September 2010 - mathfun@mailman.xmission.com

Re: [math-fun] Help wanted identifying a recursive tiling
by Bill Gosper 06 Sep '10

06 Sep '10

This is precisely the tiling behind www.tweedledum.com/rwg/pentagonfill.pdf and http://gosper.org/pentfill.pdf . (And that stick-length illusion.) Martin Gardner once fowarded me a set of ingenious but cryptic frac-tiles by Ammann. I don't know where they are, so I hope someone else has a copy. They all lead to spacefills, probably of the pentfill type, since they were based on algebraic numbers. Martin was intrigued by Ammann's constantly shifting return address, and repeatedly requested some biographical data. Ammann finally replied that he was a parking lot attendant. Uh-huh. Here's an easy one with two equal tiles. Your hemispheres are not quite alike: www.tweedledum.com/rwg/hemis.gif . I think I've finally figured out that pentagram snowflake S. Witham found (and my tutors rediscovered) in Minsky rug plots. Has anybody else worked on it? --rwg Robert Munafo wrote: Fred lunnon wrote: > [...] > Perhaps you might explain in more detail about what exactly you want to > know about it? Good question Fred. I guess what I was looking for is a web site or a book where "my" tiling is described and it tells who originally discovered or invented it. Mike Stay wrote: > This isn't really a tiling: you can take nearly any shape and "tile" > the plane with it in the way you describe--given any unfilled region, > just scale down your shape until it fits and recurse. See the > Apollonian gasket, for example. As far as I can tell, my tiling is no different in that respect from the Ammann tilings, for example. (See http://tilings.math.uni-bielefeld.de/substitution_rules/ammann_a3) So, like I said above, I want some book or website that explains how and why they are different. Some kind of organized taxonomy for recursive substitution-systems for space-filling patterns. It should include everything on tilings.math.uni-bielefeld.de but should also include more information (that website seems to be an abandoned project) > On 9/5/10, Robert Munafo <mrob27(a)gmail.com> wrote: > > The rules for my tiling can be seen here: > > > > http://mrob.com/pub/math/images/penta-tiling.jpg -- Robert Munafo -- mrob.com

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Re: [math-fun] Help wanted identifying a recursive tiling
by Dan Asimov 06 Sep '10

06 Sep '10

RWG wrote: << Martin was intrigued by [Robert] Ammann's constantly shifting return address, and repeatedly requested some biographical data. Ammann finally replied that he was a parking lot attendant. Uh-huh. >> < http://en.wikipedia.org/wiki/Robert_Ammann > Apparently he was a very smart guy who didn't have professional training. Another such story with Marjorie Rice: < http://en.wikipedia.org/wiki/Marjorie_Rice >. There's something about tiling. --Dan _____________________________________________________________________ "It don't mean a thing if it ain't got that certain je ne sais quoi." --Peter Schickele

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Re: [math-fun] Soccer balls don't just curve; they spiral
by Bill Gosper 06 Sep '10

06 Sep '10

As Adler's video demonstrates, the domed upper surface of the frisbee suppresses roll by moving the center of lift back to the center of gravity. The Aerobie achieves this with a slightly higher angle of attack of the trailing airfoil. --rwg DanA>This isn't quite the same, but in frisbee catch, it's fun to throw it in a tilted plane with some serious CW spin, aimed far to the right of the target person (I'm a righty). It will then curve in a wide arc -- at first it looks as if it will miss by a mile -- back to the target. Just as if it were sliding along a roof. --Dan RWG writes: << . . . I've probably mentioned this before, but flying, spinning disks (smoothly) zigzag. . .

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[math-fun] Help wanted identifying a recursive tiling
by Robert Munafo 06 Sep '10

06 Sep '10

I have a recursive substitution-rule to tile the plane, which seems simple enough but nevertheless does not seem to be discussed in the normal places (like the tilings encyclopedia, http://tilings.math.uni-bielefeld.de/substitution_rules) Google searches haven't helped either (example: an Image search with the keywords: tiling regular pentagon triangle). The rules for my tiling can be seen here: http://mrob.com/pub/math/images/penta-tiling.jpg I consider this a "simple" and "obvious" tiling because there are only two rules, and each rule tries to maximize the size of the pentagon(s) on the right-hand side. However, it is "non-simple" in the sense that, with repeated applications of the substitution rules, increasingly many different sizes of triangles and pentagons are produced. - Robert -- Robert Munafo -- mrob.com

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Re: [math-fun] Billiard word counts (was "Interleavings of three arithmetic progressions")
by N. J. A. Sloane 05 Sep '10

05 Sep '10

On Aug 20 Fred posted this message, which was followed by several related messages: The counts of 3-D billiard words of length n with k = 1,2,3 descendants of length n+1, for n = 0,...,19 are respectively 0, 0, 0, 6, 24, 78, 186, 372, 876, 1632, 3024, 5310, 8496, 13344, 21186, 31878, 46752, 66936, 94800, 130194, when k = 1; 0, 0, 6, 18, 36, 78, 150, 306, 420, 792, 1338, 2082, 3228, 4830, 7050, 9954, 13920, 18738, 24666, 32610, when k = 2; 1, 3, 3, 3, 9, 15, 33, 63, 153, 219, 261, 351, 585, 879, 933, 1233, 1401, 1899, 2301, 3111, when k = 3; Allan's sequence is the sum of the last two; note that his 15 should read 21. Fred Lunnon Could someone please submit these three to the OEIS (Fred?), so that I can clean up these emails? Thanks! Neil

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Re: [math-fun] Soccer balls don't just curve; they spiral
by Dan Asimov 04 Sep '10

04 Sep '10

This isn't quite the same, but in frisbee catch, it's fun to throw it in a tilted plane with some serious CW spin, aimed far to the right of the target person (I'm a righty). It will then curve in a wide arc -- at first it looks as if it will miss by a mile -- back to the target. Just as if it were sliding along a roof. --Dan RWG writes: << . . . I've probably mentioned this before, but flying, spinning disks (smoothly) zigzag. . . . . . >> _____________________________________________________________________ "It don't mean a thing if it ain't got that certain je ne sais quoi." --Peter Schickele

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[math-fun] Soccer balls don't just curve; they spiral
by Bill Gosper 04 Sep '10

04 Sep '10

You can see the kicker applying a superhuman sidespin--his feet fly out to the right after contact. (Perhaps appearing to the goalie as a blown kick.) He achieved such an airspeed and rpm that the flow remained detached until the startling midflight break to the left, like an exaggerated backdoor slider in baseball. What I find surprising is that the goalie found it surprising. It must have been a secret weapon accessible to only the few strongest kickers. I've probably mentioned this before, but flying, spinning disks (smoothly) zigzag. I first noticed it with kids flinging smashed Coke cans, but falsely ascribed it to imperfect shape. Then Scot Morris gave me a Ritz cracker thrower, and the mechanism became clear. Lift concentrates at the leading edge, applying a pitching moment to a horizontal disk. The response is delayed 90 degrees by precession, resulting in a roll. (Alan Adler has a 34 min aerodynamics video, if we can persuade him to upload it.) The roll results in banking and curvature. However once the roll passes vertical, the sign of rotation is effectively reversed! Hence there is oscillation about the vertical plane. I'm not sure how this oscillation damps. Probably equally unsure were the unfortunate targets of the Sikh chakra weapon. --rwg----------------------------------------- hgb> FYI -- I've always heard about a corkscrew pitch in softball -- perhaps it is actually real in softball, as well. BTW, the video of the shot is on Youtube accessible from the linked article. http://www.latimes.com/health/boostershots/la-heb-soccer-20100902,0,7474743… latimes.com/health/boostershots/la-heb-soccer-20100902,0,4589924.story BOOSTER SHOTS: Oddities, musings and news from the health world Physicists solve the mystery behind a 1997 soccer miracle 7:23 PM PDT, September 2, 2010 Forget about bending it like Beckham -- can you curve it like Carlos? All right, so this is a little beyond the normal health boundaries, but for more of our active, sporty, specifically soccer-loving readers, a new study in the New Journal of Physics solves the longstanding mystery of whether a dramatic goal in an international match more than a decade ago was due to athletic skill or dumb luck. The French researchers looked at a famous free kick goal made by Brazilian player Roberto Carlos against the French team in the 1997 Tournoi de France. When observed straight on, the ball makes a dramatic zigzag, flying far to the right of the goalpost before shooting back in. Some had attributed it to accuracy; others called it a fluke. But the physicists say that from that distance, at the speed the ball was traveling, the goal was practically inevitable. Study coauthor Christophe Clanet, a physicist at the Ecole Polytechnique in Palaiseu, France, said at first, the study didn?t pique his interest: After all, it was well known that a spinning ball will have a curved trajectory, a phenomenon known as the Magnus effect. To test whether that was the explanation in the soccer match, the physicists fired balls into water with a slingshot (making sure the ball?s buoyancy wouldn?t affect the results) and recorded its path. What they found surprised Clanet. The spinning ball didn?t just curve; it spiraled. In a world free of gravity, the physicist said, the ball would appear to shoot straight for a while and then suddenly coil in on itself, slowing down and stopping once it reached the center of that trajectory. Clanet said the main lesson here was for goalkeepers to watch out for strong kickers aiming from that sweet spot 35 meters away. Why hadn?t this curl been seen before? The key was distance: Carlos took his kick from 35 meters out -- a long way compared with the 18-25 meters that players tend to kick from. That means the ball usually made -- or missed -- the goal before having a chance to make its sharp turn. Carlos is known to kick hard -- in the range of 130 km per hour -- and the ball?s speed practically canceled out gravity, allowing the spiral to come into play. If Carlos had been any closer, the ball would have missed the goal before it had a chance to curve around. And if he had kicked the ball from any farther back, the ball might have curved too much and shot past the goalposts, or hit the ground and lost the spiral. So did Carlos know what he was doing when he made that kick, or was it purely accidental? Clanet pointed to the pinpoint accuracy with which these professional players could kick, and their years of experience on the field. ?I?m pretty sure he knew the trajectory at this distance was special,? he said. -- Amina Khan / Los Angeles Times

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[math-fun] Soccer balls don't just curve; they spiral
by Henry Baker 03 Sep '10

03 Sep '10

FYI -- I've always heard about a corkscrew pitch in softball -- perhaps it is actually real in softball, as well. BTW, the video of the shot is on Youtube accessible from the linked article. http://www.latimes.com/health/boostershots/la-heb-soccer-20100902,0,7474743… latimes.com/health/boostershots/la-heb-soccer-20100902,0,4589924.story BOOSTER SHOTS: Oddities, musings and news from the health world Physicists solve the mystery behind a 1997 soccer miracle 7:23 PM PDT, September 2, 2010 Forget about bending it like Beckham -- can you curve it like Carlos? All right, so this is a little beyond the normal health boundaries, but for more of our active, sporty, specifically soccer-loving readers, a new study in the New Journal of Physics solves the longstanding mystery of whether a dramatic goal in an international match more than a decade ago was due to athletic skill or dumb luck. The French researchers looked at a famous free kick goal made by Brazilian player Roberto Carlos against the French team in the 1997 Tournoi de France. When observed straight on, the ball makes a dramatic zigzag, flying far to the right of the goalpost before shooting back in. Some had attributed it to accuracy; others called it a fluke. But the physicists say that from that distance, at the speed the ball was traveling, the goal was practically inevitable. Study coauthor Christophe Clanet, a physicist at the Ecole Polytechnique in Palaiseu, France, said at first, the study didnt pique his interest: After all, it was well known that a spinning ball will have a curved trajectory, a phenomenon known as the Magnus effect. To test whether that was the explanation in the soccer match, the physicists fired balls into water with a slingshot (making sure the balls buoyancy wouldnt affect the results) and recorded its path. What they found surprised Clanet. The spinning ball didnt just curve; it spiraled. In a world free of gravity, the physicist said, the ball would appear to shoot straight for a while and then suddenly coil in on itself, slowing down and stopping once it reached the center of that trajectory. Clanet said the main lesson here was for goalkeepers to watch out for strong kickers aiming from that sweet spot 35 meters away. Why hadnt this curl been seen before? The key was distance: Carlos took his kick from 35 meters out -- a long way compared with the 18-25 meters that players tend to kick from. That means the ball usually made -- or missed -- the goal before having a chance to make its sharp turn. Carlos is known to kick hard -- in the range of 130 km per hour -- and the balls speed practically canceled out gravity, allowing the spiral to come into play. If Carlos had been any closer, the ball would have missed the goal before it had a chance to curve around. And if he had kicked the ball from any farther back, the ball might have curved too much and shot past the goalposts, or hit the ground and lost the spiral. So did Carlos know what he was doing when he made that kick, or was it purely accidental? Clanet pointed to the pinpoint accuracy with which these professional players could kick, and their years of experience on the field. Im pretty sure he knew the trajectory at this distance was special, he said. -- Amina Khan / Los Angeles Times

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Re: [math-fun] The Martin Gardner memorial issue of . . .
by Dan Asimov 03 Sep '10

03 Sep '10

<< . . . Skeptical Inquirer is on the newsstands. Anyone read it yet? >> as I wrote yesterday. Anyhow, I gave in and bought the magazine. All I can say is that if you're looking for any reference to his mathematical contributions that's more than infinitesimal, you won't find it here. --Dan _____________________________________________________________________ "It don't mean a thing if it ain't got that certain je ne sais quoi." --Peter Schickele

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Re: [math-fun] domino networks
by Dan Asimov 03 Sep '10

03 Sep '10

My standard question: What if instead of using the 4x7 rectangular network, one uses the 4x7 *toral* network? (I.e., the product of a 4-cycle and a 7-cycle.) Then can the domino question be solved? --Dan _____________________________________________________________________ "It don't mean a thing if it ain't got that certain je ne sais quoi." --Peter Schickele

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