December 2012 - mathfun@mailman.xmission.com

Re: [math-fun] Mathematics of Juggling
by Henry Baker 04 Dec '12

04 Dec '12

More "Mathematics of Juggling" April 17, 2010 Allen Knutson, Cornell professor of mathematics and former world record-holding juggler, gives a public demonstration and lecture on how juggling patterns can be represented the mathematically. The mathematics of juggling was codified about 25 years ago, when three different groups of jugglers came up with a notational system, called "siteswaps," which made it easy to record patterns, note similarities and find new juggling patterns--even by computer. 65-minute video; 400MByte download http://cornellcast.video.cornell.edu/20111114/537/1_8ks3f11f_1_bow51q0c_1.m… At 06:50 PM 4/18/2012, Bill Gosper wrote: >Colin Wright (e.g., >*Colin Wright*: The Mathematics of *Juggling* on >Vimeo<http://vimeo.com/27998521> >)

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Re: [math-fun] New lighting technology breakthrough?
by Bill Gosper 04 Dec '12

04 Dec '12

[Not math, but hopefully fun] Automotive lights: Besides the traditional yellowish white headlights, we're seeing cold white, and pale green, blue, pink, and lavender. Plus those annoying prismatic things. But crawling down I280 tonight, I was struck by the uniformity of the red in the miles of taillights stretching out before me. Both LED and conventional. Is there some strict spectral standard? Or does my color discrimination roll off fast at the red end? hgb>To a first approximation, today's "white" LED and today's "white" fluorescent light are the same: They both excite a fluorescent dye with a blue/ultraviolet light. They differ in how the blue/ultraviolet light is created. The visible "white" color (or color "temperature") is dependent upon the type of dye. The dyes can be made from "quantum dots", which can be engineered to have the appropriate spectrum. "White" light can also be made using 3-4 colored LED's (green LED's are less efficient, so you tend to need RGGB instead of just RGB); this is used in large LED billboards and for LED TV's. Since you need 2 G's anyway, you can use G's with 2 different wavelengths to get a wider "color gamut" -- RGG'B. There are some LED TV's that utilize this idea. There are now LED stage/studio lights whose colors can be computer-programmed for different colors using these combinations. I would speculate that white light made from LED's of individual colors _might_ be more efficient (in terms of electrical power) than the white light from downconverted UV, but I haven't been able to confirm this. My best guess is that downconverted UV is cheaper/simpler to use, because it requires only one type of bulb, and because it draws upon a century of fluorescent light technology. At 04:50 PM 12/3/2012, Warren Smith wrote: The news hype media is full of stuff about how David Carroll, prof. at Wake Forest Univ. in NC, has invented new wondrous lighting technology, environmentally friendly, efficient as LEDs and "at least twice as efficient as fl.tubes," but simpler and looks better, wholy solid state, simple, most any shape (sheets, rods, bent rods,...), glowing capacitor using magic nanoparticles embedded in plastic, no toxic mercury like fl.tube has, long lived (>10 years), will be commercialized with in 1 year. I tracked it to this press release http://news.wfu.edu/2012/12/03/taking-the-buzz-out-of-office-lights/ and this pre-paper of some ilk: http://www.sciencedirect.com/science/article/pii/S1566119912004831 What the media hype machine never mentioned in about 10 articles I saw, but is revealed in the latter document, is that they incorporate substantial amounts of Iridium and Indium. That sounds like an immediate major lose right there. -- Warren D. Smith Just today I daydreamed about making a platinum-iridium disk puzzle which people couldn't game thermally. --rwg

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[math-fun] New lighting technology breakthrough?
by Warren Smith 04 Dec '12

04 Dec '12

The news hype media is full of stuff about how David Carroll, prof. at Wake Forest Univ. in NC, has invented new wondrous lighting technology, environmentally friendly, efficient as LEDs and "at least twice as efficient as fl.tubes," but simpler and looks better, wholy solid state, simple, most any shape (sheets, rods, bent rods,...), glowing capacitor using magic nanoparticles embedded in plastic, no toxic mercury like fl.tube has, long lived (>10 years), will be commercialized with in 1 year. I tracked it to this press release http://news.wfu.edu/2012/12/03/taking-the-buzz-out-of-office-lights/ and this pre-paper of some ilk: http://www.sciencedirect.com/science/article/pii/S1566119912004831 What the media hype machine never mentioned in about 10 articles I saw, but is revealed in the latter document, is that they incorporate substantial amounts of Iridium and Indium. That sounds like an immediate major lose right there. -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)

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Re: [math-fun] Related probability problem: Random disjoint segments in the unit interval
by Dan Asimov 03 Dec '12

03 Dec '12

By a random maximal collection of intervals of length L in [0,1], I mean: First, pick a point at random from the uniform distribution on [0,1] repeatedly until an interval of length L with that point as midpoint fits in [0,1]. That's s_1. (*) Inductively having picked disjoint intervals s_1,...s_n of length L, then if no complementary interval of length > L remains, stop. Then n*L = |I(L)| below. Otherwise, pick a point at random from the uniform distribution on [0,1] - (s_1 + . . . + s_n) repeatedly until it's the midpoint of an interval of length L in that complementary region (so is disjoint from the previous intervals). Now go to (*). --Dan << While not wishing to be tiresome, I nevertheless feel constrained to point out that Dan's original statement again begs the question of what exactly is being "randomised". While [Renyi's] parking problem is surely the most natural interpretation, one might instead for instance consider a (scaled-up) process placing unit-length intervals sequentially, separated by gaps chosen uniformly in the unit interval. In this case the mean separation is obviously 1/2, and the limiting density 2/3. Any problem which imprecisely combines geometry and probability is liable to give rise to such Bertrand-style paradoxes (more properly, inconsistencies) . . . I wrote: << 1) For small L > 0, Let I(L) be a random maximal collection of disjoint closed intervals each of length L and lying in [0,1] in R. Let |I(L)| denote the total length of all the intervals of I(L). It seems clear that there exists some constant C(1) such that, with probability 1, the limit of |I(L)| as L -> 0 = C. Question: Find C. ----------------- One 2D version of this is: 2) For small L > 0 let D(L) denote a random maximal collection of disjoint closed geometric disks each of diameter L and lying in [0,1]^2 in R^2. Likewise, as L -> 0 what is the limit C(2) of the total area of the disks of D(L) ? ----------------- n) The nD version: In [0,1]^n, what is C(n) for any n ? ----------------- (If you don't like edge effects, these questions can be asked for the cubical n-torus R^n / Z^n instead of [0,1]^n, with the same answer C(n).) >> >> Daniel Asimov Visiting Scholar Department of Mathematics University of California Berkeley, California

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Re: [math-fun] checkerboard version of an e-related dating problem--a graph theoretic approach
by W. Edwin Clark 03 Dec '12

03 Dec '12

Concerning Allan Wechsler's approach: Note that the number of valid (jammed) configurations for n vertices on a line is the number of maximal matchings of the path graph with n vertices. And the number of singletons for a given configurations is the number of vertices not in the matching. [This same characterization holds for a grid graph or indeed any simple graph.] I don't know if there is a good algorithm in general for enumerating the set of all maximal matchings for a graph, but I wrote a crude one for this case and was able to find directly Allan's sequences a[n],b[n] for n from 0 to 20: a = 1, 1, 1, 2, 2, 3, 4, 5, 7, 9, 12, 16, 21, 28, 37, 49, 65, 86, 114, 151, 200 b = 0, 1, 0, 2, 2, 3, 6, 7, 12, 17, 24, 36, 50, 72, 102, 143, 202, 282, 394, 549, 762 I cannot find b in the OEIS. My value for b[4] is 2 which differs from Allan's b[4] = 3. I didn't work out the recurrences, but Maple's gfun:-listtorec guessed that the recurrences for a and b are given by: {a(n+3)-a(n+1)-a(n) =0, a(0) = 1, a(1) = 1, a(2) = 1} {(n+3)*b(n+4)+(-5-n)*b(n+2)+(-6-n)*b(n+1) = 0, b(0) = 0, b(1) = 1, b(2) = 0, b(3) = 2} --Edwin On Wed, Nov 28, 2012 at 3:42 PM, Allan Wechsler <acwacw(a)gmail.com> wrote: > Going back to trying to make sense of the 1D problem, the number of valid > configurations for N people satisfies a[0]=a[1]=a[2]=1, a[n+3]=a[n]+a[n+1]. > There are about half a dozen sequences in OEIS with this "tribonacci" > rule, with minor differences in offset and initial conditions. > > Let b[N] be the aggregate number of singletons in all valid N-person > configurations. I'm getting a recurrence b[n+3] = a[n] + b[n] + b[n+1]. > Now b[0] = 0, b[1] = 1, b[2] = 0, and then the recurrence gives 2, 3, 3, > 6, 7, 10, 15, 19 ... The last entry diverges from A121833, but the rule is > so simple that I can't believe it's not in OEIS, so I must have > miscalculated. > > Anyway, the wanted expected number of singletons is lim b[n]/a[n]. Peering > at the small entries I realize I must have blundered somewhere. Can > somebody with more attention to give it please repeat this reasoning but do > it correctly? > > On Wed, Nov 28, 2012 at 3:09 PM, Dan Asimov <dasimov(a)earthlink.net> wrote: > > > Good point, but it wasn't intentional -- I was just careless. > > > > So, I should've said "then restrict attention to only those outcomes > > satisfying the conditions of the problem: That each person holds > > at most one hand and no two singles are adjacent." > > > > Or, put another way: After giving each pair of adjacent hands an > > independent ½ probability of holding each other, restrict attention > > to the configurations having a maximal set of adjacent pairs of > > gridpoints each connected by the line segment between them, > > such that no line segment touches another one. > > > > (This almost suggests a way to obtain the results for 1D.) > > > > --Dan > > > > > > On 2012-11-28, at 11:44 AM, Andy Latto wrote: > > > > > On Wed, Nov 28, 2012 at 1:49 PM, Dan Asimov <dasimov(a)earthlink.net> > > wrote: > > >> Whether in the 1D or 2D grid case (or for that matter in nD), > > >> one way to give this problem well-defined probabilities > > >> (that I think is in some sense the most natural way) > > >> is to first give each pair of adjacent hands an independent > > >> probability of ½ as to whether they or not they hold each other -- > > >> and then restrict attention to only those outcomes > > >> satisfying the conditions of the problem: That each > > >> person holds at most one hand. > > > > > > That's a different problem, because you're ignoring the second > > > condition of the problem; no two adjacent single people. If you > > > restrict attention to those, you have a formulation equivalent to > > > Allan's, I think, but different from both of mine. > > > > > > Andy > > > > > >> > > >> I'm not sure if this agrees with the probabilities > > >> in one of Andy's two methods. > > >> > > >> The problem becomes more symmetrical if the 1D case is a circle > > >> instead of a line (and in nD a K^n grid on an n-dimensional torus) -- > > >> but this won't affect the asymptotic expected fraction of singles. > > >> > > >> --Dan > > >> > > >> > > >> On 2012-11-28, at 6:49 AM, Andy Latto wrote: > > >> > > >>> On Wed, Nov 28, 2012 at 12:17 AM, Gary Antonick > > >>> <gantonick(a)post.harvard.edu> wrote: > > >>>> Cornell mathematician Steven Strogatz thinks this is an unsolved > > problem > > >>>> but am wondering if there's something being overlooked. > > >>>> > > >>>> Here's the basic -- and solved -- problem. There's a long line of > > people > > >>>> standing near each other. They're told to hold hands with someone - > > either > > >>>> the person to their right or to their left, if available, but not > > both. > > >>>> The ratio of the resulting singles to all the people in line > > approaches > > >>>> 1/e^2 as the line increases. > > > > > > _______________________________________________ > > math-fun mailing list > > math-fun(a)mailman.xmission.com > > http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun > > > _______________________________________________ > 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] Staples Announces In-Store 3-D Printing Service
by Henry Baker 02 Dec '12

02 Dec '12

FYI -- Fractal sculptures for all ? http://www.wired.com/design/2012/11/staples-goes-3-d/ WIRED Staples Announces In-Store 3-D Printing Service By Mike Senese 11.29.12 1:10 PM Pretty soon youÂll be able to print your 3-D projects at the local Staples. A new service called ÂStaples Easy 3DÂ will allow customers to upload their designs to StaplesÂ website, then pick up the printed objects at their local office supply megastore, or have them shipped to their home or business Â not unlike the photo- and document-printing service the company already offers. The project was announced today at Euromold 2012 by 3-D printer manufacturer Mcor Technologies, who is partnering with Staples to provide its new Iris printers for the service. The Iris printers employ an innovative method to generate objects, using reams of paper that are cut and printed while being stacked and glued together. This technique allows for a high-resolution layer thickness of 100 microns, similar to that of the MakerBot Replicator 2, but not quite as fine as the 25-micron capability of the Form 1. The new printers also incorporate the ability to add photorealistic coloring Â something that more common plastic printers canÂt yet achieve. But while the glued paper is said to have a wood-like hardness, the arrangement of the layered paper grain will require special consideration for certain design layouts (this can affect other types of 3-D printers as well). And while the company says it is able to be drilled, tapped or screwed, its material properties are unknown compared to traditional materials like real wood or steel. Still, the move by an established corporation to offer 3-D printing further legitimizes the adoption of the rapidly growing field by the mass market. Similar services currently exist, being offered by companies like Shapeways and Sculpteo, but this is the first to be made available from a chain retailer. Staples Easy 3D will launch in the Netherlands and Belgium in the first quarter of 2013, and will be rolled out to other countries shortly afterword. No word yet on pricing or when it will reach the United States. Mike Senese Mike Senese is the editor of Wired Design. He's also a TV host who discusses technology, teaches science, and inspires people to build things (and make pizza) through his DIY website, mikesenese.com/DOIT

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[math-fun] Astro-news
by Warren Smith 01 Dec '12

01 Dec '12

1. Incredibly, Mercury allegedly has ice (!!) on it. Also organic material. It's in deep polar craters. http://arstechnica.com/science/2012/11/craters-on-mercury-are-oases-for-wat… 2. Almost as incredibly, they've found a galaxy NGC1277 which is 250M lightyears away and contains a black hole massing 14% of the whole mass of that galaxy. (17 billion solar mass black hole). http://mcdonaldobservatory.org/news/releases/2012/1128.html -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)

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