December 2012 - mathfun@mailman.xmission.com

Re: [math-fun] Gamow syndrome (Aleph_0 and omega)
by Adam P. Goucher 08 Dec '12

08 Dec '12

> http://mrob.com/pub/math/images/gamow-123i-p23-fig8.jpg Oh dear! The first box is definitely correct, the third box is incorrect, and the second box depends on the continuum hypothesis (as does the caption). > Aleph_1 is identified as "the number of all geometrical points on a > line", and the caption called Aleph_0, Aleph_1 and Aleph_2 "the first > three infinite numbers" (which implies there are no other infinite > numbers in between). Aleph_1 is defined to be the second infinite number, which is only the same as Beth_1 (cardinality of the reals) if you assume CH. Also, the number of curves (by which I mean countable unions of continuous curves) is just Beth_1, same as the cardinality of the reals. This includes everything you can justifyably call a curve (including the boundary of the Mandelbrot set!). Beth_2 (= 2^Beth_1 = 2^2^Aleph_0) is the number of ways you can colour the plane with two colours (which I think is what Gamow meant by 'geometrical curves'). > > Yes, this is one of those things which are obviously false but > > actually true. Then there are things which are obviously either > > true or false, but turn out to be equivalent to the continuum > > hypothesis instead. > > I suppose that's what got Gamow. Hmm, yes. My favourite example of a CH-equivalent problem is this (I was given the problem by Prof. Imre Leader): "Can we colour the points in R^3 with the colours red, green and blue such that every line parallel to the x-axis contains only finitely many red points, every line parallel to the y-axis contains only finitely many green points, and every line parallel to the z-axis contains only finitely many blue points?" Sincerely, Adam P. Goucher http://cp4space.wordpress.com

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Re: [math-fun] Antikythera device -- could it have been intended for navigation?
by Henry Baker 08 Dec '12

08 Dec '12

Thanks, Warren, for following up on this. The good new is that if you are correct in its use as a navigational device, then this particular instance may not be that rare after all, and there might be similar instruments in other wrecks in the Mediterranean & Black Seas. Since there are thousands of such wrecks, there may be tens/hundreds of devices to find. Perhaps some will be in better condition than the Antikythera device. Perhaps other such ancient devices have _already_ been found, but subsequently mistakenly discarded as corroded pieces of modern junk. At 09:21 AM 12/7/2012, Warren Smith wrote: >I finally talked myself into sending an email to the people behind the > http://www.antikythera-mechanism.gr/ >web site (who also have authored numerous articles about the device). > >Here's the email I sent them. > > I was stimulated by recent speculation the antikythera device was perhaps > designed by Archimedes, seems to have predicted calandar, moon, sun, planets, > and knew (via a "pin and slot drive" mechanism) an approximate version > of Kepler's ellipse fact of non-uniform speed and non-circular motion. > Then I found your web site > http://www.antikythera-mechanism.gr/ > and some popular press articles. > > What I want to begin exploring in this email is the speculation -- which I hardly saw > anyplace -- that the device could have been intended by its designer for navigational use.

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Re: [math-fun] Antoine's necklace
by Bill Gosper 07 Dec '12

07 Dec '12

mrob> Adam's mention of Antoine's necklace reminded me of a topology question I've had for a while. According to the Wikipedia description (which is admittedly basic, but agrees with my own perception of the situation), the necklace remains linked through all finite stages of the construction, yet in the limit (of the countably infinite number of steps) what remains is a set of single points. That's the "paradox" (which I know is just my limited perception) : 1. Since single points are not tori, it seems that the necklace is a set of disjoint points, so the necklace is no longer linked. 2. But the description clearly says that the set's complement is not simply connected, which implies that the necklace at stage Aleph_0 is still linked. Transitions from a finite property to an infinite property appear everywhere. For example, every term in the sequence 1, 2, 3, 4, ... (the positive integers) is finite, but the limit of the series (Aleph_0) is infinite. I can grasp that. Similarly, the normal (1-dimensional middle-thirds) Cantor set transitions from being a set of intervals with nonzero measure to a set of points. That doesn't seem to bother me either. But the necklace one bothers me. That transition from linked set of tori to unlinked set of points seems impossible. ------ rwg>A similar thing happens with increasingly fine polygonal approximations to spacefilling functions. In http://www.tweedledum.com/rwg/rad2fill.htm the upper figure defines the iteration. For any finite order, six of these equipartition the area near the center of the lower figure. But at the limit, the areas disappear and everything is boundary. I don't get the many-torus illustrations in http://en.wikipedia.org/wiki/Antoine%27s_necklace . Doesn't the text say four tori? Shouldn't three work? (Bogus Borromean rings.) --rwg Mrob> So the question is, does anyone have another explanation of how the necklace becomes unlinked, which might make it seem less of a paradox? The answer is probably here, but my mind's a little too foggy to get it: http://en.wikipedia.org/wiki/Alexander_horned_sphere http://en.wikipedia.org/wiki/Schoenflies_problem http://en.wikipedia.org/wiki/Knot_theory#Knotting_spheres_of_higher_dimensi… Other things I referred to: http://en.wikipedia.org/wiki/Simply_connected_space http://en.wikipedia.org/wiki/Cantor_set http://en.wikipedia.org/wiki/Lebesgue_measure http://en.wikipedia.org/wiki/Aleph_number#Aleph-naught On 12/6/12, Adam P. Goucher <apgoucher(a)gmx.com> wrote: [...] Theoretically, you could assemble these magic roundabouts recursively to produce an even larger clockwise one. http://en.wikipedia.org/wiki/Antoine%27s_necklace

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[math-fun] fusible numbers
by James Propp 07 Dec '12

07 Dec '12

I knew about the shoelace-timer problem, but I didn't know about fusible numbers. Very cool. But: There seems to be some controversy about their properties (see http://arxiv.org/abs/1202.5614) Can anyone tell me what the current situation is? Jim Propp

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Re: [math-fun] Antikythera device -- could it have been intended for navigation?
by Warren Smith 07 Dec '12

07 Dec '12

I finally talked myself into sending an email to the people behind the http://www.antikythera-mechanism.gr/ web site (who also have authored numerous articles about the device). Here's the email I sent them. I was stimulated by recent speculation the antikythera device was perhaps designed by Archimedes, seems to have predicted calandar, moon, sun, planets, and knew (via a "pin and slot drive" mechanism) an approximate version of Kepler's ellipse fact of non-uniform speed and non-circular motion. Then I found your web site http://www.antikythera-mechanism.gr/ and some popular press articles. What I want to begin exploring in this email is the speculation -- which I hardly saw anyplace -- that the device could have been intended by its designer for navigational use. On the website the only mention of this idea is the following disparaging quote by Martin Allen: "Another common suggestion is that it could have been an aid to navigation, especially as the ancient inhabitants of Rhodes were notorious for their navigational skills. However this seems unlikely as it would be an impractical device to use at sea, and seems excessive for the purposes of navigation. It would have been a more useful tool to astronomers, for assisting in making land based astronomical measurements. Indeed, with knowledge of eclipse timings it would assist in making measurements on a world scale, for example longitude. The first Mechanisms could have been built purely on the basis of seeing if it could be done. Indeed researchers are still making models of the mechanism today, to see if the theory can be put into practice. Unfortunately this does not apply in the case of the Antikythera Mechanism, as it is not thought to be the first of its kind." OK. First of all, nothing at all like this mechanism has ever been found. It is the unique exemplar. So I don't see how you know it is not the first of its kind. The only thing we know for sure is, it did not catch on and "stay invented." It instead was "lost" and its discovery millennia later in the shipwreck "stunned the world" hence I conclude it was really very very lost, not a soul had the faintest clue any such device ever existed. You cannot get more lost than that. So if it was not the first, it had to have been one of the nearly-first, for the simple reason that if this device had caught on big time with 1000 copies produced, then it would have been too well known to have become lost. So in fact it WAS the first, or nearly so -- Allen's claim just was wrong. Second, it was in roughly the right place and time that it may have been designed by Archimedes. (300 ton Roman ship guessed to have sunk 85 to 60 BC. Archimedes lived about 287 to 212 BC in Syracuse, from which it is believed the ship came.) So one might speculate that Archimedes or someone associated with him designed it and for a reason, and that reason was as an experimental navigation device for sea travel. The fact the planets would have been "excessive" is irrelevant -- this was an experimental device. The claim this would have "impractical for shipboard use" is based on nothing whatever that I can see. What the heck is impractical? See below for realistic practicality remarks. The experiment (we could continue speculating) was a failure and/or terminated by Archimedes' assassination and/or the theft of the device by the Romans and/or their ship's sinking, and hence the whole idea was lost, not to be rediscovered for 1700 years. The Greeks knew the Earth was round, and they knew how big the Earth was. Eratosthenes of Cyrene (lifespan about 276 to 194 BC, and a known friend of Archimedes) famously measured the Earth's circumference accurate to about 2% using trigonometry. This knowledge was later largely lost, for example Columbus famously had the idea, which was famously controversial, that the Earth was round, but he had a completely wrong idea of its size. Now much later, the British famously had trouble with sea navigation and in 1714 offered a huge monetary prize to anybody who could solve the longitude problem. This was solved by the invention of the first accurate chronometers by John Harrison during 1730-1767, and by 1825 their use was widespread. This was technically difficult and I would think beyond what the Greeks could have hoped to do pre-industrial era. At that time (1700) mainstream thought was that it would in fact be infeasibly difficult to design an accurate-enough chronometer, and therefore the top candidate method (which never was able to compete with the chronometer based method) was based on astronomy. The motions of the moon and planets in fact ARE a chronometer of more-than-sufficient accuracy, provided you can (a) predict and (b) measure them, and then (c) make the necessary computations to use the predictions and measurements to deduce your location. All of these things are not trivial, especially if the measurements have to be conducted accurately from a ship at sea, but they are possible. In fact, the chronometer-based method requires sextant measurements to be conducted from ship, and it was very successful for 100 years, and was even used by Ernest Shackleton from a 20 foot rowboat in stormy seas during a desperate 800 mile journey from Antarctica to S.Georgia in 1916. If it could be done from such a rowboat in those seas, it simply is flat out wrong to say it was impractical from much larger Ancient Greek and Roman ships (remember, the wreck it was found on was 300 tons) in the comparatively friendly waters of the Mediterranean. And while I am at it, let me say that a chronometer is a hell of a lot more delicate than the antikythera device, but nevertheless was practical for use, even from a 20 foot rowboat in 800 mile journey in worst seas in world. So once again, I flat out dispute any entirely cavalier and entirely unsupported assertion the antikythera device could not have been intended for ship navigation because it would have been "impractical for ship at sea." And since I am postulating device experimental, it was not even necessary, for me to be right, that it be practical anyhow. Now, the fact that Archimedes personally knew Eratosthenes who knew the Earth was round and measured it, using astronomical+trig methods, accurate to 2%, and the fact Archimedes who was the son of an astronomer and known to have constructed a brass "planisphere," also surely was aware of the "epicycles" celestial motion prediction methods known to Aristotle (384 - 322 BC) which are highly "gearlike" in nature, all lead to the suspicion Archimedes could well have thought of the idea of trying to do navigation via astronomy. To solve the prediction and perhaps some part of the computation problem, a gear-device, the antikythera, would hit the spot. The observation problem would be solved by use of a sextant or primitive analogue thereof. If there was such a device, it was lost, but there must have been devices of this nature for at least 50 years prior to Archimedes since otherwise it would have been simply impossible for the Greeks to have measured with enough accuracy to see the need for their theories of "epicycles." Also I point out Archimedes according to legend designed+built devices involving giant solar mirrors, suggesting sextant-like devices would not have been ridiculously far from his ken. I might also point out that there is a hell of a lot of economic incentive to navigate, and not much incentive to do astronomy. So, let us summarize, in view of that Martin Allen quote above about why this was not a navigation device: ALLEN CLAIM1. it would be an impractical device to use at sea, and seems excessive for the purposes of navigation. ANSWER1: these both refuted. ALLEN CLAIM2: It would have been a more useful tool to astronomers, for assisting in making land based astronomical measurements. Indeed, with knowledge of eclipse timings it would assist in making measurements on a world scale, for example longitude. ANSWER2: Who cares about astronomers? Which is more likely to be a hugely greater economic driving motivation -- astronomers or seamen? Wasn't Archimedes interested in practical engineering (pulley, lever, allegedly war weapons including sea-related ones) And anyhow I do not see that the two are incompatible, and am fully happy with the comment re longitude. However, the "eclipse timings" remark seems like largely bullshit to me, in the sense that those are not useful for much of anything since eclipses very rare and do not occur when & where you want, they occur when & where they want. ALLEN CLAIM 3. The first Mechanisms could have been built purely on the basis of seeing if it could be done. ANSWER3: well, again, this comment is fine with me, albeit it you in trying to use this argument are conflicting with your own statement about how this device was "thought not to be the first." So anyhow, in summary, the lame arguments by Martin Allen in that quote, all either support my hypothesis, or are refuted, and some to some extent are self-contradictory. They have no force against my hypothesis. Now what would it take to make a device good enough for navigation? Earth circumference is 25000 miles. To be accurate to 25 miles, which is good enough to be useful, then, you want accuracy of order <=1 part in 1000. So we'd want to measure some subset of the 6 angles between, e.g. sun, moon, fixed stars, and earth-horizon accurate to 1 part in 1000, i.e. 1/3 of a degree. Was this something Greeks could hope to do? I say "yes." Indeed, their knowledge of the calendar already was of that order of accuracy. Next, you want prediction accuracy for those angles from your antikythera device, to be accurate to <=1 part in 1000 or better. Was that something the Greeks could hope to accomplish? Again I think yes -- maybe not immediately, but after enough astronomical observation, record-keeping, and device development, then yes. And keep in mind the device could be re-set back to the truth whenever the ship came to a place like Greenwich where there were better astro-measuring devices and a known location on Earth. I.e. it did not require high accuracy for years, only for the duration of sea voyage <2 months at most, so error-growth thus was limited. To deduce latitude: measure angle(sun, horizon) at noon ("noon" means when min-angle maximized) and know the calendar day. That is all you need. Also measuring angle to North Star will do. To deduce longitude, you need the time and angles, and the time is in fact related to the angles by a mathematically known relationship hence need not be measured with any kind of mechanical clock. So... this all is hardly a proof, but I think is enough to make it an interesting and somewhat plausible speculation, that Archimedes or somebody of his ilk, built device to explore possible use for navigation. Arguments against this hypothesis have been confused, to say the least. This does not mean I'm right and they are wrong, but it does mean those counterarguments cannot be taken seriously at this time. If there were a pointer somehow related to "your longitude" on the device, that would clinch the case, but that might be too much to ask even if the whole hypothesis is true. Some of the broken-off pieces claimed by some to be for the planets -- were they really? How much evidence is there those really did planets? Maybe they had other functions related to longitude. (I have no idea, I'm just wondering. If you do not know the gear ratios corresponded to planet-period ratios, then we are free to speculate other speculations.) PS. 1. As far as I can tell, there is no convincing evidence the other stuff was planets, except perhaps for Mars (which incidentally would have been the planet most useful for navigation). 2. Wikipedia has this: "The device is rather small, indicating that the designer was aiming for compactness and, as a result, the size of the front and back dials is unsuitable for public display. A simple comparison with the size of the Tower of the Winds in Athens would suggest that the Antikythera mechanism manufacturer designed the device for mobility rather than public display in a fixed location."

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[math-fun] Gamow syndrome (Aleph_0 and omega)
by Robert Munafo 07 Dec '12

07 Dec '12

On 12/6/12, Adam P. Goucher <apgoucher(a)gmx.com> wrote: > The limit of the series is the *ordinal* omega. You're only allowed > to use aleph-null to refer to the sizes of sets. Thanks for the pedantry Adam, this is math and we need the precision. I am one of the many sufferers of George Gamow's over-simplifications (or some would say outright falsehoods) such as this howler: http://mrob.com/pub/math/images/gamow-123i-p23-fig8.jpg Aleph_1 is identified as "the number of all geometrical points on a line", and the caption called Aleph_0, Aleph_1 and Aleph_2 "the first three infinite numbers" (which implies there are no other infinite numbers in between). > Yes, this is one of those things which are obviously false but > actually true. Then there are things which are obviously either > true or false, but turn out to be equivalent to the continuum > hypothesis instead. I suppose that's what got Gamow. -- 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] rubik's cube, a very special way to look at it.
by Simon Plouffe 07 Dec '12

07 Dec '12

Hello, maybe some of you have seen this amazing video : http://www.youtube.com/watch?v=kN5_1vJbW9M&feature=player_embedded which comes from this : http://www.gravitation3d.com/magictile/ ps : have fun. Simon Plouffe

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[math-fun] Antoine's necklace
by Robert Munafo 06 Dec '12

06 Dec '12

Adam's mention of Antoine's necklace reminded me of a topology question I've had for a while. According to the Wikipedia description (which is admittedly basic, but agrees with my own perception of the situation), the necklace remains linked through all finite stages of the construction, yet in the limit (of the countably infinite number of steps) what remains is a set of single points. That's the "paradox" (which I know is just my limited perception) : 1. Since single points are not tori, it seems that the necklace is a set of disjoint points, so the necklace is no longer linked. 2. But the description clearly says that the set's complement is not simply connected, which implies that the necklace at stage Aleph_0 is still linked. Transitions from a finite property to an infinite property appear everywhere. For example, every term in the sequence 1, 2, 3, 4, ... (the positive integers) is finite, but the limit of the series (Aleph_0) is infinite. I can grasp that. Similarly, the normal (1-dimensional middle-thirds) Cantor set transitions from being a set of intervals with nonzero measure to a set of points. That doesn't seem to bother me either. But the necklace one bothers me. That transition from linked set of tori to unlinked set of points seems impossible. So the question is, does anyone have another explanation of how the necklace becomes unlinked, which might make it seem less of a paradox? The answer is probably here, but my mind's a little too foggy to get it: http://en.wikipedia.org/wiki/Alexander_horned_sphere http://en.wikipedia.org/wiki/Schoenflies_problem http://en.wikipedia.org/wiki/Knot_theory#Knotting_spheres_of_higher_dimensi… Other things I referred to: http://en.wikipedia.org/wiki/Simply_connected_space http://en.wikipedia.org/wiki/Cantor_set http://en.wikipedia.org/wiki/Lebesgue_measure http://en.wikipedia.org/wiki/Aleph_number#Aleph-naught On 12/6/12, Adam P. Goucher <apgoucher(a)gmx.com> wrote: > [...] Theoretically, you could assemble these > magic roundabouts recursively to produce an even larger clockwise one. > > http://en.wikipedia.org/wiki/Antoine%27s_necklace -- 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] Antoine's necklace
by Adam P. Goucher 06 Dec '12

06 Dec '12

> That's the "paradox" (which I know is just my limited perception) : > 1. Since single points are not tori, it seems that the necklace is a > set of disjoint points, so the necklace is no longer linked. > 2. But the description clearly says that the set's complement is not > simply connected, which implies that the necklace at stage > Aleph_0 [sic, see below] is still linked. I like to refer to these as 'infinitely complex dust clouds'. I have some more familiar examples. The rationals are 'disjoint points', but they can't be separated; it's impossible to continuously deform the real line to introduce a nice empty space between the negative rationals and positive rationals. I guess that's the most familiar example, so hopefully the most intuitive. > Transitions from a finite property to an infinite property appear > everywhere. For example, every term in the sequence 1, 2, 3, 4, ... > (the positive integers) is finite, but the limit of the series > (Aleph_0) is infinite. The limit of the series is the *ordinal* omega. You're only allowed to use aleph-null to refer to the sizes of sets. http://en.wikipedia.org/wiki/Ordinal_number http://en.wikipedia.org/wiki/Cardinal_number Sorry to appear pedantic, but it's an important distinction. The naturals and the fusible numbers are both equally large in terms of cardinal size (you can biject between them) -- they have cardinality aleph-null. But the naturals have order type omega (they can be bijected in a nice, order-preserving way with the set of ordinals below omega, namely the naturals themselves!), whereas the fusible numbers have order type epsilon-nought (they can be bijected beautifully to the ordinals below epsilon-nought such that order is preserved). > I can grasp that. Similarly, the normal > (1-dimensional middle-thirds) Cantor set transitions from being a set > of intervals with nonzero measure to a set of points. That doesn't > seem to bother me either. > > But the necklace one bothers me. That transition from linked set of > tori to unlinked set of points seems impossible. Yes, this is one of those things which are obviously false but actually true. Then there are things which are obviously either true or false, but turn out to be equivalent to the continuum hypothesis instead. Sincerely, Adam P. Goucher http://cp4space.wordpress.com

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[math-fun] Period of antikythera device
by Robert Munafo 06 Dec '12

06 Dec '12

On 12/6/12, Henry Baker <hbaker1(a)pipeline.com> wrote: > Does anyone here know what the fundamental _period_ of the antikythera > device is? > > I.e., by all reports, this device is a "planetarium in a box". So how many > years of simulated time does it take for this device to return to its > initial configuration ? 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 Apart from the sun position, which is a base unit of your question ("how many years..."), the known/proven functions are moon position and phase, saros x4, eclipse prediction, and exeligmos. All of the other stuff (metonic, olympiad, planets, etc.) is speculative and/or based on gears of unknown size. But the stuff we know for sure includes these named gears: b1=b2 = years c1=c2 = (64/38) b2 d1=d2 = (48/24) c2 e1=e2=e3=e4 = (127/32) d2 f1=f2 = (188/53) e4 g1=g2 = (30/54) f2 (saros x4 scale) h1=h2 = (20/60) g2 i1 = (15/60) h2 (exeligmos dial) The number of teeth in gears l1, ma1, mb1, mb2 isn't shown and there are more gears that are known but are not linked to any other known gears, so there must be additional unknown gears (a lot of which are the subject of the paper). There is also the known "pin-and-slot epicyclic lunar mechanism" which implements the first anomaly of the moon without affecting its rotational period. Nevertheless we can at least start to answer the question by looking at the known ratios above. The most complex fraction (highest value of numerator*denominator) is h1=h2 = 238760/27189 years: that gear turns 238760 times each 27189 years. The other denominators are all factors of 27189, so the overall cycle for those known gears is 27189 years. The answer to your question (period of the entire device) would have to be a multiple of 27189. > [...] Also, if the antikythera device really is a "planetarium in a box", and if > it really does do a good job of modeling the known universe at the time, how > come we can't tell _the exact date_ of its demise ? > [...] > This is essentially equivalent to being able to determine the exact time of > an earthquake from the position of the hands of clocks which stopped as a > result of the earthquake. The pointers (marking the positions on the scales and dials) were lost, so we don't know what moon position, date, year, etc. was displayed on it when the ship sank. In other words, the hands of the clocks came off the dials because the earthquake was really strong. Maybe I don't understand your question. -- 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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