August 2011 - mathfun@mailman.xmission.com

[math-fun] computer algebra
by Veit Elser 24 Aug '11

24 Aug '11

Most of you know lots more about computer algebra than I do. I'm interested in finding ALL solutions over the complex numbers of the following five equations: {1 + 1/b + c + b/d + e/c + d/f + f/e == 0, 1 + b + 1/c + d/b + c/e + e/f + f/d == 0, 1 + c/b + 1/d + d/e + e + b/f + f/c == 0, 1 + b/c + d + 1/e + e/d + c/f + f/b == 0, 1 + c/d + d/c + b/e + e/b + 1/f + f == 0} Here's what I already know: There exist solutions where all the variables are 6th roots of 1. There exist solutions where all the variables are 7th roots of 1. Numerically I can find solutions different from either of the above to very many digits. When I use Reduce or GroebnerBasis in Mma7 I quickly run out of memory. Any suggestions? Veit

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[math-fun] computer algebra
by Bill Gosper 24 Aug '11

24 Aug '11

Veit> Most of you know lots more about computer algebra than I do. I'm > interested > in finding ALL solutions over the complex numbers of the following five > equations: > > {1 + 1/b + c + b/d + e/c + d/f + f/e == 0, > 1 + b + 1/c + d/b + c/e + e/f + f/d == 0, > 1 + c/b + 1/d + d/e + e + b/f + f/c == 0, > 1 + b/c + d + 1/e + e/d + c/f + f/b == 0, > 1 + c/d + d/c + b/e + e/b + 1/f + f == 0} > > Here's what I already know: > > There exist solutions where all the variables are 6th roots of 1. > > There exist solutions where all the variables are 7th roots of 1. > > Numerically I can find solutions different from either of the above > to very many digits. > > > When I use Reduce or GroebnerBasis in Mma7 I quickly run out of memory. > Any suggestions? > > Veit You can get a lot of slutions just by treewalking with Factor[Resultant[...]]. Down the e=d=1 branch, {{f -> 1, b -> 1, c -> 1/2 (-5 - Sqrt[21])}, {f -> 1, b -> 1, c -> 1/2 (-5 + Sqrt[21])}, {f -> 1, b -> 1/2 (-5 - Sqrt[21]), c -> 1}, {f -> 1, b -> 1/2 (-5 + Sqrt[21]), c -> 1}, {f -> 1/2 (-5 - Sqrt[21]), c -> 1, b -> 1}, {f -> 1/2 (-5 + Sqrt[21]), c -> 1, b -> 1}, {f -> 1/10 (17 - 3 Sqrt[21]), c -> 1/10 (-11 - Sqrt[21]), b -> 1/10 (-11 - Sqrt[21])}, {f -> 1/10 (17 - 3 Sqrt[21]), c -> 1/10 (-11 + Sqrt[21]), b -> 1/10 (-11 + Sqrt[21])}, {f -> 1/10 (17 + 3 Sqrt[21]), c -> 1/10 (-11 - Sqrt[21]), b -> 1/10 (-11 - Sqrt[21])}, {f -> 1/10 (17 + 3 Sqrt[21]), c -> 1/10 (-11 + Sqrt[21]), b -> 1/10 (-11 + Sqrt[21])}, {f -> 1/10 (-11 - Sqrt[21]), c -> 1/10 (17 - 3 Sqrt[21]), b -> 1/10 (-11 - Sqrt[21])}, {f -> 1/10 (-11 - Sqrt[21]), c -> 1/10 (-11 - Sqrt[21]), b -> 1/10 (17 - 3 Sqrt[21])}, {f -> 1/10 (-11 - Sqrt[21]), c -> 1/10 (-11 + Sqrt[21]), b -> 1/10 (17 + 3 Sqrt[21])}, {f -> 1/10 (-11 - Sqrt[21]), c -> 1/10 (17 + 3 Sqrt[21]), b -> 1/10 (-11 + Sqrt[21])}, {f -> 1/10 (-11 + Sqrt[21]), c -> 1/10 (17 - 3 Sqrt[21]), b -> 1/10 (-11 - Sqrt[21])}, {f -> 1/10 (-11 + Sqrt[21]), c -> 1/10 (-11 - Sqrt[21]), b -> 1/10 (17 - 3 Sqrt[21])}, {f -> 1/10 (-11 + Sqrt[21]), c -> 1/10 (-11 + Sqrt[21]), b -> 1/10 (17 + 3 Sqrt[21])}, {f -> 1/10 (-11 + Sqrt[21]), c -> 1/10 (17 + 3 Sqrt[21]), b -> 1/10 (-11 + Sqrt[21])}} plus two more apparently intractable sets with f = irreducible sextics. --rwg

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Re: [math-fun] does anyone recognize ... ?
by Dan Asimov 22 Aug '11

22 Aug '11

Fwiw, Mathematica gives its value to 20 places as 1.5586704364253890428, which has a 0 where the last digit below is 4. --Dan Rich wrote: << Does anyone recognize the sum (for n>=1) of log(n+1) -------- n! The numerical value is roughly 1.5586704364253894. My attempts to use ISC found no matches, but I'm no expert on looking for non-obvious matches. >> Sometimes the brain has a mind of its own.

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Re: [math-fun] a new bad habit?
by Dan Asimov 19 Aug '11

19 Aug '11

RWG: << Does a simple plane (finite n)gon always contain some triangle defined by two consecutive sides? >> Yes, and I can prove it. --Dan Sometimes the brain has a mind of its own.

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[math-fun] a new bad habit?
by Bill Gosper 19 Aug '11

19 Aug '11

Just as smoking provides a convenient means to dispose of excess health, we may similarly dispose of excess creativity. Generate ten or so random points in the plane. "Connect the dots" <http://gosper.org/rangons11.pdf> (511K) for the simple polygons of maximal and minimal area. --rwg Does a simple plane (finite n)gon always contain some triangle defined by two consecutive sides?

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Re: [math-fun] (NFI) XGP/TeX dreams
by Tom Duff 19 Aug '11

19 Aug '11

John Aspinall <j(a)jkmfamily.org> says: > CAD languages based on Constructive Solid Geometry _do_ describe the > shape of an object in fairly rigorous, tractable ways. But they say > absolutely nothing about how to make the object. > > CAM languages such as G-code (http://en.wikipedia.org/wiki/G-code) > describe how to make the object by moving a (subtractive) tool through > space occupied by raw material. It's basically connect-the-dots, but > with an eraser. > > Going from one to the other is roughly like compilation. In fact, G-code is precisely a Constructive Solid Geometry language, where every model is the CSG difference of the initial billet and the Minkowski sum of the tool and the tool path. i.e. Blank-(Tool*Path) (in some notation or other.) -- Tom Duff. No proper program contains an indication which as an operator-applied occurrence identifies an operator-defining occurrence which as an indication-applied occurrence identifies an indication-defining occurrence different from the one identified by the given indication as an indication-applied occurrence.

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[math-fun] (NFI) XGP/TeX dreams
by Henry Baker 19 Aug '11

19 Aug '11

I don't recall the exact date that the XGP showed up at MIT's Project Mac, but I think it was the very early 1970's. The XGP was the first bitmapped XeroGraphic Printer, and it changed the course of printing history, including leading inevitably to the Apple Macintosh/desktop publishing industry and the now ubiquitous "multifunction printer/scanner/fax" that probably all of you have in your homes. The coolest thing about the XGP is that it was available to (nearly) anyone at 545 Tech Square, and it immediately became a huge hit. Once the appropriate page layout languages and fonts became available, people twiddled with them incessantly, until their theses looked like ransom notes! One of the important steps along the way was the development of Knuth's TeX, which subsequently became the standard for mathematical publishing -- where it remains today. --- Fast forward to 2011. 3D printers exist, but don't appear to have generated the same excitement as the original XGP. I'm not sure quite why, because a 3D printer is sooo much cooler than the XGP ever was. Maybe these printers are too expensive; maybe they're too slow; maybe they're too hard to access. What is missing that would make the 3D printer today as exciting as the XGP was in the early 1970's ? Does Knuth (or a Knuth-wannabe) have to come up with 3D TeX ? What do mathematicians want from a 3D printer, anyway ? What would be the effect of putting a free 3D printer into 100 universities ? 1,000 universities ? 5,000 high schools ? 10,000 grade schools ? 50,000 libraries ? (I have no connection with any 3D printer company, but think that it is a major revolution.)

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[math-fun] Poker hands and majorising sequences
by Adam P. Goucher 19 Aug '11

19 Aug '11

I was considering poker hands in an attempt to generalise the game to any number of suits and cards per hand. Apart from flushes, straights and combinations thereof, the poker hands are (in descending order): Four of a kind (4,1,0,0,0,0,0,0,0,0,0,0,0) Full house (3,2,0,0,0,0,0,0,0,0,0,0,0) Three of a kind (3,1,1,0,0,0,0,0,0,0,0,0,0) Two pairs (2,2,1,0,0,0,0,0,0,0,0,0,0) Pair (2,1,1,1,0,0,0,0,0,0,0,0,0) High card (1,1,1,1,1,0,0,0,0,0,0,0,0) You should be able to determine the notation used based on context. Notice that each sequence majorises all lower sequences, in the way required for Muirhead's Inequality to hold. http://en.wikipedia.org/wiki/Muirhead's_inequality Now, it is not the case that if one sequence majorises the other, then it is a more likely poker hand. For example, consider a variant with 13 cards per hand. It is obvious that (1,1,1,1,1,1,1,1,1,1,1,1,1) is less likely than (2,1,1,1,1,1,1,1,1,1,1,1,0), yet the latter majorises the former. However, if we consider the limit of poker games where the number of cards per suit increases to infinity, then A is rarer than B if sequence A majorises sequence B. What about sequences where neither one majorises the other? The minimal example is (3,3,0) compared with (4,1,1), which only occurs in six-cards-per-hand poker. Sincerely, Adam P. Goucher

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[math-fun] Morpion Solitaire: new world record
by rcs＠xmission.com 19 Aug '11

19 Aug '11

[forwarded for Christian Boyer, cboyer(a)club-internet.fr --Rich] ----- Morpion Solitaire: new world record, by Chris Rosin. 177 moves, 5 more moves than his previous record. This record will be presented at IJCAI'11 (International Joint Conference on Artificial Intelligence), July 2011. May 2011 news of www.morpionsolitaire.com Christian.

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Re: [math-fun] (NFI) XGP/TeX dreams
by Henry Baker 19 Aug '11

19 Aug '11

Some ideas for high school math (these are pretty obvious & lame): * 3D plots, so students can actually feel the shapes. Yes, you can produce 3D color plots & manipulate them with 3D glasses using current high-end PC's, but sometimes kids like to see physical objects. * experiment with reflective properties of different shapes -- e.g., lenses, reflective mirrors (need to be polished), etc. * experiment with center of gravity, etc., of different shapes * experiment with brachistochrone-type curves * experiment with curves of constant width * experiment with interlocking gears of different shapes * experiment with boat/submarine hulls of different shapes being pulled through the water etc. Perhaps Mike's point also needing a laser scanner is quite important. But the XGP appeared before any decent digital cameras! If I were a kid who had to get a CAT scan, I'd love to "print out" a copy of one of my own bones. That would be very cool! --- I haven't studied CAD/CAM languages recently, but the ones from 10+ years ago were pretty uninspiring. Perhaps an artist/mathematician _without any "computer language" skills_ will show us how to do it better. We need some "direct manipulation" ideas for how to describe objects which are very different from the traditional computer language ideas. At 03:57 PM 8/18/2011, Fred lunnon wrote: > The phrase "3-D printer" is catchy, but misleading: printing in the sense of >reproduction of text is essentially two-dimensional. > > Even in 2-D, pictorial reproduction and creation is technically more complex, >involving skills lying completely outside the domain of DTP; its 3-D >equivalent --- sculpture --- captures better what these devices do. > > One obvious area of application (for which they are apparently already >regularly used) is simply rapid prototyping of CAD/CAM designs, for which >appropriate solid-modelling languages are already highly developed, having >been utilised for many years to build virtual models for viewing on-screen, >or (imposing crucial limitations) solid milling. > > Looking beyond old-hat industrial plotting, there seems to be no obvious >reason why --- for example --- PostScript cannot be immediately generalised >to 3-D, and used to drive a laser sintering prototyper. > > The technology is anyway still in its infancy: just around the corner are surely >devices which will mix materials (cf. colour printing vs. black-and-white) and >apply surface finishes, as well as improving resolution. > > It will have arrived when some bright postgrad programs one of these engines >to reproduce itself. Now there's a prospect providing food for thought ... >[Though if Mike's 3-D photocopier hits the streets first, maybe even the >postgrad will be redundant.] > >Fred Lunnon > >On 8/18/11, Henry Baker <hbaker1(a)pipeline.com> wrote: >> I don't recall the exact date that the XGP showed up at MIT's Project Mac, >> but I think it was the very early 1970's. The XGP was the first bitmapped >> XeroGraphic Printer, and it changed the course of printing history, >> including leading inevitably to the Apple Macintosh/desktop publishing >> industry and the now ubiquitous "multifunction printer/scanner/fax" that >> probably all of you have in your homes. >> >> The coolest thing about the XGP is that it was available to (nearly) anyone >> at 545 Tech Square, and it immediately became a huge hit. Once the >> appropriate page layout languages and fonts became available, people >> twiddled with them incessantly, until their theses looked like ransom notes! >> >> One of the important steps along the way was the development of Knuth's TeX, >> which subsequently became the standard for mathematical publishing -- where >> it remains today. >> >> --- >> >> Fast forward to 2011. 3D printers exist, but don't appear to have generated >> the same excitement as the original XGP. I'm not sure quite why, because a >> 3D printer is sooo much cooler than the XGP ever was. Maybe these printers >> are too expensive; maybe they're too slow; maybe they're too hard to access. >> >> What is missing that would make the 3D printer today as exciting as the XGP >> was in the early 1970's ? >> >> Does Knuth (or a Knuth-wannabe) have to come up with 3D TeX ? >> >> What do mathematicians want from a 3D printer, anyway ? >> >> What would be the effect of putting a free 3D printer into 100 universities >> ? 1,000 universities ? 5,000 high schools ? 10,000 grade schools ? >> 50,000 libraries ? >> >> (I have no connection with any 3D printer company, but think that it is a >> major revolution.)

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