions we have so far discovered were for astronomical or for calendar purposes. I have some sailing experience and (modest) navigation qualifications - and I can confirm that none of the function would be of use in day-to-day navigation of a boat. --WDS comment: I'm dubious. Does this guy really navigate using chronometer, sextant, and spherical trigonometry? I doubt anybody has done that for decades. And one's initial reaction on seeing a chronometer (which is just a wind-up clock with high-quality innards) might well have been "no use for day to day navigation of boat." This judgment would be totally wrong. The only way to realize its use is to investigate mathematical theory, not by just looking at it and saying "I've sailed on boats." And by the way, the whole phrase "day to day" employed by Edmunds is misleading. More accurate would be "week to week." To get an idea of reality, it helps to consider the famous sea-voyage by Shackleton with 5 others in 1914 in open rowboat 800 miles. Did they do "day to day" navigation? No they did not, because it was impossible. They set off 24 April. The log of this voyage reveals that they were able to obtain successful sun sights on 26 April, 29 April, 4 May. On 8 May they spotted land (their destination). Landed 10 May. (They then did some further local sailing next few days.) Thus the entire voyage lasted 17 days during which they only accomplished 3 sun-sights. Similarly, the plan that was being investigated by the British for navigation based on the moon (this was before the success of the chronometer) would have required accurate measurement of angles from moon to horizon, stars, sun. Again, such measurements would probably have been impossible to accomplish every day. For example in bad seas or cloudy. I repeat, this whole scheme was under development by the British admiralty as a possible navigation method until it got superseded by the unexpected success of the chronometer which allowed simpler method. --Edmunds: Its possible use for determining longitude at eclipse times has been suggested before, but this is totally impractical for navigation because eclipses occur so rarely. --WDS: I agree eclipses rare, and I agree eclipses basically no use, but you do not need eclipses to determine longitude. And if you were relying on eclipses, you would be being an idiot. My whole point was not to do that and not to be an idiot. For example, did the British admiralty intend to rely on eclipses? Of course not. Edmunds appears not to have read my letter and to be highly unfamiliar with the whole idea behind this kind of navigation. --Edmunds: It has been argued that such devices might assist in setting longitude at, say, opposite ends of the Mediterranean sea in a campaign lasting several years, as may have been attempted by the Chinese in other waters in the 15th century - but again this is not day-to-day navigation. Anyway, all they would have needed is a written compilation of eclipse months! --WDS: Did the Chinese even know that the Earth was round? The Greeks were well aware of it by 5th century BC. Wikipedia http://en.wikipedia.org/wiki/Spherical_Earth has "In the 17th century, the idea of a spherical earth, now considerably advanced by Western Astronomy, ultimately spread to Ming China, when Jesuit missionaries, who held high positions as astronomers at the imperial court, successfully challenged the Chinese belief that the earth was flat and square." citing http://www.uni-tuebingen.de/uni/ans/eastm/back/cs11/cs11-4-martzloff.pdf Jean-Claude Martzloff: "Space and Time in Chinese Texts of Astronomy and of Mathematical Astronomy in the Seventeenth and Eighteenth Centuries", Chinese Science 11 (1993-94): 66-92 (69). So, I'm not terribly impressed by this Edmunds quote, because (a) eclipses are irrelevant hence everything Edmunds says about eclipses is irrelevant, and (b) also his whole side-remark about the Chinese (which also is pretty irrelevant) seems somewhere between dubious and refuted anyhow. --Edmunds: We know from a study of ancient literature sources - see e.g. http://pos.sissa.it/archive/conferences/170/019/Antikythera%20&%20SKA_019.p… - that the device was certainly not unique, indeed we have a fair number of references to such astronomical devices over the period approx 250 BC to 500 AD, and no mention whatever of a similar navigational device. The subtle mechanical details of the Antikythera device would be impossible to design without considerable previous experience, and it must come from a tradition of the making of such geared mechanisms. By far the most likely explanation for its presence on the (trading) boat is that it was being transported, possibly as part of a large consignment of goods, or simply being delivered on its own. --WDS: I'm happy to agree it was on that boat merely for transport. In fact, I had speculated the device was experimental and stolen by the Romans. So we are in agreement. So therefore this disclosure by Edmunds has no logical bearing at all on my entire hypothesis. Which he would have realized had he actually read my letter. And the URL Edmunds gave does not seem convincing to me of anything much. First of all, before Galileo invented the pendulum, it was almost utterly pointless to build a mechanical clock in the sense accuracy would have been totally pathetic. A mechanical clock making an error of say, 30 minutes per day, which probably would have been about best possible before the year 1200, would have been for navigational purposes off by 500 miles per day even if portable (which they weren't). Any clock anyone ever made pre-Galileo and especially pre-1200 AD would have been merely as a joke, toy, or mystical device intended to impress somebody with mysticism, not actually be useful. I do not see the point of it. The first pendulum clock was built by Chr.Huygens in 1656. Second, the "sphaerae" referred to in this manuscript may not have been a mechanical device with moving parts at all, at least for all the evidence I see given in the manuscript! (They mention a Cyclopaedia 1810 article, but what the heck did they know in 1810?) --Edmunds: With Best Wishes, Prof Mike Edmunds --WDS: to make a genuine assessment of whether a device like the Antikythera could have been intended for navigation (or could have been intended as experimental with possible future development into a workable nav. device) the main question that needs to be answered is an accuracy question. There is angle-measurement accuracy for observations of moon etc. And there is predictive accuracy via the device. All previous assessments of predictive accuracy for the device that I have seen, were wrong-headed and hence largely irrelevant, since they concerned multiyear predictions. In fact, if it were used for navigation, the device would be reset to truth at start of voyage, then the voyage lasts (say) 3 weeks. So what we need to assess is the accuracy during this 3 week span assuming perfect accuracy at start. As the observers of the Antikythera device have already noted, the device was built to make it easy to take apart and reset. The question is 2-fold (a) what accuracy could this device itself have achieved, and (b) what about a similar device but with optimized parameters?

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[math-fun] Eureqa fun experiments: unit step function
by Henry Baker 10 Dec '12

10 Dec '12

I gave Eureqa the following data: y=0 (x=-14..0), y=1 (x=1..18). I set the error metric to least absolute error, which eliminates many "ringing" and "horn" artifacts. When the functions allowed are only +,-,*,/, I get pretty decent rational function approximations, but because of Eureqa's "low complexity" bias, the numerator & denominator polynomials are _extremely sparse_. With / but not *, you can get continued fractions, but Eureqa doesn't like them. I changed the "complexity" of / to 1 from 2, but convergence to CF solutions is extremely slow. Eureqa seems highly biased towards "multiplying out" any polynomials. For example, I couldn't get Eureka to put out a "factored form" -- e.g., (1+x)*(2+x) -- at all. When the functions are expanded to include sines & cosines, you _don't_ get Fourier series, but weird sines composed with additional sines. When the functions include exp(x), we see functions like y = x/(x + exp(-9.45e9 * x)) With powers: y = 0^(0^x) y = x/(x + 0^x) With factorial: y = x/(x + factorial(-618*x)) With complementary error functions: y = erfc(53.8* erfc(4*x)) With error function: y = 0.5 + 0.5 * erf(21.2*x - 21.4) y = erf(x + x*erf(33*x)) With tanh function: y = tanh(x + x*tany(911*x)) y = 0.5 + 0.5 * tanh(9*x - 12.2) y = tanh(39 + 39*tanh(72.2*x - 22.4)) With Gaussian: y = gauss(5.21e22 * gauss(7+x)) With logistic function y = logistic(1.23e3 * x - 975) With floor function y = -floor(-0.0183*x) With ceiling function y = ceil(9.37e-5 * x) With abs function y = (x+|x|)/(1.03e-15 + x + |x|) -------------- Bottom line: I like the basic idea of Eureqa, but it may be necessary to reprogram it from scratch to do the sorts of "mathematical" explorations that we here at math-fun like to do. Even by adjusting the "complexity" of different operations by large amounts, I still wasn't able to force Eureqa to output expressions like nested square roots, or deeply nested divisions (i.e., continued fractions). Eureqa also seems to prefer "simple" constants, and refuses to refine the number of digits of a constant before looking for a more complex expression. There should be some user-programmable mechanism to provide a complexity function for constants.

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[math-fun] Antikythera device & lunar navigation
by Warren Smith 09 Dec '12

09 Dec '12

Been reading a bit about the history of navigation. The British were successful in locating many places, such as Jamaica, via astronomical methods before the chronometer was developed. The method they used was that some astronomical event -- Jupiter's moons and the transits of Mercury were the most used and popular -- would be clocked both in Jamaica and at Greenwich. Then they'd send each other the info. As a result, they could synchronize high accuracy pendulum clocks in both places. Thus they could deduce the longitude of Jamaica. This however required a decent observatory and high accuracy pendulum clocks, neither of which could be used on a ship. But this could be done, and was done, and was the basis of their early charts. Later, a competition arose between the clock makers, especially the genius John Harrison who developed a number of great inventions laying the basis for precise instrumentation of all kinds, versus the astronomers, in solving the ship navigation problem. The clock-based method ultimately won. However, laying his thumb very heavily on the scale trying to bias the judges was Astronomer Royal Nevil Maskelyne, who if this were a Hollywood movie, would be the bad guy. The astronomers published lunar tables for navigation purposes. Kepler published tables in 1627. Street in 1661. Note Newton's "Principia" came out in 1787 so all those were before his Laws were known. In 1753/5 solar and lunar tables were published by Tobias Mayer. The British Admiralty found the the typical error in the moon's location got from these tables was 1 arcminute, i.e. 1/216000 of a full circle. They then published Nautical Almanacs with updated tables and full instructions on how to use them to navigate. E.g. in the 1767 almanac there are tables of the moon-to-sun angle and angles to seven fixed stars, for times every 3 hours, and instructions are given to convert measurements of these things to "time at Greenwich" -- including making corrections for "parallax" and "refraction." These tables kept improving as the years went by due to further astronomical observations and corrections based on them -- as well as errors detected+corrected of which there were perhaps 100. (There also were tables concerning Jupiter's moons published 1821). The French and Spanish then published their own such almanacs and tables, then the Portuguese and Danes. However as I said chronometers won versus astronomy for ship navigation, and indeed it ultimately became popular for important ships to carry many, not just one, chronometer. The observatory at Greenwich held an annual contest to determine the best available chronometers during 1822-1835. When the trans-ocean telegraph cable came it became possible to sync clocks without need for chronometers or astronomy, but the old determinations were found to be quite correct. Prof. Edmunds sent me a paper claiming the antikythera device would have error in moon location of 20 degrees hence useless for navigation. However, his error assessment was totally irrelevant since it was for a 19-year-duration prediction. The history I have recounted makes it clear that at least in principle an antikythera-like device could have attained accuracy of 1 arcminute, although I doubt that was ever actually achieved, and even if my whole "it was intended for navigation" hypothesis is correct then still clearly the antikythera must have either been a failure or never reached widespread success as a navigation tool, and must have been only an experimental device. For it to be an actually wide-used and useful navigation device would have required a network of observatories, chart production, a small science well known to many, and very many copies of the device produced... which I think could not have been forgotten to this great an extent if it happened. So it didn't.

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