December 2010 - mathfun@mailman.xmission.com

Re: [math-fun] Newton's cradle
by Dan Asimov 11 Dec '10

11 Dec '10

I've long assumed that, ignoring quantum effects, at least Newtonian mechanics is a continuous function of its inputs. Including things like steel balls bouncing off each other. (Of course, as RWG points out, a small error like an off-center collision can propagate into a large error. But in this theoretical problem we are assuming no built-in error like this.) Is it really true that for a triple collision in the plane the result depends on which sequence of double collisions is used to calculate it? (Even in the limit as the discrepancy from a perfect 3-way collision approaches 0 ?) --Dan << I've never been convinced that these problems are even well-defined, for the same reason that a simultaneous collision between three bodies in the plane is ill-defined. In that case, it's immediately obvious that you get completely different results according to how you decompose into a sequence of 2-body collisions. >> Those who sleep faster get more rest.

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[math-fun] FW: why did everyone (except Bill) get this wrong?
by Cordwell, William R 11 Dec '10

11 Dec '10

[posted per request] I thought of it as all of the balls having very slight gaps between them. The initial, heavy, ball hits the first stationary ball in the array, knocking it forward with 4/3 of the initial ball's speed, and the initial ball continues forward more slowly (at 1/3 its initial speed). The first ball in the array acts like an initial ball on the (n-1) remaining members of the array, performing rapid (with almost unnoticeable delay) consecutive (almost) elastic collisions, propelling the final ball in the array out with the 4/3 v0. The initial, heavy, ball, catching up to the now stationary first ball, does the same thing, but starting with the reduced 1/3 v0. It should give a nice, simple geometric series for the total kinetic energy (and for the momentum, as well). Since, when all of the balls are the same mass, and if two balls are launched towards the remaining array, then two balls come off, there must be, effectively, a small gap between the initial pair. It would be interesting to know if the gap is actual, or if the shape (so that the balls touch at almost just a single point) allows compression/behavior equivalent to having a gap. Nice problem! Thanks, Bill -----Original Message----- From: gantonick(a)gmail.com [mailto:gantonick@gmail.com] On Behalf Of Gary Antonick Sent: Thursday, December 09, 2010 7:53 PM To: Cordwell, William R Subject: Re: [math-fun] why did everyone (except Bill) get this wrong? Hi Bill, Your guess is right on. You're not in the group of 20. Since you got it right.. how did you think about the problem? Can you see why the other guys went astray? - Gary On Thu, Dec 9, 2010 at 2:26 PM, Cordwell, William R <wrcordw(a)sandia.gov> wrote: > Dear Gary, > > So, if I were somewhat guessing, I'd say that the end ball comes off with speed 4/3 v0, followed almost immediately by the 2nd-to-the-end ball at maybe 4/9 v0, etc. > > Am I in with the group of 20? > > Thanks, > Bill C. > > -----Original Message----- > From: math-fun-bounces(a)mailman.xmission.com [mailto:math-fun-bounces@mailman.xmission.com] On Behalf Of Gary Antonick > Sent: Thursday, December 09, 2010 10:44 AM > To: math-fun > Subject: [math-fun] why did everyone (except Bill) get this wrong? > > Hi all, > > I recently posed a simple mechanics question to a bunch of university > faculty. Pretty much everyone got it wrong. > > What I'm wondering is.. why? Am trying to explore this a bit. Might > make an interesting story. > > Here's the question. You have a Newton's Cradle. The first ball has > twice the mass it normally would. You swing it so it hits the second > ball. What happens? > a. the last ball flies out > b. the last two balls fly out > c. something else > > Turns out the answer is c. > > No one gets this right. OK. Bill Gosper got it right. In fact, he's > completely unraveling the problem. And Neil Bickford was essentially > there. Maybe I should have asked more people in math-fun. > > But I asked 20 university faculty (physics and engineering, mainly) > and they all got it wrong. Except one guy who had to simulate it > first. > > What's going on? > > - Gary > > _______________________________________________ > math-fun mailing list > math-fun(a)mailman.xmission.com > http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun > > >

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[math-fun] why did everyone (except Bill) get this wrong?
by Gary Antonick 10 Dec '10

10 Dec '10

Hi all, I recently posed a simple mechanics question to a bunch of university faculty. Pretty much everyone got it wrong. What I'm wondering is.. why? Am trying to explore this a bit. Might make an interesting story. Here's the question. You have a Newton's Cradle. The first ball has twice the mass it normally would. You swing it so it hits the second ball. What happens? a. the last ball flies out b. the last two balls fly out c. something else Turns out the answer is c. No one gets this right. OK. Bill Gosper got it right. In fact, he's completely unraveling the problem. And Neil Bickford was essentially there. Maybe I should have asked more people in math-fun. But I asked 20 university faculty (physics and engineering, mainly) and they all got it wrong. Except one guy who had to simulate it first. What's going on? - Gary

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[math-fun] Antikythera Mechanism in Lego
by Henry Baker 10 Dec '10

10 Dec '10

FYI -- The idea from "epicycles" may have come from mechanical model implementation issues, rather than the other way 'round. http://www.guardian.co.uk/science/punctuated-equilibrium/2010/dec/10/1 The Antikythera Mechanism Two years ago, a paper was published in Nature describing the function of the oldest known scientific computer, a device built in Greece around 100 BCE. Recovered in 1901 from a shipwreck near the island of Antikythera, this mechanism had been lost and unknown for 2000 years. It took one century for scientists to understand its purpose: it is an astronomical clock that determines the positions of celestial bodies with extraordinary precision. In 2010, a fully-functional replica was constructed out of Lego. For those of you who, like me, wish to learn more about this ancient computer, here's a video interview that documents how scientists went about examining this device. After a lot of study, a group of scientists determined that the Antikythera Mechanism could predict eclipses, and that it had a dial to record the dates of the ancient Olympiads. The 2,000-year-old box of intricate gearwork provides a glimpse of the engineering prowess of the Hellenic world. The team discuss their results here, in this video: Part 1: This video is from 11 December 2008 (part 1). Nature has its own YouTube Channel. Part 2: This video is from 11 December 2008 (part 2). Nature has its own YouTube Channel. Source: Marchant, J. (2010). Ancient astronomy: Mechanical inspiration. Nature, 468 (7323), 496-498 DOI: 10.1038/468496a http://dx.doi.org/10.1038/468496a http://www.nature.com/news/2010/101124/full/468496a.html --- Published online 24 November 2010 | Nature 468, 496-498 (2010) | doi:10.1038/468496a Ancient astronomy: Mechanical inspiration The ancient Greeks' vision of a geometrical Universe seemed to come out of nowhere. Could their ideas have come from the internal gearing of an ancient mechanism? Jo Marchant Two thousand years ago, a Greek mechanic set out to build a machine that would model the workings of the known Universe. The result was a complex clockwork mechanism that displayed the motions of the Sun, Moon and planets on precisely marked dials. By turning a handle, the creator could watch his tiny celestial bodies trace their undulating paths through the sky. The mechanic's name is now lost. But his machine, dubbed the Antikythera mechanism, is by far the most technologically sophisticated artefact that survives from antiquity. Since a reconstruction of the device hit the headlines in 2006, it has revolutionized ideas about the technology of the ancient world, and has captured the public imagination as the apparent pinnacle of Greek scientific achievement. Now, however, scientists delving into the astronomical theories encoded in this quintessentially Greek device have concluded that they are not Greek at all, but Babylonian Â an empire predating this era by centuries. This finding is forcing historians to rethink a crucial period in the development of astronomy. It may well be that geared devices such as the Antikythera mechanism did not model the Greeks' geometric view of the cosmos after all. They inspired it. The remains of the Antikythera mechanism were salvaged from a shipwreck in 1901 (see 'Celestial mirror from the deep') and are now held in the National Archaeological Museum in Athens. A series of ever more sophisticated radiographic studies of the gearwheels hidden inside the corroded mass culminated in proposed reconstructions of the device from a team led by astronomer Mike Edmunds of the University of Cardiff, UK, in 2006 (ref. 1), and from London-based mechanic and curator Michael Wright in 2007 (ref. 2). The largest surviving piece of the Antikythera mechanism contains some of its internal workings, including one of the major gearwheels and several smaller gears that have been crushed underneath. The device, which dates from the second or early first century BC, was enclosed in a wooden box roughly 30 centimetres high by 20 centimetres wide, contained more than 30 bronze gearwheels and was covered with Greek inscriptions. On the front was a large circular dial with two concentric scales. One, inscribed with names of the months, was divided into the 365 days of the year; the other, divided into 360 degrees, was marked with the 12 signs of the zodiac. Pointers moving around this dial were thought to show the date as well as the corresponding position of the Sun, Moon and probably the five planets known at the time. A revolving ball, painted half black and half silver, displayed the phase of the Moon, and letters marked on the zodiac scale acted as a kind of index, linking to inscriptions that described the appearances and disappearances of major stars at different times of the year. On the back of the device were two spiral dials, one above the other. The top one showed a repeating 235-month calendar, popular because after 235 months or 19 years, the distribution of new Moons in the solar year is the same. The bottom spiral represented a 223-month repeating eclipse cycle. Symbols inscribed on its month divisions told the user when to expect eclipses, and gave information about the type and timing of each event. The researchers who did the 2006 reconstruction noted that the 235- and 223-month cycles were originally derived by the Babylonians. That was to be expected: the empire's priest-astronomers, who saw astronomical events as powerful omens, had identified many such cycles over the centuries, and Greek astronomers of this period often made use of their results. But this fact did not shake the researchers' central conclusion that the device embodied the Greeks' own geometrical models of the cosmos. These models, based on spheres or circles that described the motion of the planets in three-dimensional space, had originally been qualitative and philosophically pleasing rather than accurate. But by the time of the Antikythera mechanism's construction, scholars such as Hipparchus, who worked in Rhodes in the second century BC, had been inspired by the Babylonians' precision to put numbers into the Greek models, and to insist that they fit with actual observations. Modern experts were confident that the device's zodiac display Â its centrepiece Â reflected such state-of-the-art geometrical theories. Supporting this idea were X-ray scans revealing that a mechanism hidden within the device's clockwork directly modelled the varying motion of the Moon. Because the Moon's orbit around Earth is elliptical rather than circular, it seems to travel faster at some points in its orbit than others. Greek philosophers believed that all heavenly orbits were perfect circles, so Hipparchus explained this variation in the Moon's motion by superimposing one circular orbit onto another that had a different centre Â the 'eccentric' theory. Missing gears The gearwork in the Antikythera mechanism seems to put this into practice perfectly, using a pin-and-slot mechanism that enabled one gearwheel to drive another around a slightly displaced axis. In their 2006 Nature paper1, Edmunds and his colleagues described it as "a mechanical realization" of Hipparchus' lunar theory. The team and other scholars assumed that the maker of the Antikythera mechanism must have used similar techniques to model the path of the Sun and probably the planets, too. The relevant gearing is missing, but the assumption is plausible: Greek astronomers accounted for the movement of planets Â which not only seem to speed up and slow down in the sky, but sometimes change direction Â using a theory that was mathematically equivalent to the eccentric model. The basic idea, which would be refined and made famous by the Greek astronomer Ptolemy in the second century AD, was that each planet travelled in a small circle called an epicycle, whose centre was simulta neously moving in a larger loop around Earth. To demonstrate how the Antikythera mechanism could have operated, Wright built a working model of it. His device incorporates small gear wheels riding on larger ones to model the epicycles of Mercury, Venus, Mars, Jupiter and Saturn, as well as the varying speed of the Sun. The Antikythera mechanism thus seemed to be a stunning demonstration of how the ancient Greeks had translated their most famous astronomical theory into physical wheels of bronze. But now a new team has noticed a detail that could turn this view of the mechanism on its head. Historian of astronomy James Evans at the University of Puget Sound in Tacoma, Washington, and his colleagues knew that the 360 divisions on the zodiac scale should be spaced slightly farther apart than the 365 divisions on the calendar scale that encircled it. But when they used X-ray scans provided by Edmunds' team to precisely measure the division widths on the surviving part of the dial, which encompasses 88 degrees, they found that the zodiac marks are actually closer together3. The marks on the vanished parts of the scale must have compensated somehow with a wider spacing. The researchers believe that this was done on purpose to represent the Sun's uneven progress through the sky. Instead of the device using epicyclic gearing to drive a pointer with varying speed as previously thought, Evans believes it is "extremely likely" that its maker used a pointer moving with constant speed around a circle split into two sections of equal overall size that were divided differently: a 'fast zone' in which the degree markings were closer together than normal, and a 'slow zone' in which they were farther apart. This scheme is identical to a theory of the Sun's movement used by the Babylonians, known as System A. If correct, this interpretation suggests that the astronomy encoded in the mechanism's gearwork does not represent state-of-the-art Greek theories after all. It is Babylonian through and through. Click for larger image This is a tough assertion to prove. The uneven division of the zodiac scale could have been just the result of sloppy work by the machine's creator, and its similarity to the Babylonian scheme just a coincidence. Wright, who was the first to suggest that epicyclic gearing modelled the motions of the Sun and planets, says he is "very uncomfortable" with the idea that the device modelled the Moon's motion mechanically, yet used an abstract numerical scheme to do the same for the Sun. But astronomy historian Alexander Jones of the Institute for the Study of the Ancient World in New York is taking the hypothesis seriously. He argues that Greek astronomers were more interested in convenience than consistency. Such an intimate mix of geometric and arithmetic approaches fits the spirit of the period, he says. "They were playing with different toolboxes at the same time." Key events Evans's hypothesis forces a rethink of other parts of the mechanism, too. Previously, scholars assumed that the positions of the Sun, Moon and planets were all displayed around the same zodiac scale. But if the zodiac scale had been tweaked to accommodate the varying speed of the Sun, it would no longer be accurate for showing the positions of the other bodies. Evans thinks that the five planets were instead displayed on individual smaller dials (see 'Cross-cultural computer'). He adds that these dials didn't necessarily have to show the planets' positions in the sky. He thinks the machine's maker would have been more interested in showing the timing of key events in each planet's cycle, such as changes of direction, or first and last appearance in the night sky. If so, the pointers on these dials could have been driven at constant speed by simple gear trains representing period relations derived by the Babylonians Â no epicycles required. Jones is more cautious about this suggestion, although he says it "makes sense in terms of how planetary motion was talked about at the time". Both he and Evans are hoping that more clues will come from inscriptions on the front cover of the mechanism. The surviving lettering is hidden inside the mechanism's battered and corroded remains, but it is being painstakingly reconstructed and translated from X-ray scans by Agamemnon Tselikas, director of the Centre for History and Palaeography in Athens, and Yanis Bitsakis, a physicist at the University of Athens. So far the two researchers have deciphered mentions of Mars, Mercury and Venus, along with several references to the 'stationary points' at which planets seem to change direction. Evans argues that even the clearly epicyclic gearing of the Moon display may model Babylonian arithmetic, not Greek geometry. The amplitude of the variation encoded by the pin-and-slot mechanism is larger than that used by Hipparchus in his eccentric model, he points out, and is closer to the amplitude used in the lunar algorithms of the Babylonians. "Perhaps a mechanic tried to represent the variations in the Moon's speed according to the Babylonian theory using gears," he says Â and hit upon an epicyclic arrangement. In other words, epicycles were not a philosophical innovation but a mechanical one. Once Greek astronomers realized how well epicyclic gearing in devices such as the Antikythera mechanism replicated the cyclic variations of celestial bodies, they could have incorporated the concept into their own geometrical models of the cosmos. "It is a new possibility," says Jones. "I am quite attracted to it." There is little evidence for who came up with the idea of epicycles, although it is often ascribed to third-century-BC Greek geometer Apollonius of Perga. Intriguingly, gears and epicycles seem to have arisen at about the same time, with gears perhaps a little earlier. Also in the third century BC, Archimedes used simple gears to change the size of an applied force. Some two centuries later, the Roman politician and author Cicero wrote that Archimedes built a bronze astronomical device that might have been similar to the Antikythera mechanism. "Maybe we need to rethink the connection between mechanics and astronomy," says Evans. "People think of it as purely one way, but maybe there was more of an interplay." In other words, when that Greek mechanic shaped the Antikythera mechanism's complex gear trains, he created more than a model made of bronze. He helped to forge a view of the Universe that would hold sway for nearly 2,000 years. Jo Marchant is a London-based writer and author of Decoding the Heavens, a book about the Antikythera mechanism. References 1. Freeth, T. et al. Nature 444, 587-591 (2006). 2. Wright, M. T. Interdiscipl. Sci. Rev. 32, 27-43 (2007). 3. Evans, J., Carman, C. C. & Thorndike, A. S. J. Hist. Astron. 41, 1-39 (2010). 4. Freeth, T., Jones, A., Steele, J. M. & Bitsakis Y. Nature 454, 614-617 (2008). 5. HÃ¸jte, J. M. (ed.) Mithridates VI and the Pontic Kingdom 313-320 (Aarhus Univ. Press, 2009).

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Re: [math-fun] Farey-ish fractions with weighted mediants
by James Propp 10 Dec '10

10 Dec '10

The don't-remove-common-factors version of my question is the more natural one, as Bill Thurston's posting makes clear. But the less-natural remove-common-factors version still looks like it might generate all rationals with odd denominators in [0,1]. Here's some Mathematica code I just wrote: Lengthen[L_] := Module[{i, M}, M = Table[0, {3 Length[L]}]; M[[1]] = Numerator[L[[1]]]/(2 + Denominator[L[[1]]]); M[[2]] = 2*Numerator[L[[1]]]/(1 + 2 Denominator[L[[1]]]); For[i = 1, i < Length[L], i++, M[[3 i]] = L[[i]]; M[[3 i + 1]] = (2 Numerator[L[[i]]] + Numerator[L[[i + 1]]])/(2 Denominator[L[[i]]] + Denominator[L[[i + 1]]]); M[[3 i + 2]] = (Numerator[L[[i]]] + 2 Numerator[L[[i + 1]]])/(Denominator[L[[i]]] + 2 Denominator[L[[i + 1]]])]; M[[3 Length[L]]] = L[[Length[L]]]; Return[M]] WF[n_] := WF[n] = If[n == 0, {1}, Lengthen[WF[n - 1]]] WF[n] consists of all the non-zero rational numbers in [0,1] obtained after n iterations of the insertion-of-weighted-medians procedure. I find that for all n up through 15 (the largest value of n that I tried), WF[n] contains all rationals in [0,1] with denominator 1/(2n+1). E.g., Table[MemberQ[WF[15], k/31], {k, 1, 30}] returns a list whose 30 elements all equal True, because WF[15] contains all proper fractions with denominator 31. Can anyone prove or disprove that this holds for all n? Jim Propp

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[math-fun] Farey-ish fractions with weighted mediants
by James Propp 10 Dec '10

10 Dec '10

It's well-known that if you start with the list of fractions 0/1, 1/1 and successively interpose the mediant (a+c)/(b+d) between successive elements a/b and c/d of the list, every rational between 0 and 1 will eventually appear. I just tried a variant of this today in which I interpose the weighted mediants (2a+c)/(2b+d) and (a+2c)/(b+2d) instead, like this: 0/1, 1/1 0/1, 1/3, 2/3, 1/1 0/1, 1/5, 2/7, 1/3, 4/9, 5/9, 2/3, 5/7, 4/5, 1/1 ... Does every rational between 0 and 1 with odd denominator eventually appear? I've checked this up through denominator 9. Also, if one instead interposes the weighted mediants (4a+c)/(4b+d), (3a+2c)/(3b+2d), (2a+3c)/(2b+3d), and (a+4c)/(b+4d), one does NOT obtain every rational between 0 and 1 with odd denominator; for instance, 1/3 never appears. Is there a nice way to characterize those rationals that do appear? Jim Propp

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Re: [math-fun] Farey-ish fractions with weighted mediants
by James Propp 10 Dec '10

10 Dec '10

Rich writes: >Jim, how are you treating unreduced fractions? >In the regular mediant-Farey-series construction, >unreduced fractions can't occur. In your scheme, >the mediants between 1/3 and 4/9 are 6/15 and 9/21. >Do you change these to 2/5 and 3/7 before proceeding >to the next step? Yes, I did this. I should've mentioned that; thanks for the question! Jim

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[math-fun] possible?
by Bill Gosper 08 Dec '10

08 Dec '10

Good grief, tutor Julian has plotted two infinitely differentiable surfaces intersecting in a simple arc that is not differentiable. --rwg

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Re: [math-fun] Stuff I didn't know about pendulums
by Bill Gosper 08 Dec '10

08 Dec '10

> > On Nov 29, 2010, at 8:24 PM, Bill Gosper wrote: > >> Veit, the center of oscillation seems to be the ratio of moment of >> inertia >> to some other moment. >> The math isn't the problem--it's my physics. >> --Bill > > it's all just natural philosophy ... > > Here's the equation of motion, > > I phi'' = T > > I = moment of inertia about point of support > phi = angle about point of support > T = torque due to gravity > > The torque is the same as if all the mass were located at the center of > mass. > (That's because the gravitational acceleration is a constant vector field, > approximately). > Let R be the distance between the point of support and the center of mass. > Also, let I_0 be the moment of inertia about the center of mass. By the > parallel > axis theorem I = I_0 + MR^2, where M is the mass of the pendulum. The > equation > of motion is therefore > > (I_0 + MR^2) phi'' = - MgR sin(phi). > > The ratio I_0/M = R_0^2 defines the "radius of gyration". Divide through > by MR and > you get > > (R_0^2/R + R) phi'' = - g sin{phi). > > This is the same as the equation for a point-mass pendulum on a massless > string of > length L = R_0^2/R + R. Huzzahs to the hereinbefore! As an example, a spherical shell of radius r > supported at its north > pole has R_0 = r, R = r, L = 2r (as if all the mass were placed at the > south pole). But hoohah hereat. http://en.wikipedia.org/wiki/List_of_moments_of_inertia and tutor Julian assure me that the hollow sphere moment of inertia is 2 m R^2/3. This puts the effective center 5/6 of the way to the south pole, which I find infinitely more plausible than 6/6. Based on your earlier analysis, I withdraw my claim about hypercubes. The effective center varies (slightly) depending on what plane the centroid oscillates in. --Bill > > Veit >

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Re: [math-fun] racing around a triangle
by Henry Baker 08 Dec '10

08 Dec '10

Thanks very much, Andy, for the Mathematics Intelligencer reference. I found this description: http://blog.sciencenews.org/view/generic/id/64589/title/Winning_the_World_S… Here's the full paper (basically Carroza's 2009 undergraduate thesis at Williams College): Carroza, Davide; Johnson, Stewart; Morgan, Frank. "Baserunner's Optimal Path". Math. Intelligencer 32, 1 (2010), 10-15. http://www.springerlink.com/content/t7q081384758kl5l/fulltext.pdf -- The Carrozza, et al paper is quite relevant -- especially: --- Lemma 2. A fastest C(1,1) path from one point to another in the plane, given initial velocity, final velocity, and a bound sigma>0 on the magnitude of the acceleration a, a = sigma*(At+B)/|At+B| for some constant vectors A, B, is unique. --- The cool thing about this lemma is that the optimal paths are _catenaries_ in _velocity space_ ! --- There are obviously several potential variations on the problem: 1. A path that goes _through_ all the vertices versus a shortest path that goes completely around the triangle (it may be faster for some really slim triangles to not even attempt to touch one of the vertices, but you still don't cross the triangle boundary). My initial goal doesn't require going through all the vertices (so it isn't completely analogous to the "running bases" problem), so it is conceivable that a simple circular orbit with the largest triangle side as diameter may be the best solution (for some non-obtuse triangles). 2. The path is a true cyclic "orbit": the positions & velocities match seamlessly at the beginning & end. It is conceivable that the fastest such "orbit" may actually go around the triangle twice (or k times), in which case we take the total time & divide by k. I.e., it may not be necessary to match accelerations & velocities on a single pass, so long as we match them at some fixed integer number of passes. For example, an optimal path that starts with initial velocity 0 might get faster & faster on each trip around the triangle, but converge to an optimal cyclic path. I guess that it is unlikely, but perhaps possible to have a optimal chaotic solution which _never_ matches, in which case we may have to look for a limit of the running average time. My model for this sort of chaotic behavior is the Minsky circle-drawing method from Hakmem. 3. Obvious extension to (convex) polygons -- e.g., base-running. At 09:35 PM 12/7/2010, Andy Latto wrote: >On Tue, Dec 7, 2010 at 5:58 PM, Henry Baker <hbaker1(a)pipeline.com> wrote: >> I'm still trying to solve for the optimum path around a triangle, subject to the acceleration constraint |a|<=1. > >By "around a triangle", it seems you mean any path that passes through >the three vertices of the triangle; is this correct? > >Also, I think you have an additional implied constraint that the route >be cyclic, that is, the velocity at the end of the path is the same as >the velocity at the beginning. Otherwise the best solution for the >degenerate triangle would be to start at one end of the line segment >with velocity 2, decelerate until you reach the other end with >velocity zero, and then continue to accelerate in the same direction >until you return to the original vertex, again with speed 2, but now >in the direction away from, rather than towards, the other vertex. > >The Mathematical Intelligencer had an article a month or two back >about the optimal way to run the bases in baseball. That is, the >fastest path with bounded acceleration starting at one vertex of a >square with velocity 0, passing through the other three vertices, and >returning to the original vertex (though without the requirement that >the final velocity be 0). I think much of this article is relevant to >your triangle problem. > >Andy

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