November 2012 - mathfun@mailman.xmission.com

[math-fun] cycloidal clock, pendulum sci=fi, other nuttery
by Warren Smith 26 Nov '12

26 Nov '12

> From: Henry Baker <hbaker1(a)pipeline.com> > cycloidal > escapement for a clock. > > The problem is that the rolling circle has to roll on the _ceiling_, not on > the floor, according to Lawlor's Brachistochrone paper. > > Ok, so we put a "rack" (a linear gear) on the ceiling & parallel to the > ceiling. > > Then we _hang_ a gear wheel from a rail which runs parallel to the ceiling. > The hanger has a small rolling mechanism similar to that in a "sliding" > door. > > The gear teeth of the hanging gear wheel engages the teeth in the linear > gear rack on the ceiling. > > Now, if we push the entire gear wheel slide forward, the engaged gear teeth > will rotate the gear wheel. > > So far, so good. > > We now make the wheel extremely light with as low a moment of inertia as > possible -- e.g., magnesium. --is this necessary? To be elegant you want it to work even for a nonzero-mass wheel. Would that be mathematically possible? > We also make the sliding hanger mechanism as light as possible, with as > little friction as possible. --unfortunately, in practice I think the friction would be far larger than just using a plain pendulum. Ditto for the hanging rope (rope confined by cycloidal walls) scheme. > We then mount a lead weight on the gear wheel. > > With small-to-medium displacements of the wheel, the lead weight will now > move in a cycloidal path, similar to Huygens's path. --why "small to medium"? Why not "all"? > We can now arrange for some other mechanism (light actuated, e.g.) which > notices when the amount of motion becomes too small, and gives the wheel a > small additional kick to add energy. > > This is what I came up with in a few minutes. --another attempt to rebuild obsolete technology. Somebody in England actually did build what he claimed was the world's most accurate pendulum clock, using pendulum swinging low-amplitude in vacuum, photosensor "escapement" with electromagnet kicker, old-invar rod. Whole thing built inside a special hut with heavy concrete foundation, and active temperature regulation using "box inside a box" 2-level temperature control. Pointless exercise, really, but he enjoyed it and offered evidence from comparison with atomic clocks he'd indeed achieved greater accuracy than any previous pendulum clock, and also he was able using it to detect the motion of the moon, which might make a good sci-fi story -- you know, the alien race trapped on planet shrouded in perpetual fog, nevertheless detects the existence of moon via pendulum clocks. Oh wait, good sci-fi stories also are obsolete. QUESTION WHICH ACTUALLY HAS A SLIGHT CHANCE OF MATTERING: Could this kind of detector be used to detect gravity waves?

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Re: [math-fun] antikytheras & cycloids
by Henry Baker 26 Nov '12

26 Nov '12

You're correct on both counts. Lawlor's Figure 4 shows the pendulum string coming down from between two upward curvature cycloids. But in the spirit of the antikytheras, I was trying to come up with something involving _gears_. At 06:39 PM 11/25/2012, Allan Wechsler wrote: >The oldest solution to this problem is to make the pendulum very long, >and have it describe a very small angle, so that the system is locally >linear and the period is independent of amplitude for all practical >purposes. Furthermore, every clock will turn out to have a preferred >pendulum amplitude, so it's really only important to get the time >constant right for that amplitude. > >There is another solution, which is to make the pendulum arm of a >flexible material, like a thin wire or string, and to confine its >upper reaches between two convex-downward guides which come to a cusp >at the fulcrum. This has the effect of shortening the arm for large >excursions. The correct guide shape will direct the pendulum bob to >trace a cycloid. I don't remember the correct shape: it's something >simple, like a circle or another cycloid. But it doesn't matter how >complicated the guide shape is: it's fairly easy to carve or bend it >to within tolerance by trial and error. It turns out not to be worth >the effort in increased accuracy, though.

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[math-fun] antikytheras & cycloids
by Henry Baker 26 Nov '12

26 Nov '12

All this talk about antikytheras led me to think about a cycloidal escapement for a clock. The problem is that the rolling circle has to roll on the _ceiling_, not on the floor, according to Lawlor's Brachistochrone paper. Ok, so we put a "rack" (a linear gear) on the ceiling & parallel to the ceiling. Then we _hang_ a gear wheel from a rail which runs parallel to the ceiling. The hanger has a small rolling mechanism similar to that in a "sliding" door. The gear teeth of the hanging gear wheel engages the teeth in the linear gear rack on the ceiling. Now, if we push the entire gear wheel slide forward, the engaged gear teeth will rotate the gear wheel. So far, so good. We now make the wheel extremely light with as low a moment of inertia as possible -- e.g., magnesium. We also make the sliding hanger mechanism as light as possible, with as little friction as possible. We then mount a lead weight on the gear wheel. With small-to-medium displacements of the wheel, the lead weight will now move in a cycloidal path, similar to Huygens's path. We can now arrange for some other mechanism (light actuated, e.g.) which notices when the amount of motion becomes too small, and gives the wheel a small additional kick to add energy. This is what I came up with in a few minutes. Perhaps someone here can come up with a more elegant version.

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[math-fun] complex hermitian matrix
by Warren Smith 25 Nov '12

25 Nov '12

--I'm not buying Veit's claims: The definition of "random hermitian matrix" is key. If I take it to mean "pick the eigenvalues from a symmetric real prob.distrib. then multiply resulting diagonal matrix D to get H = U^(-1) D U = hermitian with U random unitary (Haar measure) -- which is a perfectly good probability distribution -- then the proposed solution prob(all eigevalues>0) = 2^(-N) is right, regardless of what Veit says. --Veit: The claim 2^(-N) assumes much more, that the eigenvalues are independently distributed. In fact, they are very far from independent, making the actual probability much smaller: c^(-N^2). This result is asymptotic, for large N (the limit of interest in statistical mechanics). The number c is well known, but it is not 2. As Andy pointed out, the probability measure should be invariant under arbitrary unitary transformations, i.e. M -> U M U^(-1). But the Hermitian matrices live in an N^2 dimensional space while the space of unitary matrices has only N(N-1) dimensions. The extra N dimensions correspond to the eigenvalues of M. Wigner had the idea of using the maximum entropy probability distribution, constrained by just two properties: the expectation values of Tr M and Tr M^2. If we want the expectation value of Tr M to be zero, then our probability distribution is simply the Gaussian e^(-Tr M^2) times the unitary-invariant measure. --that would yield 2^(-N) as above! If you marginalize this distribution on just the eigenvalues (i.e. integrate out the unitary transformations) you get, say in the case of N=3 (unnormalized) dP = e^(-E1^2-E2^2-E3^2) (E1-E2)^2 (E2-E3)^2 (E3-E1)^2 dE1 dE2 dE3. It's the product over all eigenvalue pairs -- their differences squared -- that ruins the independence of the eigenvalue distribution. BTW, this very same distribution seems to perfectly model the distribution of Riemann zeta function zeros, but nobody understands why!

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[math-fun] Antikythera mechanism -- navigation device?
by Warren Smith 25 Nov '12

25 Nov '12

Thinking some more, it is probably unreasonable to measure angles to sun, moon, horizon accurate to 1 part in 1000 or 10000 from deck of an ancient ship at sea. (Also probably unreasonable to observe moons of Jupiter under same conditions, had they even known about them at the time.) If so, then the antikythera device would not have been a useful ship-navigation device, but might perhaps have been an experimental prototype for one, perhaps even invented by Archimedes. That would explain why only one such device has ever been found and why they clearly were not heavily used. This theory all is a total fantasy, but not necessarily wrong. It would have been a feasible idea, in principle, to do ship navigation using moon+sun predictions vs observations. Later, it actually was feasible, and actually done, to navigate using chronometers and sun observations with sextants, but this was not easy. First, building accurate chronometers was difficult. Second, making accurate sextant observations from a moving ship, is also pretty damn difficult, harder the smaller the boat. Third, it helps to have good charts.

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[math-fun] Antikythera mechanism -- navigation device?
by Warren Smith 25 Nov '12

25 Nov '12

Let me do a crude initial look (pending response form those who actually know about the a.mechanism...) at the question of whether the antikythera mechanism could have been used for navigation. I'll ignore the planets and just do the moon+sun, since apparently they do not understand what the mechanism was doing about planets and anyhow those seem much less useful (especially if you are unaware of the moons of Jupiter, etc, which could have been used as a timekeeping device if you were aware of & capable of observing them, which nobody was until the time of Galileo). Suppose you are on an N-day sea voyage. You want to know your latitude and/or longitude. Apparently the magnetic compass was not in any wide use until about the year 1100 AD, so was not available to ancient Greeks (although there seems to be documentary evidence some Chinese knew the principle much earlier). Also, at that time there were no accurate clocks. But you could observe the angle between the sun & moon in the sky, and also the angle of same to the horizon. If you could accurately predict these, you could infer your longitude. Also, if you could predict angle of sun to horizon at noon ("noon" means max-angle time) versus day-of-year, then you could tell your latitude. The accuracy is limited by (i) prediction accuracy and (ii) angle measurement accuracy. If both are assumed accurate to 1 part in 1000 then you could navigate accurate to about 0.001 earth-circumference, which is 25 miles. That seems good enough to be useful. If 0.0001 accuracy then 2.5 miles. So it it not inconceivable, at least as far as this initial crude look is concerned, that the device could have been intended and used for navigation.

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[math-fun] The antikythera device ... ellipses?
by Guy Haworth 25 Nov '12

25 Nov '12

If I remember the TV program aright, there was a peg and slot mechanism in the antikythera device to effect elliptical rather than circular motion. Does the Lego model include this feature? Regards - Guy

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[math-fun] brachistochrone for rolling ball
by Warren Smith 25 Nov '12

25 Nov '12

The underlying formula for a sliding frictionless bead was V = sqrt(2*h*g) where g=gravity acceleration, h=vertical height dropped so far, V=speed. For a ball (or other rotationally symmetric) rolling object with zero slip, a constant fraction of the energy is in rotational, and a constant fraction in translational, energy. Hence V=const*sqrt(2*h*g). This is the same formula, just with an altered value of g. Therefore, I conclude the brachistochrone, isochrone, etc will have unaltered (still cycloid) shape, although of course you want for the center of mass of the ball to move along the cycloid, not the contact point of ball with track -- hence track would need to be a "1-radius-expanded" version of cycloid. It is unnecessary to make balls have foamy outside and dense inside to reduce their moment of inertia. Any MoI is fine. Galileo was not aware of brachist. or iso property of cycloid, but he was aware of, or at least conjectured, cycloid as best possible shape for thin beam arch (constant beam width) -- which I have discussed here, the cycloid has uniform compressive stress but not uniform bending stress. Apparently Galileo invented the cycloid. The notion of "uniform stress" is something that can be defined without needing calculus -- probably for somebody as ingenious as Galileo, pure Euclidean geometry would suffice. (Archimedes similarly computed area & volume of sphere without needing calculus.)

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Re: [math-fun] Antikythera mechanism
by Warren Smith 25 Nov '12

25 Nov '12

> From: "Adam P. Goucher" <apgoucher(a)gmx.com> > And, in a guiltless act of self-promotion, you may want to explore my > Wolfram Demonstration of the Antikythera Mechanism: > > http://demonstrations.wolfram.com/AntikytheraMechanism/ > > You can view both the exterior and the internal mechanism, up to and > including the entire known arrangement of gears. Detailed information > about the demonstration and how it was created is on cp4space: > > http://cp4space.wordpress.com/2012/09/19/antikythera-mechanism/ --how accurate are the predictions from this mechanism vs reality? Could it have been useful for navigation (it was found in an ancient shipwreck)?

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Re: [math-fun] antikythera device: an ancient computer
by Adam P. Goucher 25 Nov '12

25 Nov '12

And, in a guiltless act of self-promotion, you may want to explore my Wolfram Demonstration of the Antikythera Mechanism: http://demonstrations.wolfram.com/AntikytheraMechanism/ You can view both the exterior and the internal mechanism, up to and including the entire known arrangement of gears. Detailed information about the demonstration and how it was created is on cp4space: http://cp4space.wordpress.com/2012/09/19/antikythera-mechanism/ Sincerely, Adam P. Goucher > ----- Original Message ----- > From: Robert Baillie > Sent: 11/25/12 04:48 PM > To: math-fun > Subject: [math-fun] antikythera device: an ancient computer > > a while back, someone posted a link to this youtube video of the antikythera > device made out of legos: > http://www.youtube.com/watch?v=RLPVCJjTNgk > > here is an interesting (58 minute) bbc documentary that explains in more detail > how the device actually works, as far as we know today: > http://www.youtube.com/watch?v=-rUsgL2VGeU > > many of the gears had a prime number of teeth. this was very helpful in > figuring out what the device was used for, and how it worked. > > here are several shorter videos that might also be of interest: > The Antikythera Mechanism - 2D (8 minutes) > http://www.youtube.com/watch?v=UpLcnAIpVRA > > The Antikythera Mechanism - 3D (4 minutes) > http://www.youtube.com/watch?v=L1CuR29OajI > > Antikythera mechanism working model.mov (4 minutes) > http://www.youtube.com/watch?v=4eUibFQKJqI > > in case anyone passes through bozeman, montana, there is a great computer museum > that has a replica of the antikythera device. the museum also has an old ibm > 1620 computer, a curta calculator, and many other neat old things: > http://www.compustory.com/ > note: their rather primitive website doesn't do the place justice! > > bob baillie > > _______________________________________________ > math-fun mailing list > math-fun(a)mailman.xmission.com > http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun

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