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-plan... 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@mailman.xmission.com>wrote:

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Today's Topics:

1. Re: How many spokes does a bicycle need? (Henry Baker) 2. Re: How many spokes does a bicycle need? (rwg@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)

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Message: 1 Date: Mon, 25 Oct 2010 12:06:36 -0700 From: Henry Baker <hbaker1@pipeline.com> To: Gary Antonick <gantonick@post.harvard.edu> Cc: math-fun@mailman.xmission.com, Dan Asimov <dasimov@earthlink.net> Subject: Re: [math-fun] How many spokes does a bicycle need? Message-ID: <E1PASOL-0005jZ-HT@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.

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Message: 2 Date: Mon, 25 Oct 2010 21:07:31 -0000 (UTC) From: rwg@sdf.lonestar.org To: "math-fun" <math-fun@mailman.xmission.com> Cc: Dan Asimov <dasimov@earthlink.net> Subject: Re: [math-fun] How many spokes does a bicycle need? Message-ID: <485decbb1d6cda99d27b0e6f8d3d5fb3.squirrel@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

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Message: 3 Date: Mon, 25 Oct 2010 15:43:52 -0700 (GMT-07:00) From: Dan Asimov <dasimov@earthlink.net> To: math-fun <math-fun@mailman.xmission.com> Subject: [math-fun] Negative yield? Message-ID: < 15194111.1288046632781.JavaMail.root@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.

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Message: 4 Date: Mon, 25 Oct 2010 16:03:33 -0700 From: Henry Baker <hbaker1@pipeline.com> To: Dan Asimov <dasimov@earthlink.net>, math-fun <math-fun@mailman.xmission.com> Subject: Re: [math-fun] Negative yield? Message-ID: <E1PAW95-0005FJ-Vj@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

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Message: 5 Date: Mon, 25 Oct 2010 16:08:21 -0700 From: Joshua Zucker <joshua.zucker@gmail.com> To: Dan Asimov <dasimov@earthlink.net>, math-fun <math-fun@mailman.xmission.com> Subject: Re: [math-fun] Negative yield? Message-ID: <AANLkTikbypHAiXhDCOWoEgipgoigXhTSwdff1_9pd-HE@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@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@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun

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Message: 6 Date: Tue, 26 Oct 2010 01:41:22 +0100 From: Fred lunnon <fred.lunnon@gmail.com> To: math-fun <math-fun@mailman.xmission.com> Subject: Re: [math-fun] How many spokes does a bicycle need? Message-ID: <AANLkTik42s48R98E=gXCxS-iE8OV_KYcXUWg2tr2AVk7@mail.gmail.com> Content-Type: text/plain; charset=ISO-8859-1

On 10/25/10, rwg@sdf.lonestar.org <rwg@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

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