January 2010 - mathfun@mailman.xmission.com

Re: [math-fun] another circuitous derivation
by Dan Asimov 26 Jan '10

26 Jan '10

One question: What's "Coogle" ? --Dan << . ___ _ . / __|___ ___ __ _| |___ .| (_ / _ \/ _ \/ _` | -_) . \___\___/\___/\__, |_\___| |___/ >> _____________________________________________________________________ "It don't mean a thing if it ain't got that certain je ne sais quoi." --Peter Schickele

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[math-fun] another circuitous derivation
by Bill Gosper 26 Jan '10

26 Jan '10

had among its terminal nodes In[652]:= Sum[(-1)^n*ArcTan[Csch[Pi*(n - 1/2)/Sqrt[3]]], {n, \[Infinity]}] == -5*Pi/24 Out[652]= 1 (n - -) Pi n 2 Sum[(-1) ArcTan[Csch[-------------]], n >= 1] Sqrt[3] 5 Pi = - ----. 24 It doesn't *look* that hard ... Joerg? Easy warmup: In[654]:= Product[Coth[Pi*(n-1/2)]^8, {n, \[Infinity]}] == 2 Out[654]= 1 8 Product[Coth[(n - -) Pi] , n >= 1] = 2. 2 --rwg

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[math-fun] Polynomial roots
by Joshua Zucker 22 Jan '10

22 Jan '10

A nice article on a new batch of pictures of roots of polynomials with integer coefficients. http://www.scientificamerican.com/article.cfm?id=math-polynomial-roots&sc=S… Enjoy, --Joshua

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[math-fun] sum-product matrices
by David Wilson 21 Jan '10

21 Jan '10

A matrix of distinct positive integers is sum-product if its rows have equal sum and columns have equal (possibly distinct) product. Any 1 x 1 matrix is trivially sum-product. No 1 x n or n x 1 matrix is sum-product. No 2 x 2 matrix is sum-product. In a sum-product matrix a b c d we must have a+b = c+d and ac = bd. But then ac+bc = bc+bd => (a+b)c = b(c+d) => c = b, and the elements are not distinct. There are an infinitude of 2 x 3 sum-product matrices. I believe the smallest, both in row sum and colum product, is 4 10 12 15 6 5 with row sum 26 and column product 60. I believe the following is true: For positive integers x, y, z, let u = x+y+z and v = xy+xz+yz. Then the matrix M(x, y, z) = vx vy vz uyz uxz uxy has equal row sums uv and equal column products xyzuv. If you choose x < y < z with xz < y^2 and gcd(x, y, z) = 1, form M(x, y, z), and divide this by the gcd of its elements, then I believe you will generate the "primitive" 2 x 3 sum-product matrices. I'm pretty sure that apart from 1 x 1 matrices, there are no square sum-product matrices. ? But I was pleased to find that there are indeed sum-product matrices with equal row sum and column product. The following are (all?) "primitive" 14 x 2 sum-product matrices with sum = product = 1890. By "primitive" I mean distinct up to row and column permutation. 2 3 15 45 63 70 90 105 135 189 210 270 315 378 945 630 126 42 30 27 21 18 14 10 9 7 6 5 2 5 7 21 30 45 70 105 126 135 189 210 315 630 945 378 270 90 63 42 27 18 15 14 10 9 6 3 2 5 9 14 27 45 63 90 105 126 189 270 315 630 945 378 210 135 70 42 30 21 18 15 10 7 6 3 2 5 14 15 18 27 42 63 90 189 210 270 315 630 945 378 135 126 105 70 45 30 21 10 9 7 6 3 2 6 7 18 27 30 42 90 126 135 189 210 378 630 945 315 270 105 70 63 45 21 15 14 10 9 5 3 2 7 9 15 18 27 30 45 90 135 189 315 378 630 945 270 210 126 105 70 63 42 21 14 10 6 5 3

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[math-fun] Multimagie update
by Christian Boyer 21 Jan '10

21 Jan '10

www.multimagie.com/indexengl.htm: an update is now online, click on News of January 2010 Among the news you will see that my 5 enigmas of 2008 are still unsolved! And several news on additive-multiplicative magic squares, magic when you add the cells, and still magic when you multiply the cells: - a new 6th enigma, construct a 5x5 add-mult square - several new add-mult squares including my big 1024x1024 square, its magic sum is 274878169600, but its magic product has 8356 digits... kindly checked by two persons of [math-fun], thanks to Joshua Zucker and W. Edwin Clark - J. Dénes / A. D. Keedwell / P. Erdös had already posed in 1974 the problem on the possible add-mult squares - help needed, any person having some biographical info on the first author of this kind of squares: Walter W. Horner, Pittsburgh??? Direct access to the unsolved enigmas: www.multimagie.com/English/Enigmas.htm Christian.

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[math-fun] Triangle puzzle
by Dan Asimov 19 Jan '10

19 Jan '10

Let a, b, m > 0, with m < min{1/a, 1/b}. Let T be the triangle in R^2 with vertices A = (-a,0), B = (b,0), and C = (0,1). Draw the NE-ish segment at slope m from L(0) = (0,0) to BC, and denote its right endpoint as R(1). Draw the NW-ish segment at slope -m from R(1) to AC and denote its left endpoint as L(1). Repeat infinitely many times, obtaining R(2), L(2), etc. What is the total length of all the segments so drawn? --Dan _____________________________________________________________________ "It don't mean a thing if it ain't got that certain je ne sais quoi." --Peter Schickele

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[math-fun] Polynomial Spirals
by Kerry Mitchell 12 Jan '10

12 Jan '10

Hi all, I came across polynomial spirals as generalizations of the Cornu spiral: http://www.2dcurves.com/spiral/spiralps.html At the bottom of this page are 4 examples where the curvature is a polynomial function of the arc length. How can one generate a pair of parameterized equations for x and y, given the fact that the curvature is a polynomial function of the arc length? Thanks, Kerry -- lkmitch(a)gmail.com www.kerrymitchellart.com http://spacefilling.blogspot.com/

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Re: [math-fun] comparing homogeneous vectors
by Dan Asimov 12 Jan '10

12 Jan '10

I was going to answer this until I noticed Fred had written: << [N]onzero vectors may possibly be isotropic --- |X| = 0 with X /= 0. Suddenly I have no idea what this means. One of the axioms of a metric (or rather a norm on which a metric is based) is ||X|| = 0 <=> X = 0. So I am confused as to what sort of "metric" is being considered here, or else what the notation ||X||, er |X|, means. --Dan Fred wrote: << Suppose we have a space with points represented by homogeneous coordinate vectors --- that is, {F - 0} factored out, where F is some continuous ground field, in practice \R or \C. How do we tell whether two points are (approximately) the same? If the metric is Euclidean (e.g. real projective), a slightly clumsy solution is to rescale both vectors X,Y to length unity, and check for either |X-Y| and |X+Y| small; unless |X| or |Y| is small to start with, when it can be snapped back to zero (representing no point). But when the metric is more general (e.g. complex projective, conformal inversive), nonzero vectors may possibly be isotropic --- |X| = 0 with X /= 0. Ignoring the local metric, performing the comparison using instead a Euclidean metric, offends against my aesthetic scruples; in particular failing to be invariant under local isometries. This apparently trivial algorithmic conundrum has irritated me for years. Has anybody got a better idea? >> _____________________________________________________________________ "It don't mean a thing if it ain't got that certain je ne sais quoi." --Peter Schickele

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[math-fun] In a Series, Nickelodeon Will Focus on Math
by Henry Baker 12 Jan '10

12 Jan '10

FYI -- http://www.nytimes.com/2010/01/11/business/media/11math.html January 11, 2010 In a Series, Nickelodeon Will Focus on Math By ELIZABETH JENSEN Nickelodeon, whose preschool shows focus on teaching social skills as much as letters and numbers, will move squarely into the academic realm, with the introduction of ÂTeam Umizoomi,Â which it said is the only preschool series centered entirely on teaching math to children. The half-hour show, a mix of animation and live action that has been in development since late 2005, will go on the air on Jan. 25 in the networkÂs midday lineup. Cyma Zarghami, NickelodeonÂs president, said that the network saw an opening and took it. ÂEverybodyÂs trying to teach preschoolers how to read and nobody is trying to teach them how to do math,Â she said. The move puts Nickelodeon more squarely in competition with ÂSesame StreetÂ on PBS, which has math skills as a major component. ÂWeÂre not deliberately trying to get into their space, but I do believe the opportunity in this space has been left wide open,Â said Ms. Zarghami. Â ÂSesame StreetÂ is iconic and will live on forever,Â she added. ÂBesides them thereÂs just a smattering of othersÂ that incorporate teaching math skills to the age group. PBSÂs ÂCyberchaseÂ is math-focused, but is aimed at older children, ages 8 to 12. Nickelodeon has ordered 20 episodes of ÂTeam Umizoomi.Â If it catches on, it could also open a new revenue source for the company, in ancillary educational products. Ms. Zarghami said that Nickelodeon is exploring introducing teaching videos, workbooks and Âlearning tools that go with the show, as opposed to the more traditional toys and plush.Â Currently the company licenses some of its properties, including ÂDora the ExplorerÂ and ÂGo, Diego, Go!Â for tie-in math and reading games made by others, but it has no shows whose product lines are specifically education-oriented. In some ways the new program is standard childrenÂs fare. The main characters are miniature superheroes Â a boy, a girl and a break-dancing robot Â who zoom about fixing simple crises in their city, whether a shortage of milk, a lost kite or a subway system stalled by a dropped mitten. But the show is infused in all aspects Â down to character MilliÂs pony tails that turn into a ruler Â with concepts from an interactive math curriculum that the characters tap to solve their problems, including shape-matching, counting, simple computation and measurement. The Nick Jr. Web site, nickjr.com, will have a learning section for parents, including the curriculum and activities to help preschoolers master the concepts, as well as games and videos. Soo Kim, who worked on ÂBlueÂs CluesÂ and is one of ÂTeam UmizoomiÂsÂ creators and writers and its design director, said the show was built on the idea that Âpreschoolers naturally encounter their world by using their math,Â matching buttons to holes or counting out pancakes for breakfast. The message to children, she said, is ÂyouÂre already great at math.Â Pop songs, like one with the refrain ÂShapes all over the placeÂ in the first episode, are modeled after ABCÂs ÂSchoolhouse RockÂ ditties, which endured long after they left the air. NickelodeonÂs programming has become Âincreasingly more educational as time has passed,Â Ms. Zarghami said. ÂBlueÂs CluesÂ promotes interactivity, with its pauses for children to find the clues, and ÂDora The ExplorerÂ teaches short Spanish phrases while the character is on her adventures. With ÂTeam Umizoomi,Â she said, there is Âno disguising that itÂs math.Â The show does not signal a major shift in the networkÂs shows. ÂI believe we have enough equity with moms based on our track record to take on more educational stuff head-on, and they would trust us to do it well,Â said Ms. Zarghami, but Âsocial skills are still really important.Â

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[math-fun] comparing homogeneous vectors
by Fred lunnon 11 Jan '10

11 Jan '10

Suppose we have a space with points represented by homogeneous coordinate vectors --- that is, {F - 0} factored out, where F is some continuous ground field, in practice \R or \C. How do we tell whether two points are (approximately) the same? If the metric is Euclidean (e.g. real projective), a slightly clumsy solution is to rescale both vectors X,Y to length unity, and check for either |X-Y| and |X+Y| small; unless |X| or |Y| is small to start with, when it can be snapped back to zero (representing no point). But when the metric is more general (e.g. complex projective, conformal inversive), nonzero vectors may possibly be isotropic --- |X| = 0 with X /= 0. Ignoring the local metric, performing the comparison using instead a Euclidean metric, offends against my aesthetic scruples; in particular failing to be invariant under local isometries. This apparently trivial algorithmic conundrum has irritated me for years. Has anybody got a better idea? WFL

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