May 2010 - mathfun@mailman.xmission.com

[math-fun] Finite automata game
by Jason 24 May '10

24 May '10

Here's an amusing game about creating state diagrams: http://jayisgames.com/games/manufactoria/#

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[math-fun] If the chairman is now the "chair"
by Bill Gosper 24 May '10

24 May '10

is the boatman now the boat? The boatperson? Actual puzzle: What is the order of the symmetry group for "chair" cyclohexane? http://gosper.org/chair.pdf (or .nb) shows why "chair" is a terrible name for this. --rwg Apropos "correct" language, somebody should tell that Cellular Automata guy that the term "Garden of Eden" has Judaeo-Christian connotations that some of us may find offensive. He should consider "Recovering Fruit-of-Knowledge Taste Test Support Environment."

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[math-fun] The goemboec (gömböc)
by Fred lunnon 23 May '10

23 May '10

There's been some discussion of rollers on this list recently, so I'll pass on an URL which Google saw fit to advertise on my post concerning Gardner: http://www.gomboc-shop.com/app/urwfilter/mathinfo/do/index.html Fred Lunnon

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Re: [math-fun] Martin Gardner, October 21, 1914 ? May 22, 2010
by William Marshall 23 May '10

23 May '10

I still have the letters I received from Martin. In an article on polyhexes and polyaboloes reprinted in Mathematical Magic Show, Martin had written about tetraboloes, "Rectangles with a side of 1h are clearly impossible except, of course, for the 1h-by-1h." He must have been astonished when I pointed out to him that 1h X 3h and 1h X 4h rectangles could be constructed. (Why did nobody else ever tell him in all the years before I did?) I don't know how his death will affect the reprinting of his fifteen Mathematical Games collections by the Cambridge University Press. I know that Martin intended including further updated material and corrections for the entire series but don't know how much of the updating he has written for it. The first three books have already been published, and the Cambridge site (http://www.cambridge.org/series/sSeries.asp?code=NGML) shows the cover art for the fourth book in the series (Knots and Borromean Rings, Rep-Tiles, and Eight Queens). See also: http://www.richarddawkins.net/articles/472686-martin-gardner-rest-in-peace-… > From: "George W. Hart"<george(a)georgehart.com> > > (Not fun.) > > It is with great sadness that I report Martin Gardner died today. > > > http://blogs.discovermagazine.com/badastronomy/2010/05/22/martin-gardner-19… > > When I visited him last October in Norman, OK, he was sharp, alert, and > happy to offer a list of great ideas he thought would be good for the > Museum of Mathematics. He demonstrated some card tricks, calculator > tricks, and a rope illusion. He told a number of interesting stories > about his life and said he was starting to work on his autobiography --- > a book project which he predicted would take three years to complete. I > thought that it took enormous vision to start a three-year project at > the age of 95 and am sad that it will never be finished. > > George Hart > http://momath.org > http://georgehart.com

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Re: [math-fun] Regular spherical polyhedra
by Dan Asimov 23 May '10

23 May '10

I believe the condition that "No two adjacent polygons have the same image under F" takes care of Andy's concern, and in fact that is the reason this condition was included. --Dan Andy wrote: << I wrote: << . . . * For any edge e of any one polygon, there are exactly two polygons that have e as a full edge. Two such polygons sharing exactly one full edge are called "adjacent". No two adjacent polygons have the same image under F. >> If I were making the definition, I would add the condition here that the image under X of the two polygons adjacent to e includes an open neighborhood of every point in the image of e except possibly the vertices e connects. Or roughly speaking, the two faces adjacent to e are on "opposite sides" of the edge. Did you intend to include this condition? Are there polygons that satisfy your definition that do not satisfy this condition? >> ________________________________________________________________________________________ "Outside of a dog, a book is man's best friend. Inside of a dog, it's too dark to read." --Groucho Marx

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[math-fun] gaskets redux
by Bill Gosper 23 May '10

23 May '10

Fourteen years seems like a good time to dust off one of the strangest and potentially most interesting things I ever dug up--the Fourier series for the gasket-filling curve. The coefficients are easily computed by a rapidly convergent 2x2 matrix product which is not termwise equivalent to a sum nor anything else familiar. It seems an absolute poster child for the superiority of matrix products. In fact, it overperforms, computing in its four elements the desired A(x) four different ways: A(x) -A(x) ( ). A(-x) -A(-x) (Unfortunately, you don't gain a factor of 2 from this because you typically want A(k+1/m) and A(-k+1/m), rather than A(-k-1/m), for some integer m>1. But it makes sense to cache both anyway.) I was able to express A(x) as a *limit* of a sum of 3^n/2 terms, but divided by 2^n instead of 3^n/2--a sort of cockeyed Stieltjes integral. But not really, because the summand involved a stutter-free integer sequence d(n) (A108964) with a weird rule of formation individually, and a different weird rule of formation for the set {d(n)}. So there was no taking the limit for the integral anyway. The final form is oo gasket(t,m) = Sum(A(k+1/m) exp(i pi t (k + 1/m))) k=-oo which traces |m| gaskets on the sides of a regular |m|-gon as -n < t < n. (Inside for m<0.) We usually think of (and draw) this fractal as a bunch of line segments representing the boundaries of the deleted triangles. But this is a set of measure zero in the true fractal, and seems to be the values of gasket(t,3) for which the ternary expansion of t has only finitely many 0s, 1s, or 2s. Formulae and pix: http://gosper.org/gasky.pdf . --rwg Some new weirdness has crept in. The case m=2 used to draw two gaskets base-to-base as expected, but Mathematica now draws crazy squiggles. Also, does anybody know how to prevent Mma from idiotically spending *minutes* trying to find a closed form or a fancy (but disastrous) numerical method when all you want is a Sum of a few dozen terms? I wind up having to Total a Table! PS: Are there uncountably many "squarefree" (stutter-free) strings base 3? PPS, There are no straight lines in Julian's gasket since no segments in the uninverted gasket extend through the origin.

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Re: [math-fun] Martin Gardner, October 21, 1914 – May 22, 2010
by Dan Asimov 23 May '10

23 May '10

I'm deeply saddened to learn at this turn of events. It had to happen eventually, and every week or three that would occur to me, such a presence he has been in my life. When I was 12 in 1959, someone gave me as a present his "Scientific American Book of Mathematical Puzzles and Diversions". I have never loved a book as much as that one. He was an important inspiration to my becoming a mathematician. I read his Mathematical Games column religiously. And was deeply impressed with his other writings, which I now feel a strong desire to read in full (though that would take a few years). He is a role model for me: I aspire to be at least as incisive a philosopher and as inspiring a mathematics explicator as he was. (That's an impossible goal, but still worth aiming for.) We exchanged about four letters, the last one or two rather long, and one phone call (after he erroneously wrote a positive quote used on the jacket of Marilyn Vos Savant's disaster, "The World's Most Famous Math Problem", about Fermat's Last Theorem). George Hart wrote: << (Not fun.) It is with great sadness that I report Martin Gardner died today. http://blogs.discovermagazine.com/badastronomy/2010/05/22/martin-gardner-19… When I visited him last October in Norman, OK, he was sharp, alert, and happy to offer a list of great ideas he thought would be good for the Museum of Mathematics. He demonstrated some card tricks, calculator tricks, and a rope illusion. He told a number of interesting stories about his life and said he was starting to work on his autobiography --- a book project which he predicted would take three years to complete. I thought that it took enormous vision to start a three-year project at the age of 95 and am sad that it will never be finished. >> ________________________________________________________________________________________ "Outside of a dog, a book is man's best friend. Inside of a dog, it's too dark to read." --Groucho Marx

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[math-fun] Regular spherical polyhedra
by Dan Asimov 20 May '10

20 May '10

Suppose we define a slightly generalized type of regular "spherical" polyhedron (RSP) as follows. (They are a kind of metric and regular definition of a branched covering.) ---------------------------------------------------------- Definition: An RSP is an abstract metric space X that is a finite union of identical *copies* of spherical polygons called faces {P_k | 1 <= k <= N} (on say the unit sphere), equipped with a mapping F: X -> S^2 that is an isometry when restricted to any one P_k. Note that the P_k are not necessarily either disjoint or distinct, and they may have interior angles > pi. Note that the polygons P_k are subsets of X but not of S^2. X must satisfy: * For any vertex v of any one polygon, the sum of the angles at v, over all the polygons having v as a vertex, is an integer multiple of 2pi. * For any edge e of any one polygon, there are exactly two polygons that have e as a full edge. Two such polygons sharing exactly one full edge are called "adjacent". No two adjacent polygons have the same image under F. * For any two polygons, there is a sequence of polygons starting with one and ending with the other, with any two consecutive polygons in the sequence being adjacent. * (Regularity): For any triple (v,e,f) with vertex v in edge e in face f, and any other such (v',e',f'), there is an isometry I: X -> X that carries v |-> v', e|-> e', f |-> f'. * Any two RSP's are called "equivalent" if there is an isometry of one onto the other that carries each vertex, edge or polygon of one to a vertex, edge, or face, respectively, of the other. ---------------------------------------------------------- I know the complete classification of RSP's, but I wonder if anyone has seen this definition before, or such a classification. Examples of nonstandard polyhedra that are RSP's include: * Taking any of the Platonic solids projected radially to the sphere S^2, and replacing each face of the projection with the closure of its complement in S^2. (E.g., starting with a cube, the complement of a projected face has angles = 4pi/3.) * Project the regular tetrahedron to the sphere and choose any two disjoint edges e, e' of it. Now take two identical copies T_1, T_2 of this spherical tetrahedron and "slit" each of them them along each edge corresponding to e and e', creating 8 loose edges. Now identify these in 4 pairs (as when creating a Riemann surface) so that one side of e on T_1 is identified to the other side of e on T_2, etc., so that no loose edges remain. (In this case the topology of X is that of a torus.) --Dan ________________________________________________________________________________________ "Outside of a dog, a book is man's best friend. Inside of a dog, it's too dark to read." --Groucho Marx

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Re: [math-fun] Quadratic equations --- still getting it wrong!
by Dan Asimov 19 May '10

19 May '10

<< I have never seen this done in a textbook. But Bob Churchhouse in his Numerical Analysis course at Cardiff always used to discuss the behaviour of the coefficients as functions of the roots. I thought it a most useful exercise, and did not hesitate to appropriate it for my own course when the time came. WFL >> Isn't this relationship -- for a monic polynomial P(x) = (x-r_1)...(x-r_n) = x^n + c_(n-1) x^(n-1) + ... + c_0 just given by the elementary symmetric functions: c_(n-k) = (-1)^k S_k(r_1,...,r_n) where S_k is the sum of all products of the r_j for k distinct indices j ??? But also I wonder how much light this would shed on numerical problems of calculating the reverse relationship: finding the roots from the coefficients. --Dan ________________________________________________________________________________________ "Outside of a dog, a book is man's best friend. Inside of a dog, it's too dark to read." --Groucho Marx

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[math-fun] Cellular Automata [Fwd: a CA repository]
by Fred lunnon 18 May '10

18 May '10

---------- Forwarded message ---------- From: Genaro <genaro.juarez.martinez(a)googlemail.com> Date: May 18, 2010 10:25 PM Subject: a CA repository To: genarojm(a)gmail.com Dear friends and collages, Late of a long time, a small/basic CA repository is compiled, enumerating a list of subjects, as follows: - CA researchers (includes historic pages) - free software (old and recent) - CA books (several all they with on-line access) - CA research groups, etc. If you have some feedback or proposal, I will very happy to know. The idea is concentrate an useful and practical information/references related to CA to get a most complete relation of CA resources. I hope this become be useful for you, students and related guys. Please you can visit such repository from ICUC-UWE in: http://uncomp.uwe.ac.uk/genaro/CA_repository.html All best, Genaro -- Genaro Juarez Martinez Unconventional Computing Group, UWE, UK http://uncomp.uwe.ac.uk/genaro/

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