April 2003 - mathfun@mailman.xmission.com

Re: [math-fun] brick tilings
by reid＠math.arizona.edu 24 Apr '03

24 Apr '03

(thanks to michael kleber for reviving this.) > I just looked at Gosper's pix of tilings of rectangles by 1x3s. > All seem to have at least one substructure of 3x3 as three 1x3s. > Are there tilings without this substructure? check out "tiling a polygon with rectangles" by claire and richard kenyon, proceedings of the 33rd foundations of computer science (FOCS), 1992, pp. 610-619. theorem 3 of that paper asserts that, given a simply connected polyomino region tiled by m x 1 (vertical) polyominoes and 1 x n (horizontal) polyominoes, then any two such tilings can be obtained from one another by steps of the form replace an m x n subrectangle tiled by horizontal tiles by one tiled by vertical tiles or the inverse operation. this easily implies that the answer to rich's question is "no". mike

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Re: [math-fun] brick tilings
by Dan Hoey 24 Apr '03

24 Apr '03

reid(a)math.arizona.edu (michael reid) wrote: > check out "tiling a polygon with rectangles" by claire and richard > kenyon, proceedings of the 33rd foundations of computer science (FOCS), > 1992, pp. 610-619. theorem 3 of that paper asserts that, given a simply > connected polyomino region tiled by m x 1 (vertical) polyominoes > and 1 x n (horizontal) polyominoes, then any two such tilings can > be obtained from one another by steps of the form > replace an m x n subrectangle tiled by horizontal > tiles by one tiled by vertical tiles > or the inverse operation. Neat! And the restriction to simply connected regions is necessary: _______ _______ | |_____| |_____| | | | | | | | | | |_|___| | | |___|_| |_____|_| |_|_____| Dan

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[math-fun] brick tilings
by Richard Schroeppel 22 Apr '03

22 Apr '03

I just looked at Gosper's pix of tilings of rectangles by 1x3s. All seem to have at least one substructure of 3x3 as three 1x3s. Are there tilings without this substructure? Rich rcs(a)cs.arizona.edu

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[math-fun] some tiling facts
by Thane Plambeck 22 Apr '03

22 Apr '03

Here's a collection of results that are related to Michael Kleber's nice argument, although they don't seem to imply his result, as far as I can see... deBruijn showed that 1xa bricks will tile an mxn only when m or n is a multiple of a for example, you can't tile a 6x6 box with 1x4 bricks his proof is by a coloring argument that uses "a" colors in a checkerboard-like pattern ron graham showed that a fault-free tiling of a pxq rectangle with axb tiles exists (where we assume pq>ab and (a,b)=1) if and only if each of a and b divides p or q; each of p and q can be expressed as a linear combination of a and b into at least two ways; for {a,b} = {1,2}, (p,q) is not equal to (6,6), ie there is no fault-free tiling of a 6x6 by 1x2's (argument due to Golomb and Jewett) Thane Plambeck 650 321 4884 office 650 323 4928 fax http://www.qxmail.com/home.htm

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Re: [math-fun] math writing
by asimovd＠aol.com 21 Apr '03

21 Apr '03

Rich asks: << Anyone care to ... come up with an accurate explanation of manifold, understandable at the level of a college graduate who majored in, say, chemistry? >> Since manifolds are central to the kind of math I usually do, I've often been asked by a non-mathematician to explain them. I usually start by depicting a few curves and surfaces, and then point out that they don't need to "be anywhere" to make sense. Rather, they are the result of putting together pieces of 1- or 2-dimensional "modeling clay" according to certain instructions. This idea can be illustrated with the circle, the sphere and/or torus, and then the Klein bottle or projective plane, the last two of which can be described by sewing together certain edges of a square. I then point out that these two don't have any way of existing in ordinary space (except with flaws), much as a knot can't be flawlessly depicted in 2D. If someone follows this far, then the idea of modeling clay naturally can be elevated to of 3-manifolds (since the actual stuff is 3D, anyway). So it's then possible to explain the 3-sphere as two solid balls with corresponding points on the boundary "glued together" ("Rome to Rome, Paris to Paris, etc."), and/or the 3-torus as a cube with opposited faces identified. This approach seems to work with poets no worse than with chemists. (Note: Since someday I hope to put this kind of explanation in a book, I'll ask that if anyone wants to quote it then please attribute it to me -- thanks.) --Dan Asimov

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Re: [math-fun] NYTimes article
by asimovd＠aol.com 21 Apr '03

21 Apr '03

Hi, David. Re that math article: I read it Sat. evening when it was posted on the NYT website, and found it full of errors (such as assuming that n-manifolds are subsets of n+1 space). Wrote Johnson a polite note pointing some of them out, and he responded saying mostly that inaccuracies are justified when writing for a lay audience. What do you think? Regards, Dan

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[math-fun] An illegal California lottery game
by Thane Plambeck 21 Apr '03

21 Apr '03

These two paragraphs are from a 11/30/98 judgement by the California 2nd Appelate District Court in a case called "Jack M Janis et. al. vs the California State Lottery Commission" ------- In November 1992, the California State Lottery (CSL) began operating a Keno game. Several private gaming interests immediately challenged the legality of Keno, seeking to have its operation enjoined and the game declared illegal. During the litigation, CSL continued to operate Keno. The trial court and the court of appeal upheld the legality of Keno. On June 24, 1996, however, the California Supreme Court ruled the Keno game constituted illegal gambling. CSL stopped operating Keno later that day. On June 25, 1996, Janis filed an administrative claim with State Board of Control (BOC). Janis purported to represent a class of individuals who played CSL's Keno. Janis sought the return of all funds wagered on Keno, arguing CSL unlawfully promoted and profited from the illegal game. In mid-July 1996, the BOC notified Janis the administrative claims were incomplete and requested additional information. Janis did not respond to BOC's request.... ------ I remember when this Keno game came out. The way I recall it, you could buy a ticket for $1. The exact numbers involved may be inaccurate in what follows, but basically it was like this: You selected from 2 to 10 numbers out of 80 possible. Whether you chose 2, 3,4,5,6,7,8,9, or 10 numbers was up to you. Then twenty numbers were chosen at random by the lottery. The thing that caught my attention was that the game published an exact, guaranteed and fixed prize for all possible outcomes: for example, you knew that if you picked 7 numbers, and got 6 right, you would win a particular $ prize. Naturally the highest prize was obtained if you elected to select 10 numbers and then got them all right. There was no dependency on how many others might also have gotten it right---the prize was published and fixed. This made it possible to calculate the expected value of a ticket, once purchased. I calculated them for each possible choice 2,3,4,5,6,7,8,9,10 and found them to differ almost by a factor of two, for example, one ticket was only worth about 32 cents, while another was worth 64 cents or so. I have an indistinct memory that the reason the challenge to this game was successful was that one's ability to calculate a 'better' play amongst possible presented options made the game into a "game of skill," and not a "lottery." However I never found a good write up of the arguments before the California Supreme Court in the newspapers Anyway, does anyone else remember this thing? Thane Plambeck 650 321 4884 office 650 323 4928 fax http://www.qxmail.com/home.htm

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[math-fun] NYTimes article
by David H Bailey 21 Apr '03

21 Apr '03

A World of Doughnuts and Spheres April 20, 2003 By GEORGE JOHNSON Though you might not guess it from trying to read some of the research papers, the whole point of mathematics is to make things simpler. No one has taken this more seriously than the topologists, a rarefied breed of thinkers who insist that the world, however messy and diverse it may appear, is really made of just two basic shapes, the doughnut and the sphere. Actually it's a bit more complicated than that - the doughnuts can have more than one hole, for example, and the topologists don't limit themselves to the usual three dimensions. Lately, they have been preoccupied with claims that a Russian mathematician has solved a famous century-old problem involving what might be called hyperdoughnuts and hyperspheres existing in an imaginary four-dimensional space. Grappling with such slippery abstractions, Dr. Grigori Perelman of the Steklov Institute of Mathematics in St. Petersburg says he has found a proof of the Poincaré Conjecture, which seeks to explain how some of these airy higher-dimensional objects behave. He outlined his approach earlier this month in a series of lectures at the Massachusetts Institute of Technology. If he is right, it will be the biggest mathematical news since 1995, when Dr. Andrew J. Wiles, a Princeton professor, proved Fermat's Last Theorem. Sweetening the victory, Dr. Perelman would be eligible for a $1 million prize, sponsored by the Clay Mathematics Institute in Cambridge, Mass., for solving what it considers one of the seven most important problems of the millennium. The money is almost beside the point. That grown men and women can make a living pondering such matters is a sign that civilization, as fragile as it may sometimes seem, remains intact. "When you spend years closeted away with these things, they are as real to you as your family," said Dr. Michael Freedman, a mathematician at Microsoft who made his mark 22 years ago proving the Poincaré Conjecture for objects in five-dimensional space. Before that it had been proven for all dimensions beyond five. "Ironically the higher dimensions turned out to be easier than the lower ones," Dr. Freedman said. They offered more wiggle room. Topology is the study of that which remains constant as an object is bent, stretched or squeezed. A coffee cup with a looped handle, a bugle and a garden hose can each be transformed into a doughnut (more formally called a torus). Likewise, anything without a hole through it - a pencil, a brick, a piece of spaghetti (but not rigatoni, which is a very long and skinny doughnut) - can be molded into a sphere. The topological cookbook does not permit tearing an object or joining two unconnected points. That would be cheating and would allow anything to be transformed into anything else. Try as you might, you cannot turn a sphere into a doughnut or a doughnut into a sphere. Topologically they are as immiscible as oil and water. Having cataloged all the possible shapes in this realm, topologists have been reaching further. A sphere can be thought of as the three-dimensional world's version of a circle. So, going one level higher, what would be the four-dimensional equivalent of a sphere? And the five-dimensional version and so on? Seeking to find some order, the French mathematician Henri Poincaré proposed almost a century ago that the world of four dimensions obeys a rule similar to the one that prevails down here: Things without a hole are just different squishings of some canonical four-dimensional answer to the sphere. The technical name for this impossible object is the 3-sphere. Just as an ordinary sphere is a two-dimensional surface curving to form an enclosed object in three-dimensional space, a 3-sphere is a three-dimensional surface curving in on itself in four dimensions. Every few years someone claims to have tamed this monster, coming forth with a proof of the conjecture that is subsequently torn to shreds. "It's a famous problem, and at any time maybe a dozen people are working on it," Dr. Freedman explained. "Statistically one or two of them will be convinced that they almost have it." Hearing that a competitor may be on the verge of a breakthrough, "you work for four nights in a row, and then in some crazed state you claim that you've also proved it," he said. Dr. Perelman has raised the stakes even higher, claiming not only to have finished off the Poincaré Conjecture but to have listed every possible kind of object that can exist in the four-dimensional world - 3-spheres and who knows what else, an atlas of an invisible neighboring realm. His approach is innovative enough to make many topologists optimistic that the answer is finally in sight. If so, when the celebrations are over, the result may be met with a bit of melancholy. "Say you're a graduate student and you're picking a subject that will become your career," Dr. Freedman said. Do you really want to pick an area whose main problem has just been solved? "It's not a tragedy, because these people will go into other things," he said. "But it's a small sorrow for this particular branch of topology. You won't have the brilliant young people you have now." http://www.nytimes.com/2003/04/20/weekinreview/20JOHN.html?ex=1051909579&ei= 1&en=d9bfd625f0906cea

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Re: [math-fun] An illegal California lottery game
by asimovd＠aol.com 21 Apr '03

21 Apr '03

Thane writes: << I have an indistinct memory that the reason the challenge to this game was successful was that one's ability to calculate a 'better' play amongst possible presented options made the game into a "game of skill," and not a "lottery." >> I know there are many state laws against various games of chance where money is involved. But it seems so bizarre, if true, that a game involving some degree of skill with monetary prizes is illegal!!! Awards for skill are given All The Time in California as elsewhere. Can the skill factor *really* be why a court ruled this game illegal? (Undoubtedly, Nevada gambling casinos were behind the legal challenge. They are the ones most threatened by each additional California lottery game, and they have the deep pockets to challenge such.) --Dan

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Re: [math-fun] four-by-four
by Henry Baker 20 Apr '03

20 Apr '03

Does anyone remember the designations of how (steam?) locomotive wheels are driven? At 11:28 AM 4/17/03 -0400, Bernie Cosell wrote: >On 17 Apr 2003 at 11:19, asimovd(a)aol.com wrote: > >> David Wilson asks: >> >> << >> Does anyone know the etymology of the term four-by-four or 4x4 in relation to >> an off-road vehicle? >> >> >> >> I have to agree with Bernie here: The 4-speed transmission, and the 4-wheel >> drive. > >I think the etymology has changed. I believe that the first designations >of it appeared back 30 yrs ago with the first "four on the floor" and so >Jeeps and other vehicles were advertised as "four by four" [transmission >and drive wheels]. I think that that usage has paled in favor of the >wheels/driven wheels use. [BTW: I think the mention here had it >backwards. I think the now-common usage is: > totalwheels x drivenwheels >and so a normal passenger car would be a 4x2. > > /Bernie\ > >-- >Bernie Cosell Fantasy Farm Fibers >mailto:bernie@fantasyfarm.com Pearisburg, VA > --> Too many people, too few sheep <--

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