September 2007 - mathfun@mailman.xmission.com

[math-fun] Re: collective
by Steve Witham 25 Sep '07

25 Sep '07

>From: Jon Ziegler <jonz(a)alumni.caltech.edu> > >Oddly, a quick perusal uncovers no specific word for a collection of >collective nouns. I have to assume I've just missed it. And a >collection of those words would be...? A plurality? Maybe a pluralism. --Steve

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[math-fun] Correction re: jaggies vs. Brezenham
by Steve Witham 24 Sep '07

24 Sep '07

> >From: "R. William Gosper" <rwg(a)osots.com> >> >>http://gosper.org/jaggies.png is a 4x blowup of a detail of a Golly >>(Game of Life) >>screen, where each pixel is 2^24 x 2^24 cells. The diagonal line >>marked by arrows > >is a row of equally spaced gliders, (dx,dy) = (14412108,16685960). Another way of explaining that horizontal jog, even though dy > dx: +-----+-----+ | | * | glider very close to the top of a pixel | | | | * | | glider very close to the bottom +-----+-----+ >From: Steve Witham <sw(a)tiac.net> > >Yeah but Bresenham is just an >optimized way of drawing the same jaggies. I was wrong. Bresenham works by setting one of the step sizes to one pixel and the other to <= 1, instead of stepping the x and y by independent fractions. Bresenham would never get the kind of jaggy pointed to in that Golly screen (but the Golly screen is correctly showing which pixels have gliders). Trying to imagine an antialiasing method for Life that would show things with such low density (~2^-46). All I can think of is, color based on the distance from the pixel center to the closest live cell within +/- 1 pixel. Intuition can be so demanding. --Steve

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Re: [math-fun] Compute a Function From its Fourier Series?
by Eugene Salamin 24 Sep '07

24 Sep '07

----- Original Message ---- From: Mike Stay <metaweta(a)gmail.com> To: math-fun <math-fun(a)mailman.xmission.com> Sent: Monday, September 24, 2007 9:00:01 AM Subject: Re: [math-fun] Compute a Function From its Fourier Series? On 9/23/07, mcintosh(a)servidor.unam.mx <mcintosh(a)servidor.unam.mx> wrote: > 2) The variance (or better, the standard deviation) of a function > is inverse to the variance of its Fourier Transform. Thus, many > large coefficients are needed to confine a function to a small > region and conversely. Which is a general form of Heisenberg's uncertainty principle. -- Mike Stay metaweta(a)gmail.com http://math.ucr.edu/~mike It is not correct to refer to this as "Heisenberg's" uncertainty principle. Heisenberg's uncertainty relates the variances of position and momentum. In quantum mechanics there is a connection between momentum and spatial frequency, and so the Fourier uncertainty principle can then be applied. The Heisenberg principle is exclusively quantun mechanical in nature. A classical electromagentic pulse has an exactly calculable momentum and an exactly calculable center of mass position. This is true because the energy density (E^2 + B^2) and momentum density (E x H) depend only on the fields, and not on the frequency or wavelength. Gene ____________________________________________________________________________________ Fussy? Opinionated? Impossible to please? Perfect. Join Yahoo!'s user panel and lay it on us. http://surveylink.yahoo.com/gmrs/yahoo_panel_invite.asp?a=7

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[math-fun] Compute a Function From its Fourier Series?
by Robert Baillie 24 Sep '07

24 Sep '07

One of the first examples of Fourier series that a student encounters is something like this: f(x) = x/2, for -Pi < x < Pi. The Fourier series is Sum[ (-1)^(n+1)/n * Sin[n x] ]. A second, more complicated example is: f(x) = -Pi/2, for -Pi < x < 0, f(x) = Pi/2, for 0 < x < Pi. This Fourier series is Sum[ (1 - (-1)^n)/n * Sin[n x] ]. Can one work backwards, from the coefficients to the function? That is: given an expression in n, say, c(n) = (-1)^(n+1)/n, can one find the function that has c(n) as its coefficients? This seems difficult, especially because in the second example, the function consists of two distinct pieces over [-Pi, Pi], and as part of the calculation, you'd have to find the point (x = 0) where the function jumps. One could plot 1000 terms of Sum[ c(n) * Sin[n x] ] for -Pi < x < Pi, then try to guess what the expressions and points of discontinuity of f(x) are. Or, one could compute 1000 terms for many x's and do curve fitting, but guessing would be involved here, too. But is there a systematic method to do this? Bob Baillie

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[math-fun] Problems that led to branches of science ...
by Guy Haworth 24 Sep '07

24 Sep '07

Did the 'Bridges of Koningsberg' lead to topology or just to Graph Theory? I was at a talk on Euler by Prof. Im Hof, and he didn't mention Topology in this context. Guy

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[math-fun] Re: Jaggies
by Steve Witham 23 Sep '07

23 Sep '07

>From: "R. William Gosper" <rwg(a)osots.com> > >http://gosper.org/jaggies.png is a 4x blowup of a detail of a Golly >(Game of Life) >screen, where each pixel is 2^24 x 2^24 cells. The diagonal line >marked by arrows >is a row of equally spaced gliders, (dx,dy) = (14412108,16685960). The gap at >the green arrow is a legitimately missing glider, but every other dot on that >line represents one glider. I.e., Golly darkens a pixel iff there's >anything in >the 2^24 square it represents. Note that dy>dx. >Puzzle: How can the nonmonotonicity at the black arrow be anything but >a display bug? (Does anybody remember BRMs and DDAs?) Both dx and dy are < 2^24. So, going up-right, the next glider can either be on the same row of pixels, or one row up; on the same column, or one column to the right. That means it might be up, up-right, right, or in the same pixel. The arrow points to a place where it goes right (but not up). The places where you don't go up are relatively rare (once every ~184 gliders) because 16685960 is close to 16777216=2^24. Row number and column number as functions of glider number are related to the continued fractions for dy/2^24 and dx/2^24. They will look like fractals until you get to the LCM. The jaggies are like the beating of those two fractals against each other, and resolve to a pattern when you get to the LCM of all three numbers. That is, when the number of gliders in a string is equal to the LCM. For a similarly incoherent explanation of this same integer-division jaggy phenomenon (only with irrational numbers) see the section called "Why the Fractal Patterns?" on this page: http://www.tiac.net/~sw/2005/03/Mandala/index.html Off Google: "BRM -. Binary Rate Multiplier,. and DDA - Digital. Differential. Analyzer" Don't know if a DDA analyzes the way I just did. related to Bresenham's algorithm. Yeah but Bresenham is just an optimized way of drawing the same jaggies. --Steve

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[math-fun] Null polarity group in 3-space
by Fred lunnon 23 Sep '07

23 Sep '07

I have recently been inspecting the Lie group of geometric transformations generated by the "null polarities" in real projective 3-space associated with nonsingular, nondegenerate "linear complexes" [sets of lines subject to single linear constraints on their Pluecker coordinates]. Amongst other things, I have ascertained that: This group is isomorphic to the indefinite orthogonal group O(3,3) [preserving the quadratic form (x1)^2+(x2)^2+(x3)^2-(y1)^2-(y2)^2-(y3)^2] over the real numbers; it has dimension 15. It includes the dimension 6 Euclidean group SE(3) of proper (continuous) isometries; also the dimension 3 "axial dilations" transforming a sphere into an ellipsoid. Are these properties well-known? Are they interesting or useful? What else is known about this group? [I am particularly interested in characterising the continuous component synthetically; as well as establishing its relationship to the other three.] Can anybody direct me to pertinant existing references? Fred Lunnon [23/09/07]

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[math-fun] El Matemático question
by James Buddenhagen 22 Sep '07

22 Sep '07

In 1918 (or 1919?) the acclaimed Mexican artist Diego Rivera painted a painting called "El Matemático" (the mathematician). You can see it here: http://www.abcgallery.com/R/rivera/rivera16.html I am trying the find out who is depicted in that painting, which, by the way, was done in Paris. Can anyone help? Thanks. Jim Buddenhagen

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[math-fun] jaggies
by R. William Gosper 20 Sep '07

20 Sep '07

http://gosper.org/jaggies.png is a 4x blowup of a detail of a Golly (Game of Life) screen, where each pixel is 2^24 x 2^24 cells. The diagonal line marked by arrows is a row of equally spaced gliders, (dx,dy) = (14412108,16685960). The gap at the green arrow is a legitimately missing glider, but every other dot on that line represents one glider. I.e., Golly darkens a pixel iff there's anything in the 2^24 square it represents. Note that dy>dx. Puzzle: How can the nonmonotonicity at the black arrow be anything but a display bug? (Does anybody remember BRMs and DDAs?) --rwg

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Re: RE : [math-fun] First differences spoiler
by Michael Kleber 19 Sep '07

19 Sep '07

Eric Angelini wrote: > O N E N I N E N I N E F I V E F I V E > 1 9 9 5 5 9 9 5 5 9 1 3 13 17 1 3 13 17 --> first (letters rank) diff. is seq. itself Very nice! The existence of "one", with alphabetically adjacent first two letters, is a lucky coincidence -- it's the only place you could start ("two", "five", and "thirteen" are all off by one!). Aha! So the proper French translation of "One, nine, nine, five, five, ..." is "Quatre, vingt, dix-neuf, deux, treize, ..."? But wait! "Neuf, seize, quinze, treize, quatorze,..." works too. Ah, those lucky French. --Michael Kleber -- It is very dark and after 2000. If you continue you are likely to be eaten by a bleen.

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