October 2010 - mathfun@mailman.xmission.com

Re: [math-fun] Spikey Spheres
by Dan Asimov 30 Oct '10

30 Oct '10

I've dealt with the need to explore the set of all directions (say, points on the unit sphere) in high-dimensional Euclidean space, and it is certainly true that it's *much* harder to get close to all possible directions. This is known as the "curse of dimensionality" (a phrase that probably originated with John Tukey). For the case in point, Mathematica tells me that a spherical cap in R^n of radius = pi/4 has a relative solid angle of 2.22 x 10^(-273). ----------------------------------------------------------------- In[1]:= alpha[n_] := 2.0 * Pi^(n/2) / Gamma[n/2] In[2]:= scap[theta_,n_] := alpha[n-1] * NIntegrate[Sin[u]^(n-2), {u,0,theta}] In[3]:= fraccap[theta_,n_] := scap[theta,n] / scap[Pi,n] In[4]:= fraccap[Pi/4,1800] -273 Out[4]= 2.223961284661308 10 ----------------------------------------------------------------- But despite the interesting geometrical phenomenon that, starting in 9 dimensions, the sphere tangent to the 2^n "corner" spheres of an n-cube (each of diameter = half the side of the cube) has diameter >= 1 (with strict inequality for dimensions > 9), I would *not* call high-dimensional spheres "spiky". This is imo misleading and not helpful to one's intuition. Rather, I'd call them "roomy". --Dan << http://www.penzba.co.uk/cgi-bin/PvsNP.py?SpikeySpheres " ... I've recently been working on an optimisation problem, and I've come to realise that I can consider it as wandering around on a smooth landscape in 1800 dimensions. Strange number, I know, but that's the way it's worked out. The problem is that the usual visualisation of hill-climbing, or hill-descending, is that you're on a gently undulating vista, and that some directions are up, some are down, and it's easy to decide which is which. You write the code, set off, and somehow the system never finds a good solution. Part of the problem is that there is simply a lot of space to explore. If you discretise space and have 1000 places to be in each dimension, 2 dimensions gives you a million places to be. That's not so bad. 1800 dimensions gives you 10^5400 places to be. That's not good. You definitely need to move in moderately large strides, and then hone your solution by using binary chop or similar techniques. But it's worse than that. The problem is that the error function may be "smooth," but your intuition of what this means is wrong. ... " >> Those who sleep faster are sooner rested.

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[math-fun] Spikey Spheres
by Ray Tayek 30 Oct '10

30 Oct '10

http://www.penzba.co.uk/cgi-bin/PvsNP.py?SpikeySpheres " ... I've recently been working on an optimisation problem, and I've come to realise that I can consider it as wandering around on a smooth landscape in 1800 dimensions. Strange number, I know, but that's the way it's worked out. The problem is that the usual visualisation of hill-climbing, or hill-descending, is that you're on a gently undulating vista, and that some directions are up, some are down, and it's easy to decide which is which. You write the code, set off, and somehow the system never finds a good solution. Part of the problem is that there is simply a lot of space to explore. If you discretise space and have 1000 places to be in each dimension, 2 dimensions gives you a million places to be. That's not so bad. 1800 dimensions gives you 10^5400 places to be. That's not good. You definitely need to move in moderately large strides, and then hone your solution by using binary chop or similar techniques. But it's worse than that. The problem is that the error function may be "smooth," but your intuition of what this means is wrong. ... " --- co-chair http://ocjug.org/

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Re: [math-fun] How many spokes does a bicycle need?
by Dan Asimov 30 Oct '10

30 Oct '10

>From various comments made, especially by Fred, I think the intended problem is something like this: Given two finite cylinders C_r, C_R of radii r < R and equal heights in R^3, how many constraints of the form ||x_j - y_j|| = c_j, for x_j in C_r and y_j in C_R in order to require C_r to be fixed in a uniqconcentric I wrote: << In the biography of H.S.M. Coxeter "King of Infinite Space", one section describes his interaction with Buckminster Fuller. It states that Fuller erroneously claimed that a bicycle wheel needs 12 spokes "to hold it rigid", whereas the correct number is 7. This seems to be stated as a fact of geometry, rather than one of structural engineering. Anybody have any insight into this claim? >> Those who sleep faster are sooner rested.

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[math-fun] 3D monotiles
by rcs＠xmission.com 29 Oct '10

29 Oct '10

Has anyone tackled the 3D version of tiling space with copies of a single convex polyhedral tile? The 2D problem turned out to be surprisingly complicated. Does anyone know the smallest non-tiling polycube? Rich

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Re: [math-fun] pi cf record
by Bill Gosper 29 Oct '10

29 Oct '10

Denote by a[0], a[1], ... the continued fraction terms of pi (= 3,7,15,1,... = A001203). If there are rationals A,B,C,D such that pi = (Ae+B)/(Ce+D), then there is a quasiperiod q and a preamble p such that the second differences a[n+2q]-2a[n+q]+a[n] = 0 for all n>p. For e itself, q=3 and p=1. (Or any positive multiple of 3 and any positive integer). E.g., define a second-differencer: In[13]:= dif2[L_List, q_Integer:1] := Drop[L, 2*q] - 2*Drop[Drop[L, -q], q] + Drop[L, -2*q] In[16]:= dif2[{a, b, c, d, e}, 2] Out[16]= {a - 2 c + e} Then In[163]:= dif2[ContinuedFraction[E, 22], 3] Out[163]= {1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0} (p=1,q=3). In[166]:= dif2[ContinuedFraction[E^(2/3), 22], 5] > Out[166]= {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0} > (p=0,q=5). > In[171]:= dif2[ContinuedFraction[(E - 1)/(1 - E/6), 333], 130] > > Out[171]= {2, 1, 11, -1, 0, 62, 1, 3, 0, -1, 0, 1, -5, 5, 5, 1, -2, \ > -153, -2, 0, 1, -58, 5, 2, 1, 152, 2, 57, -6, 3, -2, -21, 6, -10, 3, \ > 22, -18, 3, 18, 4, 0, 3, 1, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, \ > 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0} > (p=44, q=130) > > By second-differencing Neil's 458585379 terms one can contrive at most > seven(!) > terminal 0s, with, e.g., p=458585379-2q, q=16400301. But not even these > seven > can signal an incipient quasiperiod, because annihilating all future second > differences requires that some yet-to-be-computed terms be nonpositive! A > program has confirmed that *all* terminal bursts, even of length one, are > similarly doomed, thereby precluding any e-like (Hurwitz) behavior with > p+2q < 458585379. Thus at least one of A,B,C,D has a big numerator or > denominator. (How big?) > --rwg > Maybe not huge. For 3.141595... = 4-6e/19, p=447 and q=676. > > >

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Re: [math-fun] simple curves on punctured planes
by Dan Asimov 29 Oct '10

29 Oct '10

Fred wrote: << On 10/24/10, Dan Asimov <dasimov(a)earthlink.net> wrote: << [Most of this was written before several posts that explain much of what I've described below, but for clarity I'm leaving this unedited.] Let C be a simple closed curve in R^3 - {p,q,r}, where #{p,q,r} = 3. >> Should read " R^2 - {p,q,r} "? >> Yes, indeed. Thank you for pointing that out. << ... In the case of the simple closed curve of the painting, it's fun to let the exterior "ooze into" the curve, by shading that in. What remains is a very skinny and folded snake with one eye at each end (namely, the two punctures in its interior). Once this becomes clear, it's easy to imagine how a sequence of Dehn twists might result in unfolding the snake. The inverse of the corresponding word is, I'm guessing, what was painted on the wall. (But the resolution of the photo is too low for me to be able to read that word.) >> a b c' b' a' b' a b c b' a' b c' b' a' b' a b c' b' a' b a b c b' --- the last "a" being a trifle dubious. WFL >> Wow, when I magnify that picture on my computer I can't make head or tail of those letters. But if there are three of them, it's clear that my guess (that it was a word in the usual two Dehn twists that generate that group) was wrong. --Dan Those who sleep faster are more rested sooner.

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Re: [math-fun] simple curves on punctured planes
by Dan Asimov 28 Oct '10

28 Oct '10

This is essentially what I suspect, also. The curve C, as is easy to see, has winding number +1 about 2 of the holes, and 0 about the other one. (Didn't Allan mention this, though?) The curve C can be obtained by starting with a simple closed curve C_0 enclosing Q and R but not P. Then some homeomorphism h: R^2 - {P,Q,R) -> R^2 - {P,Q,R} (determined up to isotopy) takes C_0 to C. The group of orientation-preserving homeomorphisms of R^2 - {P,Q,R} -> R^2 - {P,Q,R}, up to isotopy, is generated by Dehn twists about {P,Q} and about {Q,R} . . . though I neglected to say "orientation-preserving" earlier. These twists if chosen well will leave a basepoint * fixed, and thus the group Isot(R^2 - {P,Q,R}) will define a homomorphism phi: Isot(R^2 - {P,Q,R}) -> Aut(Pi_1(R^2 - {P,Q,R}, *) via phi([h])([L]) = h_#([L]), (h_# denoting h on the level of pi_1 for any homeomorphism h: R^2 - {P,Q,R} -> R^2 - {P,Q,R} that is the composite of a finite sequence of these Dehn twists, and any loop L based at *). If C_0 contains *, then so will C, which can then be expressed as an element of pi_1(R^2 - {P,Q,R}, *), i.e., the free group on the loops a, b, c based at *, going around P, Q, R -- presumably the word accompanying the painting. (Which certainly was a lot of words to say very little about one.) --Dan Rich wrote: << I'll hazard a guess that the curve in the picture is supposed to be the homotopic sum of loops around the individual punctures, and that the Word is the formula for the curve. Perhaps A is a simple counterclockwise loop around the first puncture, etc. If so, the transcription has a couple of problems: For the winding number of the curve to be 0 around each puncture, the total exponent of A should be 0 (it is), and ditto for B and C. (Nope: There are extra B' and C'.) IIRC, a contractable loop would simplify to Identity when A,B,C are interpreted in the (non-commutative) free group on three generators. I'm uneasy about this interpretation, since the rules for homotopy of curves allow the curve to cross itself, while the puzzle at hand forbids this. >> Those who sleep faster are sooner rested.

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Re: [math-fun] math-fun Digest, Vol 92, Issue 20
by Stuart Anderson 26 Oct '10

26 Oct '10

How many spokes in this 'fractal' bike? It was constructed outside the building where I work; http://sydney-city.blogspot.com/2010/10/martin-place-bike-bike.html it has survived drunken revellers and bike thieves (they couldnt get a single bike off the structure). The penny farthing bike has a large wheel with 24 spokes This got me thinking about wheels on wheels and how to connect them using regular polygon tiling. a 24-gon, an octagon and an equilateral triangle can tile a vertex sum of 2Pi radians http://www.squaring.net/gfx/bikes/3-8-24.png some of them look like designs for Ferris wheels; http://www.squaring.net/gfx/bikes/3-9-18disc.png polygon tilings which work; 3, 8, 24 http://www.squaring.net/gfx/bikes/archimedean3-8-24tile.svg 3, 7, 42 This has the largest regular polygon, would make a good penny farthing wheel; start by tiling 3 and 7 around the rim; http://www.squaring.net/gfx/bikes/3-7-42polygons.png add spokes; http://www.squaring.net/gfx/bikes/3-7-42spokes.png You can go further, creating spokes which extend to make an even larger wheel; http://www.squaring.net/gfx/bikes/3-7-42project.png and even further again; http://www.squaring.net/gfx/bikes/3-7-42-extended-more.png 3, 9, 18 http://www.squaring.net/gfx/bikes/3_9_18.png 3, 10, 15 (15 is odd so cant go all the way around the rim without overlapping polygons) http://www.squaring.net/gfx/bikes/3-10-15.png but i made a wallpaper (by leaving some irregular polygon 'gaps') http://www.squaring.net/gfx/bikes/1st-2nd-3rd-18.9.3-reg-polygon-vertex-pla… 3, 12, 12 http://www.squaring.net/gfx/bikes/3-12-12.png 4, 5, 20 as a tiling http://www.squaring.net/gfx/bikes/4-5-20.png some good software for polygon tiling here from Melinda Green; http://www.superliminal.com/geometry/tyler/Tyler.htm cheers, Stuart Anderson On Tue, Oct 26, 2010 at 11:41 AM, <math-fun-request(a)mailman.xmission.com>wrote: > Send math-fun mailing list submissions to > math-fun(a)mailman.xmission.com > > To subscribe or unsubscribe via the World Wide Web, visit > http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun > or, via email, send a message with subject or body 'help' to > math-fun-request(a)mailman.xmission.com > > You can reach the person managing the list at > math-fun-owner(a)mailman.xmission.com > > When replying, please edit your Subject line so it is more specific > than "Re: Contents of math-fun digest..." > > > Today's Topics: > > 1. Re: How many spokes does a bicycle need? (Henry Baker) > 2. Re: How many spokes does a bicycle need? (rwg(a)sdf.lonestar.org) > 3. Negative yield? (Dan Asimov) > 4. Re: Negative yield? (Henry Baker) > 5. Re: Negative yield? (Joshua Zucker) > 6. Re: How many spokes does a bicycle need? (Fred lunnon) > > > ---------------------------------------------------------------------- > > Message: 1 > Date: Mon, 25 Oct 2010 12:06:36 -0700 > From: Henry Baker <hbaker1(a)pipeline.com> > To: Gary Antonick <gantonick(a)post.harvard.edu> > Cc: math-fun(a)mailman.xmission.com, Dan Asimov <dasimov(a)earthlink.net> > Subject: Re: [math-fun] How many spokes does a bicycle need? > Message-ID: <E1PASOL-0005jZ-HT(a)elasmtp-kukur.atl.sa.earthlink.net> > Content-Type: text/plain; charset="iso-8859-1" > > FYI -- I'm not trying to hijack this thread, but the frame in this video is > pretty cool... > > http://www.sciencedaily.com/videos/2008/1201-hitech_cycling.htm > > Hi-Tech Cycling > > Engineers Create A Strong But Lightweight Isotruss Bike Using Carbon Fibers > > December 1, 2008 ? Engineers used elements of architecture and geometry to > create a strong but lightweight triangle-based isotruss bicycle frame. To > make a road bike or mountain bike, the isotruss is first wound with carbon > fiber using a sheet that holds the tension constant. The engineers then > hand-wind Kevlar strands over the isotruss. The process creates a bike with > a large strength-to-weight ratio. > > Almost every kid has at one time or another asked for one for Christmas. > Now, engineers have developed what may be the most technologically advanced > bike to hit the road yet. It took ten years to develop a new incredibly > light and strong model that will take cyclists into the future. > > Karl Vizmeg has ridden his Delta 7 Arantix bike 1,700 miles. He has raced > dozens of bikes, but says a new see-through model is the strongest and > lightest. > > "This is phenomenal," said Vizmeg. "I've had so much fun this year, > particularly with the 'wow' factor, but [also] because it's such a great > racing bike." > > Vizmeg's $8,500 bike was handmade in Utah using geometry and architecture. > To make bikes like his, workers first make an isotruss, a form made from > isosceles triangles. Then, they wind carbon fiber around the form -- > creating a great strength-to-weight ratio. > > "We go back afterwards and hand-wind all the little Kevlar strands, > inch-by-inch, over each isotruss," said Tyler Evans, program manager at > Delta 7 Sports in Payson, Utah. > > They then bake the bike to bond the materials. The mountain bike frame > weighs 2.6 pounds. The new Ascend road bike weighs 1.8. > > "This bike rides like bikes that are much heavier and stronger and built > like a tank, but it's still in the featherweight category," Evans said. > > You might think the open-lattice design wouldn't be aerodynamic, but Delta > 7 says wind-tunnel tests prove the bikes are as aerodynamic as traditional > ones. The Ascend bike has another advantage. > > "You definitely feel like a rock star, like you're famous, like you belong > in the Tour de France or some high-end race," cyclist Dan Weller told > Ivanhoe. > > Right now, that feeling requires patience. It takes about 100 hours to > build each IsoTruss bike. Delta 7 produced only 200 IsoTruss models in 2008, > but is working on ways to mass-produce them in the near future. To get one > right now, you have to add yourself to the waiting list and put down a > $1,000 deposit. > > A SEE-THROUGH BIKE FRAME? The Arantix mountain bicycle and Ascend road > bicycle have frames made from carbon fiber, shaped into a form called > IsoTruss. The lattice structure is woven by hand into the form of > pyramid-like shapes made of isosceles triangles (the kind with two sides of > equal length). The design is specially designed to make the bicycle > resistant to bending and twisting, with a greater ratio of strength to > weight than metal frames. This technology is currently promoted as an > alternative to heavier, weaker materials in everything from automobiles to > building materials and utility poles. > > HOW TO WEAVE A BICYCLE: To construct the bike, the artisans take a single > strand of carbon fiber and wind it back and forth (by hand) over a > cylindrical mandrill until it is the right size, then wrap Kevlar around the > fibers to bundle it. Then they bake it in an oven, which bonds all the > carbon together. > > > > > ------------------------------ > > Message: 2 > Date: Mon, 25 Oct 2010 21:07:31 -0000 (UTC) > From: rwg(a)sdf.lonestar.org > To: "math-fun" <math-fun(a)mailman.xmission.com> > Cc: Dan Asimov <dasimov(a)earthlink.net> > Subject: Re: [math-fun] How many spokes does a bicycle need? > Message-ID: <485decbb1d6cda99d27b0e6f8d3d5fb3.squirrel(a)gosper.org> > Content-Type: text/plain;charset=UTF-8 > > > FYI -- I'm not trying to hijack this thread, [...] > That's OK, the thread was misnamed anyway. Should have been > "How many spokes does a unicycle need?" > --rwg > > > > > ------------------------------ > > Message: 3 > Date: Mon, 25 Oct 2010 15:43:52 -0700 (GMT-07:00) > From: Dan Asimov <dasimov(a)earthlink.net> > To: math-fun <math-fun(a)mailman.xmission.com> > Subject: [math-fun] Negative yield? > Message-ID: > < > 15194111.1288046632781.JavaMail.root(a)elwamui-huard.atl.sa.earthlink.net> > > Content-Type: text/plain; charset=UTF-8 > > Apparently some treasury bonds have just sold, at government auction, with > negative yields: > > "Inflation Bonds Are Sold With Negative Yield for First Time" at > < http://www.nytimes.com/2010/10/26/business/26markets.html >. > > But what does "negative yield" mean? > > I feel confident that it does *not* mean that the purchaser of the bond > must make periodic coupon payments to the seller. > > But perhaps it means that the redemption value of the bond at maturation is > *less* than the price it is purchased for? (If so, it's hard for me to > imagine why it's not better to just keep that money in a safe all those > years. Maybe a good safe is too expensive?) > > --Dan > > Those who sleep faster are sooner rested. > > > > ------------------------------ > > Message: 4 > Date: Mon, 25 Oct 2010 16:03:33 -0700 > From: Henry Baker <hbaker1(a)pipeline.com> > To: Dan Asimov <dasimov(a)earthlink.net>, math-fun > <math-fun(a)mailman.xmission.com> > Subject: Re: [math-fun] Negative yield? > Message-ID: <E1PAW95-0005FJ-Vj(a)elasmtp-mealy.atl.sa.earthlink.net> > Content-Type: text/plain; charset="us-ascii" > > The redemption value is less, unless there has been inflation. NB: I > believe that TIPS are indexed to "core" inflation, which does not include > some important things like energy prices, so TIPS won't protect against > rapid increases in the price of oil. > > Bernanke hasn't ruled out negative coupons (or equivalent). See his 2002 > speech here: > > http://www.federalreserve.gov/BOARDDOCS/SPEECHES/2002/20021121/default.htm > > --- > > Countries with a habit of too much inflation also have a habit of declaring > old currency worthless, from time to time, just to keep the amount stuffed > in mattresses under control. > > Due to the popularity of U.S. currency in the international drug trade, > other countries have occasionally called upon the U.S. to "update" our > currency to new, crisp, version 2.0 bills, after which the old currency > would become worthless. Drug dealers & money launderers would then have to > scramble to exchange their millions of bills into new ones, producing lots > of chances for them to be noticed. Despite our willingness to spend > enormous amounts of money to stop the drug trade, the U.S. hasn't (yet) > taken this relatively cheap step. > > The U.S. government did sever the relationship between gold & dollars back > in the 1930's, and severed the relationship between silver & dollars in the > 1960's. Harvard&MIT-educated Bernanke would simply take this sequence to > its limit, severing the relationship between anything of value and dollars. > > At 03:43 PM 10/25/2010, Dan Asimov wrote: > >Apparently some treasury bonds have just sold, at government auction, with > negative yields: > > > >"Inflation Bonds Are Sold With Negative Yield for First Time" at > >< http://www.nytimes.com/2010/10/26/business/26markets.html >. > > > >But what does "negative yield" mean? > > > >I feel confident that it does *not* mean that the purchaser of the bond > must make periodic coupon payments to the seller. > > > >But perhaps it means that the redemption value of the bond at maturation > is *less* than the price it is purchased for? (If so, it's hard for me to > imagine why it's not better to just keep that money in a safe all those > years. Maybe a good safe is too expensive?) > > > >--Dan > > > > > ------------------------------ > > Message: 5 > Date: Mon, 25 Oct 2010 16:08:21 -0700 > From: Joshua Zucker <joshua.zucker(a)gmail.com> > To: Dan Asimov <dasimov(a)earthlink.net>, math-fun > <math-fun(a)mailman.xmission.com> > Subject: Re: [math-fun] Negative yield? > Message-ID: > <AANLkTikbypHAiXhDCOWoEgipgoigXhTSwdff1_9pd-HE(a)mail.gmail.com> > Content-Type: text/plain; charset=ISO-8859-1 > > These are inflation-indexed bonds, so their yield today is negative, > but if inflation rates go up the yield will become positive. > > The way I understand it, they actually are paying now, essentially > buying insurance against future inflation. > > --Joshua > > > On Mon, Oct 25, 2010 at 3:43 PM, Dan Asimov <dasimov(a)earthlink.net> wrote: > > Apparently some treasury bonds have just sold, at government auction, > with negative yields: > > > > "Inflation Bonds Are Sold With Negative Yield for First Time" at > > < http://www.nytimes.com/2010/10/26/business/26markets.html >. > > > > But what does "negative yield" mean? > > > > I feel confident that it does *not* mean that the purchaser of the bond > must make periodic coupon payments to the seller. > > > > But perhaps it means that the redemption value of the bond at maturation > is *less* than the price it is purchased for? ?(If so, it's hard for me to > imagine why it's not better to just keep that money in a safe all those > years. ?Maybe a good safe is too expensive?) > > > > --Dan > > > > Those who sleep faster are sooner rested. > > > > _______________________________________________ > > math-fun mailing list > > math-fun(a)mailman.xmission.com > > http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun > > > > > > ------------------------------ > > Message: 6 > Date: Tue, 26 Oct 2010 01:41:22 +0100 > From: Fred lunnon <fred.lunnon(a)gmail.com> > To: math-fun <math-fun(a)mailman.xmission.com> > Subject: Re: [math-fun] How many spokes does a bicycle need? > Message-ID: > <AANLkTik42s48R98E=gXCxS-iE8OV_KYcXUWg2tr2AVk7(a)mail.gmail.com> > Content-Type: text/plain; charset=ISO-8859-1 > > On 10/25/10, rwg(a)sdf.lonestar.org <rwg(a)sdf.lonestar.org> wrote: > >> FYI -- I'm not trying to hijack this thread, [...] > > That's OK, the thread was misnamed anyway. Should have been > > "How many spokes does a unicycle need?" > > --rwg > > Quite so. In any case, an obscure firm somewhere in the Mid-West is > reported to have designed a single quantum-theoretical spoke, which > suffices to furnish a complete team racing cyclists. > > Presumably it only works until one of the riders carelessly looks at > a wheel, leading to the entire equipe promptly falling off their bikes > ... WFL > > > > ------------------------------ > > _______________________________________________ > math-fun mailing list > math-fun(a)mailman.xmission.com > http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun > > > End of math-fun Digest, Vol 92, Issue 20 > **************************************** >

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Re: [math-fun] How many spokes does a bicycle need?
by Dan Asimov 26 Oct '10

26 Oct '10

Good point, Allan. So: What is the answer to that math problem: --------------------------------------------------------------------------- Given rigid cylinders C_r and C_R in R^3 of equal heights, and radii r < R, what's the smallest number of constraints (*_j) of the form (*_j) ||x_j - y_j|| <= K_j for some x_j in C_r, and some y_j in C_r, and for some constant K_j, such that the (*_j)'s fix C_r in a unique position concentric with, and having the same bounding planes as, C_R. --------------------------------------------------------------------------- And: If <= is replaced by =, what is the answer in that case? --Dan Allan wrote: << It is possible that spokes are intended to be only tensile, in which case the equality in the constraint should be replaced by "less than or equal", and the answer is almost certainly different. >> (*&^$#@ earthlink sent my previous post prematurely, sorry): I wrote: << In the biography of H.S.M. Coxeter "King of Infinite Space", one section describes his interaction with Buckminster Fuller. It states that Fuller erroneously claimed that a bicycle wheel needs 12 spokes "to hold it rigid", whereas the correct number is 7. This seems to be stated as a fact of geometry, rather than one of structural engineering. Anybody have any insight into this claim? >> Those who sleep faster are sooner rested.

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Re: [math-fun] How many spokes does a bicycle need?
by Henry Baker 26 Oct '10

26 Oct '10

FYI -- I'm not trying to hijack this thread, but the frame in this video is pretty cool... http://www.sciencedaily.com/videos/2008/1201-hitech_cycling.htm Hi-Tech Cycling Engineers Create A Strong But Lightweight Isotruss Bike Using Carbon Fibers December 1, 2008 Â Engineers used elements of architecture and geometry to create a strong but lightweight triangle-based isotruss bicycle frame. To make a road bike or mountain bike, the isotruss is first wound with carbon fiber using a sheet that holds the tension constant. The engineers then hand-wind Kevlar strands over the isotruss. The process creates a bike with a large strength-to-weight ratio. Almost every kid has at one time or another asked for one for Christmas. Now, engineers have developed what may be the most technologically advanced bike to hit the road yet. It took ten years to develop a new incredibly light and strong model that will take cyclists into the future. Karl Vizmeg has ridden his Delta 7 Arantix bike 1,700 miles. He has raced dozens of bikes, but says a new see-through model is the strongest and lightest. "This is phenomenal," said Vizmeg. "I've had so much fun this year, particularly with the 'wow' factor, but [also] because it's such a great racing bike." Vizmeg's $8,500 bike was handmade in Utah using geometry and architecture. To make bikes like his, workers first make an isotruss, a form made from isosceles triangles. Then, they wind carbon fiber around the form -- creating a great strength-to-weight ratio. "We go back afterwards and hand-wind all the little Kevlar strands, inch-by-inch, over each isotruss," said Tyler Evans, program manager at Delta 7 Sports in Payson, Utah. They then bake the bike to bond the materials. The mountain bike frame weighs 2.6 pounds. The new Ascend road bike weighs 1.8. "This bike rides like bikes that are much heavier and stronger and built like a tank, but it's still in the featherweight category," Evans said. You might think the open-lattice design wouldn't be aerodynamic, but Delta 7 says wind-tunnel tests prove the bikes are as aerodynamic as traditional ones. The Ascend bike has another advantage. "You definitely feel like a rock star, like you're famous, like you belong in the Tour de France or some high-end race," cyclist Dan Weller told Ivanhoe. Right now, that feeling requires patience. It takes about 100 hours to build each IsoTruss bike. Delta 7 produced only 200 IsoTruss models in 2008, but is working on ways to mass-produce them in the near future. To get one right now, you have to add yourself to the waiting list and put down a $1,000 deposit. A SEE-THROUGH BIKE FRAME? The Arantix mountain bicycle and Ascend road bicycle have frames made from carbon fiber, shaped into a form called IsoTruss. The lattice structure is woven by hand into the form of pyramid-like shapes made of isosceles triangles (the kind with two sides of equal length). The design is specially designed to make the bicycle resistant to bending and twisting, with a greater ratio of strength to weight than metal frames. This technology is currently promoted as an alternative to heavier, weaker materials in everything from automobiles to building materials and utility poles. HOW TO WEAVE A BICYCLE: To construct the bike, the artisans take a single strand of carbon fiber and wind it back and forth (by hand) over a cylindrical mandrill until it is the right size, then wrap Kevlar around the fibers to bundle it. Then they bake it in an oven, which bonds all the carbon together.

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