February 2007 - mathfun@mailman.xmission.com

RE: [math-fun] Life on the Edge
by Eric Angelini 21 Feb '07

21 Feb '07

Hello Frank, > I have discovered an interesting cellular automaton. ... nice idea ! ... could you test one by one, starting with zero, the fate of the natural numbers (up to 1000) ? You could design them like this (fixed font): _ _ _ _ _ _ _ | _| _| |_| |_ |_ | |_| |_| | |_ _| | _| |_| | |_| _| Have a look here for "Game of Life Integers": http://www.cetteadressecomportecinquantesignes.com/GOLife.htm Best, É.

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[math-fun] Ovaltine Dreams
by Steve Witham 21 Feb '07

21 Feb '07

For the Ovaloid where |P - F1| + k|P - F2| = c... I think it's the "simplest," even though it has fourth powers in its equation. But, you know, scare quotes. I have a hard time believing it never received interest before. The questions below are phrased as if it were a new thing. Has anyone thought of a workable mechanical construction to plot it, analogous to the string around two pins? I have been designing things with levers and pulleys and holes through the paper with no success. I think maybe a pantograph is called for... Has anyone thought of an interesting optical property, analogous to the ones ellipses and parabolas have? Maybe something about materials with different refractive indices, or coupling between two parallel eggs with different speeds of sound. Or, a dog jumps swims from island to shore with the frisbee, drops off the frisbee, and then swims faster (no frisbee drag) to the second island-- and it takes the same time no matter what point on the shore he swims to. When k = -1 it's a hyperbola. Do other negative numbers give mismatched-pair hyperbola-oids? Do they go asymptotic to straight lines, asymptotic to something else, or what? What is it a cross-section of? Does it correspond to an orbit in a strange force field? If so, is there an analog of the equal-area rule? If you make pictures before I do, I wouldn't mind if you posted them somewhere on the web. --Steve ovaltine levation olive-tan velation violante not-alive vital-one elan-vito toenail-V a-violent intel-ova naive-lot ovate-nil ion-valet ten-viola ain't-love evil-NATO

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[math-fun] employing asinh for false perspective
by Steve Witham 20 Feb '07

20 Feb '07

I was plotting a fractal and wanted a false-perspective view, where the magnification is really high at one point, and decreases in a log-like way further out from there. Like that cartoon map of the U.S. from the point of view of Manhattan. The mapping needs to be like a log until you get close to zero, then flatten out. I don't know whether it was Henry Baker's comment from last August that triggered it, but I thought of the asinh function. Here is the nice result: http://www.tiac.net/~sw/2007/02/asinh_perspective In the process of programming this, I needed an asinh for Postscript. The one I found is in a package called "pst-math.pro". (.pro is apparently a suffix for Postscript libraries.) The definition basically means asinh( x ) = ln( x + sqrt( x^2 + 1 ) ) which is the standard definition, but there is a precision-slash- overflow problem when x < 0 if you actually calculate it that way in floating point. The link above has a trippy graph of the problem. There is Postscript source code there for both pictures. --Steve >From: Henry Baker <hbaker1(a)pipeline.com> >Date: Sun, 06 Aug 2006 21:39:29 -0700 >BTW, "asinh" now has a real paying job. I searched using Google and >I found that astronomers are now using asinh instead of log to >compute brightness of stars in a star catalog. > >I have advocated the use of asinh in audio & video processing to do >"dynamic range compression" in a way which is somewhat smoother than >current ad hoc methods and floating point methods. > >Another job is its use in genetic engineering & statistics to smooth >data better than log when very small magnitudes are encountered, which >would otherwise drive log nuts (i.e., very substantially negative).

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Re: [math-fun] You Can Balance a Wobbly Table by Rotating It
by Daniel Asimov 20 Feb '07

20 Feb '07

Robert Baillie writes: << On NPR's Saturday Weekend Edition, they did a story with "Math Guy" Keith Devlin. He described a recent (last year) proof that, if the slope of the ground is at most about 35 degrees (actually, ArcTan[1/Sqrt[2]]), then you can balance a wobbly table by rotating it. (This assumes a mathematically perfect table, with the ground being a continuous function of x and y). >> A closely related old problem that seems to be a "folk theorem" (but *only* a folk theorem, apparently without a written proof), is this: Claim: Any simple closed curve in the plane contains the four corners of some square. ------------------------------------------------------------------------------------- I guess the Claim that Robert refers to can be stated rigorously as follows: Any continuous space curve of the form z = f(exp(i*theta)) (i.e., {cos(theta), sin(theta), z(theta), 0 <= theta <= 2pi}, where z has period theta) contains 4 points in space that form the corners of a square . . . as long as z = f(x+iy) is differentiable and satisfies (dz/dy)/(dz/dy) <= sqrt(1/2). -------------------------------------------------------------------------------------- A[n apparently as yet unproved] theorem that would imply both of these is this: (*) Claim: Any simple closed curve in R^3 contains the 4 corners of a (planar) square. --Dan P.S. I will say, smugly, that I believe I know how to prove (*).

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RE: [math-fun] You Can Balance a Wobbly Table by Rotating It
by Andy Latto 20 Feb '07

20 Feb '07

> -----Original Message----- > From: > math-fun-bounces+andy.latto=pobox.com(a)mailman.xmission.com > [mailto:math-fun-bounces+andy.latto=pobox.com@mailman.xmission > .com] On Behalf Of Daniel Asimov > Sent: Monday, February 19, 2007 8:46 PM > To: math-fun > Subject: Re: [math-fun] You Can Balance a Wobbly Table by Rotating It > > > Robert Baillie writes: > > << > On NPR's Saturday Weekend Edition, they did a story with "Math Guy" > Keith Devlin. He described a recent (last year) proof that, > if the slope of the > ground is at most about 35 degrees (actually, > ArcTan[1/Sqrt[2]]), then you can > balance a wobbly table by rotating it. (This assumes a > mathematically perfect > table, with the ground being a continuous function of x and y). > >> > > I guess the Claim that Robert refers to can be stated > rigorously as follows: > > Any continuous space curve of the form z = f(exp(i*theta)) > (i.e., {cos(theta), sin(theta), z(theta), 0 <= theta <= 2pi}, > where z has period theta) contains 4 points in space that form > the corners of a square . . . as long as z = f(x+iy) is differentiable > and satisfies (dz/dy)/(dz/dy) <= sqrt(1/2). > -------------------------------------------------------------- > ------------------------ I think this is a completely different conjecture than the "putting a table on the floor so it doesn't wobble" conjecture. In particular, most floors are surfaces, rather than curves, and most tables have a fixed distance between their legs. I think the "putting a table on the floor" theorem is a theorem of the form "Given a differentiable function of two variables, z = f(x,y), and a distance H, there exist four points in the plane, (x1,y1), (x2, y2), (x3, y3) (x4, y4) such that the four points in space, (x1, y1, f(x1, y1)), (x2, y2, f(x2, y2), (x3, y3, f(x3, y3)), (x4, y4, f(x4, y4)) form a square of side H. > > A[n apparently as yet unproved] theorem that would imply both > of these is this: > > (*) Claim: Any simple closed curve in R^3 contains the 4 > corners of a (planar) square. > > --Dan > > P.S. I will say, smugly, that I believe I know how to prove (*). I'd be interested to see a proof of this. But I don't see any connection between this theorem and the "placing a table on the floor so it doesn't wobble" theorem. Andy Latto andy.latto(a)pobox.com

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Re: [math-fun] You Can Balance a Wobbly Table by Rotating It
by Daniel Asimov 20 Feb '07

20 Feb '07

<< On NPR's Saturday Weekend Edition, they did a story with "Math Guy" Keith Devlin. He described a recent (last year) proof that, if the slope of the ground is at most about 35 degrees (actually, ArcTan[1/Sqrt[2]]), then you can balance a wobbly table by rotating it. (This assumes a mathematically perfect table, with the ground being a continuous function of x and y). >>

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[math-fun] You Can Balance a Wobbly Table by Rotating It
by Robert Baillie 20 Feb '07

20 Feb '07

On NPR's Saturday Weekend Edition, they did a story with "Math Guy" Keith Devlin. He described a recent (last year) proof that, if the slope of the ground is at most about 35 degrees (actually, ArcTan[1/Sqrt[2]]), then you can balance a wobbly table by rotating it. (This assumes a mathematically perfect table, with the ground being a continuous function of x and y). Links: Audio of NPR story: http://www.npr.org/templates/story/story.php?storyId=7466072 Keith Devlin's website with a description: http://www.maa.org/devlin/devlin_02_07.html Technical article: http://arxiv.org/PS_cache/math/pdf/0511/0511490.pdf Scientific American article: http://www.sciam.com/article.cfm?articleID=5B1CAEBA-E7F2-99DF-3CD6B81A81B51… Mathematica animated movie: http://www.maths.monash.edu.au/~bpolster/table.mov

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CORRECTION Re: [math-fun] Simplest Ovals (WAS: sections of quadratic surfaces)
by Daniel Asimov 20 Feb '07

20 Feb '07

I wrote: << An interesting family of ovoids (a word I prefer to oval, cuz oval has a deeply ingrained casual meaning) is given by applying the transformation (r,theta) -> (r^c, theta) (c > 1) to the circle r= cos(theta), i.e., with radius = 1/2 and center at (1/2,0). >> The next few lines need correction. They should read: The resulting curve has the equation r = cos(theta)^c, |theta| <= pi/2. When c = p/q for integers p, q > 0 in lowest terms, then the curve is given by the polynomial equation (x^2 + y^2)^(p+q) = x^(2p), x >= 0. --Dan

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[math-fun] Simplest Ovals (WAS: sections of quadratic surfaces)
by Daniel Asimov 19 Feb '07

19 Feb '07

As per David Cantrell's suggestion, I've been trying do define, using the simplest possible equations, an "oval" that has one curvature maximum and one minimum, such as k(s) = 2 + sin(s) where s is arclength, and k must of course be periodic. But without more critical points, such a k is impossible for a simple closed planar curve, thanks to the Four-Vertex Theorem. (Cf. < http://en.wikipedia.org/wiki/Four-vertex_theorem >.) So, can anyone think of a real-analytic simple closed convex planar curve as a candidate for the Simplest Oval ? (Perhaps there is a polynomial P(z) such that alpha(t) := P(exp(it)), 0 <= t <= 2pi works? In fact any solution must be of the form alpha(t) = f(exp(it)) for some analytic f.) Here's a possibility (that I haven't checked, since I'm about to leave on a 3-day trip back to Philly from Vancouver Island): Take an ellipse with semiaxes = 1 and 2 in the plane, with the end of a major semiaxis at the origin, and then apply the polar-coordinate transformation (r,theta) |--> (r^2, theta) to the ellipse. (I.e., (x,y) |--> (x sqrt(x^2+y^2), y sqrt(x^2+y^2).) --Dan

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[math-fun] Lizard's sequence
by Christian Boyer 19 Feb '07

19 Feb '07

In the March 2007 issue of "Pour La Science" (French edition of Scientific American), very interesting paper by Jean-Paul Delahaye on the Eric Angelini's sequences. Paper titled "La suite du lézard et autres inventions". OEIS is of course mentioned. For example, after the "prime" numbers, we can define the "second" numbers http://www.research.att.com/~njas/sequences/A123929 the "third" numbers, and so on... http://www.cetteadressecomportecinquantesignes.com/ThousandZetas.htm Other Angelini's sequence discussed in the paper, called by Delahaye "la suite du lézard" = "the lizard's sequence": http://www.cetteadressecomportecinquantesignes.com/Decimation.htm Eric Angelini is photographed showing a pizza. On his pizza, we can read OEIS, written with anchovies! Christian.

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