March 2014 - mathfun@mailman.xmission.com

[math-fun] G4G question
by James Propp 12 Mar '14

12 Mar '14

Do any you funsters who've attended one or more G4G's happen to know whether any of the talks or gift exchanges in the past half-dozen years involved the games Gobblets, Gobblet Junior, Gobblet X4 or Gobblet Gobblers? I am thinking of making one or all of these games the subject of my gift exchange item, on the (uninformed) supposition that, like me until a month ago, most G4G attendees won't have heard of these games before. But I've been living under a rock for the past decade as far as game-playing goes, so it's entirely possible that there's an ongoing Gobblets craze in the world at large, and that the game has already been discussed extensively at recent G4G's --- in which case a write-up of the game, pitched at people who've never of it, would be inappropriate as a G4G gift-exchange item. (I'll post something about the game here either way.) Thanks, Jim Propp

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[math-fun] Projections -> sphere <-> rectangle
by Bill Gosper 11 Mar '14

11 Mar '14

Hilarie> A while back I asked if there was illustration of a project of a dodecahedron to a sphere to a rectangle. Once reasonable way to do the calculation is to compute the vertices on the sphere (a nice collection of 1, phi, and 1/phi for the coordinates), the great circle arcs between the vertices, and to use a Miller projection of that to a rectangle. I've not found a picture of this for a dodecahedron, though. The Internet has pictures of almost every other conceivable representation of a dodecadhedron, just not this one. A somewhat related question is how would you shrink wrap a ping-pong ball? A rectangle of ordinary kitchen clear plastic wrap stretches nicely across a hemisphere, but the lower part bunches up along longitudinal lines and creates a dense knot at the pole. How would you even describe a solution that minimizes the surface area, short of saying "use a spherical layer of plastic wrap"? <Hilarie Speaking of such matters, and older ones, here's a three-piece baseball cover: gosper.org/IMG_0212.JPG , to go along with our one-piece (sphericon) and the usual two. Background: https://www.youtube.com/watch?v=pvT5ySMnO8U --rwg

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Re: [math-fun] Projections -> sphere <-> rectangle
by Stuart Anderson 11 Mar '14

11 Mar '14

There is another way to go from a dodecahedron to a rectangle. It's not a projection but a combinatorial mapping with software by J. W. Cannon, W. J. Floyd, and W. R. Parry. http://www.math.vt.edu/people/floyd/research/software/subdiv.html (see bottom of the page for the dodecahedron data) The first program subdivide.c prompts you for a subdivision rules file and a tiling file for a quadrilateral, and returns the tiling file for the subdivision. Once you have the tiling file you can convert it with a second program tilepack.c to use with Ken Stephenson's circle packing program, CirclePack http://www.math.utk.edu/~kens/CirclePack/, to draw the tiling to be Euclidean or hyperbolic. Alternatively you can use the third program squarect.c to create a squared rectangle from the dodecahedron. see http://www.math.vt.edu/people/floyd/research/papers/fsr.pdf for more Stuart Anderson > > > On Mon, Mar 10, 2014 at 1:58 PM, Hilarie Orman <ho(a)alum.mit.edu> wrote: > > > A while back I asked if there was illustration of a project of a > > dodecahedron to a sphere to a rectangle. Once reasonable way to do > > the calculation is to compute the vertices on the sphere (a nice > > collection of 1, phi, and 1/phi for the coordinates), the great circle > > arcs between the vertices, and to use a Miller projection of that to a > > rectangle. I've not found a picture of this for a dodecahedron, > > though. The Internet has pictures of almost every other conceivable > > representation of a dodecadhedron, just not this one. > > > > A somewhat related question is how would you shrink wrap a ping-pong > > ball? A rectangle of ordinary kitchen clear plastic wrap stretches > > nicely across a hemisphere, but the lower part bunches up along > > longitudinal lines and creates a dense knot at the pole. How would > > you even describe a solution that minimizes the surface area, short > > of saying "use a spherical layer of plastic wrap"? > > > > Hilarie > > > > _______________________________________________ > > 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] Projections -> sphere <-> rectangle
by Hilarie Orman 11 Mar '14

11 Mar '14

A while back I asked if there was illustration of a project of a dodecahedron to a sphere to a rectangle. Once reasonable way to do the calculation is to compute the vertices on the sphere (a nice collection of 1, phi, and 1/phi for the coordinates), the great circle arcs between the vertices, and to use a Miller projection of that to a rectangle. I've not found a picture of this for a dodecahedron, though. The Internet has pictures of almost every other conceivable representation of a dodecadhedron, just not this one. A somewhat related question is how would you shrink wrap a ping-pong ball? A rectangle of ordinary kitchen clear plastic wrap stretches nicely across a hemisphere, but the lower part bunches up along longitudinal lines and creates a dense knot at the pole. How would you even describe a solution that minimizes the surface area, short of saying "use a spherical layer of plastic wrap"? Hilarie

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[math-fun] number walls? bricks?
by Warren D Smith 10 Mar '14

10 Mar '14

Actually, Lewis Carroll's determinant method has a "number brick" i.e. involves a 3D array of numbers, with nearby ones related by a simple formula, allowing the whole "brick" to be deduced from one "face." Similarly to the way certain 2D arrays of numbers can be deduced from those in just "one row" via "lozenge algorithms." -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)

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[math-fun] Lewis Carroll determinant method & Toeplitz & Sylvester resultant matrices
by Warren D Smith 10 Mar '14

10 Mar '14

> For Toeplitz matrices, there should be FFT methods. --yes, but I am not sure these available for determinant. I know they work for solving Toeplitz linear systems. (Also, superfast algorithms can in practice be less desirable than mere "fast" algorithms due to complexity of the programming. Depends on your application.) > For circulants := diag(vec(v)) we certainly have > det = prod of elements of Fourier transformed vec(v) > The following should be useful: > Victor Y. Pan: > Structured matrices and polynomials: Unified Superfast Algorithms, > Springer-Verlag, (2001) > --jj --Sylvester "resultant" matrices have small "displacement rank." Therefore, superfast algorithms should be available for many operations with this kind of matrix, despite my failure to find a fast algorithm based on Carroll's condensation. I do not have Pan's book but section 5.2 seems (?) to sketch superfast algorithm for computing some kind of representation of a factored form of a structured matrix, which should allow computing its determinant. > Hankel determinants (of Toeplitz matrices) can be evaluated via what > Conway > & Guy christened the "wall of numbers" in their "Book of Numbers". > This algorithm copes with zeros in the table, making it applicable also to > finite field computations --- the |F_2 case is particularly simple. > > Or at least it would cope, if it weren't for a couple of nasty misprints > noticed by Jim Propp; for the correct version see > https://cs.uwaterloo.ca/journals/JIS/VOL4/LUNNON/numbwall10.html > > That paper also discusses parallel algorithms with time and space > complexity > of the same order n as the matrix, involving 1-D arrays of > finite-state automata. > > [ Incidentally, the proof given there is analytical, rather lengthy and > delicate. It does at least possess the virtue of presumable correctness. --that paper is rather immensely long and annoying to read,... but I suspect some of this "number wall" stuff must be equivalent to Carroll condensation applied to Toeplitz matrices, i.e. the latter was already known under other names. Incidentally there are still more names such as "lozenge algorithms" out there.

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[math-fun] "great walls" stop tornados?
by Warren D Smith 10 Mar '14

10 Mar '14

http://www.bbc.com/news/science-environment-26492720 http://www.usatoday.com/story/weather/2014/02/25/giant-walls-tornado-alley/… Physics Prof. Rongjia Tao claims if we build 3 giant walls 1000 feet high across a large part of the USA midwest, then tornados will diminish a lot in frequency. While it seems somewhat plausible to me, I doubt this question can be answered either by armchair theory or perhaps not even by computer modeling. More of an experimental question. And there might be many other side effects (good or bad?) of such a construct and its cost would be enormous.

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[math-fun] Jim Propp's parties in the Paris Review
by Thane Plambeck 10 Mar '14

10 Mar '14

I enjoyed this https://www.theparisreview.org/blog/2014/02/27/its-my-party/ -- Thane Plambeck tplambeck(a)gmail.com http://counterwave.com/

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[math-fun] Lissajous physob
by Bill Gosper 09 Mar '14

09 Mar '14

Alan Adler (gosper.org/alan.JPG . "I'm not an inventor, I'm an engineer, dammit," ) is motorizing my Lissajous illusion for G4G11. Lo res (7MB): gosper.org/MVI_0200.mp4 Hi res (314MB): gosper.org/MVI_0200.MOV You can see it on the worktable in the JPG, upended with a pulley removed. Also note the (unrelated) Euler disc. --rwg Poor Leo: To Google translate trompe l'oeil from French to English, spell it the same but say "Tromp Leo".

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Re: [math-fun] Lissajous physob
by Bill Gosper 08 Mar '14

08 Mar '14

MikeStay> I'd like to see one roll across a table: it will appear to float and twist. gosper.org/MVI_0202.MOV --rwg

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