Bicycle Riding on Square Wheels

Roland Piquepaille writes "Before starting our long working week, let's relax with this story of a bicycle with square wheels. No, it's not a joke. And it even rides smoothly. But there is a trick: the road must have a specific shape. The Math Trek section of Science News Online tells us more about this strange bicycle -- actually a tricycle with two front wheels and one back wheel. Read this overview for some excerpts and a picture of the tricycle, or the original article for an additional animation."

Smooth ride

[gid13]

I'll bet it stays smooth on turns. :P

Read the whole article?

[baudilus]

From the article:

Steering remains difficult, however. If you turn the square wheels too much, they get out of sync with the inverted catenaries.

I wonder what shape my wheels have to be to ride smoothly over the screwed up roads that my town refuses to fix?

Re:Read the whole article?

[Hydrogenoid]

Well, circular, with a diameter approaching infinity, of course. :-)

Re:Read the whole article?

[strictnein]

Well, circular, with a diameter approaching infinity, of course. :-)

But what would the radius be then?

Re:Read the whole article?

[red floyd]

<humor>
Duh! Infinity over 2!
</humor>

Re:Read the whole article?

[alptraum]

What if they did a pattern like sine waves in both the x and y directions, making a surface of small "hills", perhaps turning would be a bit easier if taken at the right angle?

Also, looking at this pic:
http://www.sciencenews.org/articles/20040403/f4720 _4958.jpg It was originally stated in the original post that:
actually a tricycle with two front wheels and one back wheel

From this picture it looks like it's the other way around, two back wheels and one front.

Re:Read the whole article?

[Lodragandraoidh]

I wonder what shape my wheels have to be to ride smoothly over the screwed up roads that my town refuses to fix?

Actually, several non-wheel ideas come to mind:

Tracked vehicles - make your own potholes - with a smooth ride...
Walking vehicles - who cares about potholes - or roads for that matter?...
Flying vehicles - this is my personal favorite - where is that flying car they promissed us?...

Smartwheels!

[Behrooz]

I vote for the smartwheels with zillions of radar-guided extending foot-spokes, a'la Hiro's motorcycle or Y.T's skateboard in Snow Crash.

I'd say being able to skateboard smoothly down stairs would probably give you the upper hand in the simpler conditions of municipal roadway battles.

Allrighty then

[JSkills]

I'll get right on that change-the-shape-of-all-of-the-roads project right away ...

Re:Allrighty then

[ehiris]

And I'll get on the go from point A to point B without ever turning project right away. I wonder who will be done first.

Re:Allrighty then

[merlin_jim]

Actually, this research does have real world applications.

In the early parts of the industrial age it was found that a certain shape [allelectronics.com] of gear teeth (both along the axle of the gear, and the tooth's profile seen edge on), removed gear slip allowing for much smoother operation, to the point where bevelled gears are used in all car transmissions today.

This research may lead to innovative and new ways to mesh gears together; for instance, I could imagine one application to allow gears with teeth numbers that aren't strictly in ratio to their diameters to mesh properly. If that were the case, then we could make transmissions and gear boxes an order of magnitude or so smaller...

Junior school physics

[Space cowboy]

The reason the trike has smooth motion is simple - the centre of mass (where the axle is attached) doesn't move vertically. It's exactly the same reason as for a hoop rolling on a plane surface except the hoop is more obvious.

When you turn, the square shape doesn't fit so well, so the c.o.m oscillates vertically, and you get a more bumpy ride - the larger the angle you turn through, the worse the fit, and the bumpier the ride. Wheels (round ones) don't have this turning problem so much; my vote goes to the round wheels :-)

I remember doing a 'Granada power game' (schoolkid teams are set problems to do, and compete to produce the best solution). For the challenge in the year we took part, we had to construct (entirely from cardboard) a device that would travel forward under its own power for 5m, turn through 45 degrees, forward 1m, turn back through 45 degrees and throw a ball-bearing into a target, accuracy being rewarded. There were 2 walls at given positions that you had to get over as well, at 2.5m and 5.5m from the start. We just cut slots in our wheels - there were some really outlandish solutions to getting over the walls though :-)

Simon

Re:Junior school physics

[greenhide]

For the challenge in the year we took part, we had to construct (entirely from cardboard) a device that would travel forward under its own power

So, how'd you make it move on it's own power? I'm intrigued.

Re:Junior school physics

[Binestar]

So, how'd you make it move on it's own power? I'm intrigued.

Cardboard fueled boiler for the steam engine I would assume.

Re:Junior school physics

[JaredOfEuropa]

the centre of mass (where the axle is attached) doesn't move vertically

Minor nitpick: the smoothness of the ride has nothing to do with the vertical motion of the center of mass, but with the motion of the axle. The c.o.m of a wheel is usually at the axle, but it doesn't have to be. If it isn't, you'll get vibrations at high speeds, at on this bike you probably wouldn't notice it.

huge nitpick: you are both wrong

[mattyp]

movement of the center of mass nor axle has anything to do with smoothness: it's movement of the rider.

20 or 30 years ago (i searched the web, sorry, couldn't find) honda (an engineer there, for an internal contest) built a bicycle with square wheels that rode smoothly on a flat surface. It worked with a cam on the swingarm, so the axle could move up and down while rolling, and the bike frame (and rider) stayed level. I'm sure the center of mass also moved.

Bigger nitpick, you're confused.

[black mariah]

What you're talking about is, in essence, a suspension system. Which is used to overcome a rough ride. What you're all trying to say is "The smoothness of a ride is determined by how much axle movement is passed along to the rider". Or something like that.

Re:Bigger nitpick, you're confused.

[Cecil]

You're a victim of definition creep. What you're actually talking about a shock absorber, which is a specfic subset, or perhaps even just a specific component, of a modern suspension system. There is nothing saying that a suspension system must be flexible, nor that it must do anything other than suspend the rider and frame in the air. A chromoly fork is a suspension system, albeit one that is poor at shock absorption (And conversely excellent at shock transmission!)

A rotating cam designed to smooth out bumps inherent to the wheel isn't fundamentally much different than a spring designed to smooth out bumps inherent to the road, except that because the bumps inherent to the wheel are calculated and predictable, a spring would be a poor solution. The road bumps, on the other hand, can't really be predicted, so it needs a more flexible (no pun intended) method of shock absorption.

Pointless little sidenote: As far as I can see, if you had a square-wheel bike with a correcting cam, and ran it over the bumpy road described in the article, the wheel would ride smoothly, and the cam would overcorrect, so you would still need a shock absorber to go over that road. It'd just be bumping in the opposite phase compared to a normal tire. ;)

Re:Junior school physics

[JaredOfEuropa]

There's no contradiction if you read it right (ok... if I had written it better).

I should have written: "The vertical motion of the c.o.m. does not necessarily have a noticable effect on the smoothness of the ride".

Better>

I guess...

[Stu Catz]

they did re-invent the wheel, not a good invention though...

Re:I guess...

[rokzy]

it's like Microsoft saw the wheel and thought "we gotta get us some of that!"

Re:I guess...

[aardvarkjoe]

Within 4 years, the Open Source community will come up with a trike featuring round wheels but a suspension good enough for it to also interoperate with MSroads. Unlike the MS product, it will work with all versions of MSroads.

Unfortunately, two years before Microsoft introduced flying cars, so nobody's using MSroads anymore anyway. However, OS users get to feel superior because, even though they get to their destination slower, they're not nearly as likely to blow up in midair.

hot dish?

[garcia]

See picture here [sciencenews.org]

Ya yew betcha! I wonder if that basket on the bike is to hold the hot dish? Only in Minnesota would we spend the time determining if square wheels would work... Perhaps from the potholes on 494?

I reside in Minnesota so I am permitted to make these important scientific observations :)

It's open to the public -- you can go ride it!

[melquiades]

I believe it's still sitting in the basement-level lobby of the Olin/Rice building at Macalester. You can just walk up and give it a ride.

In practice, it doesn't work perfectly: the wheels slip a bit on the upslope. But if you get a bit of speed, it rolls along nicely! Quite fun.

Re:hot dish?

[haroldK]

Bah! The potholes on 494 are nothing. The potholes on 94 at 35W, now those are potholes!

ingenious concept

[brad3378]

Perfect for Michigan roads.

Re:ingenious concept

[Roofus]

I guess you have to be from Michigan to get that joke =) Are your roads made of half cylinders?

Re:ingenious concept

[Kyont]

Let's not forget to cc: the Railroad Commission on the Island of Misfit Toys!

Woohoo!

[spellraiser]

Great!

Now all we need is to lay some curved roads all over the place, make loads of these bikes, and we can all ride bicycles with square-shaped wheels!

I'm calling my Senator right now!

Could it be...

[wviperw]

The successor to the overly hyped Segway?

Wheels? Who needs wheels when rhombuses work perfectly fine!

The answer is - A circle!

[DR SoB]

The question at the bottom that states they don't have a wheel the same shape as the surface, I tend to disagree, wouldn't a common circular wheel, while going over a steep hill both be circular shapes? What about tank tracks? They are both flat? A flat wheel and a flat surface = the same!

Re:The answer is - A circle!

[jd142]

wouldn't a common circular wheel, while going over a steep hill

No, because the hill is really at best a half circle.

I'm not sure if tank tracks count as a wheel since they don't orbit a central axis. Even if they were, the tank treads aren't flat when they are in use. They're a sort of oblong shape. You might as well say a wheel is also a line because if you cut the inner tube in half it lies down and becomes a line.

If the wheel is circular, then the road would have to be circular as well. Like goin

Re:The answer is - A circle!

[Havokmon]

wouldn't a common circular wheel, while going over a steep hill

No, because the hill is really at best a half circle.

Yeah, but the Earth is a circle ;)

Re:The answer is - A circle!

[Bun]

"I'm not sure if tank tracks count as a wheel since they don't orbit a central axis."

Actually, the wheels are inside the tank tracks. The tracks themselves are a sort of 'moving platform' on which the wheels roll. It's like they make their own traction, which is why they work so well in all sorts of terrain.

Re:The answer is - A circle!

[mbbac]

A normal bicycle rolling on the Earth is a wheel with the same shape as the surface.

Cities Will Be Redesigned Around This...

[crazyaxemaniac]

It's the next Segway!

reinventing the wheel

[GillBates0]

literally

Re:reinventing the wheel

[System.out.println()]

Hell, that's nothing. They also reinvented the workweek.
Before starting our long working week,

The "normal" person's workweek in ET started about 4 hours earlier.... 1 hour earlier in PST, but they still missed the "before we start" window for most people.

What next?

[Bobdoer]

Will be be seeing pentagonal wheels or maybe even octogonal wheels? Or better yet n-gonal wheels where n is an incredibly large number?

Spirograph

[brundlefly]

This is basically the same principle as the odd-shaped pieces in your old Spirograph set [hasbro.com]....

Re:Spirograph

[sahrss]

Woah, that link crashes my Firebird! That's a first!

Only when JavaScript is enabled...hmm. Where do I go to submit this as a bug?

OT I know...

Re:Spirograph

[pair-a-noyd]

Damn it! You guys /.'ed Hasbro!
And I wanted to see the new Spirograph stuff!

Now the road....

[ericlp]

Today in the news: Inventors discover new way to make road construction ( and repair ) even more expensive....

Before the square wheel...

[theendlessnow]

Stan Wagon invented "clippy" the Microsoft Paper Clip!! Genius! Sheer Genius!

He's working on a water powered car I hear... just requires a really big hill.

No word if the car will support square wheels or not.

If you want to build your own bicycle...

[SmackCrackandPot]

...obviously you'll need a square drill [us.edu].

Re:If you want to build your own bicycle...

[mdouglas]

Actually, aren't mortising kits [deltawoodworking.com] for drilling square holes?

The wonder of assumptions...

[MosesJones]

Economics

"The following theory assumes there are no external factors"

External Factor = People

Sociology

"The following theory is based on a majority sample"

Majority = 50 in a sample of 99.

Slashdot

"The following company/technology categorisation is correct given the sample data"

Sample data = Slashdot

And now we have

"The following design is correct for a given definition of road"

Reminds me of the old maths joke

"1+2=4 for sufficiently large values of 2 and small values of 4"

From the article

[sczimme]

A catenary is the curve describing a rope or chain hanging loosely between two supports. At first glance, it looks like a parabola. In fact, it corresponds to the graph of a function called the hyperbolic cosine.

Yeah, I always get those confused...

[frink]Oy, with the wheels and the squares and the riding and the graphing, ng'hey, glaven.[/frink]

Re:From the article

[ajs318]

A catenary would look a lot like a parabola if you were considering only a narrow range of values of x. Expand the series for y = (e ** x) + (e ** (-x)) and you will see why this is so.

Re:From the article

[daeley]

I was told there would be no math in these comments!

Re:From the article

[mrgeometry]

As a calculus and pre-calculus teacher, I have found that students will call *any* concave-up graph a parabola. x^4, definitely a parabola, if you're a pre-calc student. I could draw a graph of e^x for x > 0 and some of them would still say it was a parabola.

zach

So much for

[Nom du Keyboard]

So much for reinventing the wheel.

And I was always told that was the wrong approach to use.

Proprietary Roads!

[serutan]

Seems to me this is a good analog to proprietary file formats. Instead of having people pay tolls, maybe the government should build roads with inverted caternary bumps and sell the square wheels!

Isle of misfit toys

[dr_dank]

Where's the squirt gun that shoots grape jelly?

Re:Isle of misfit toys

[Rallion]

Nice reference. Now I have to go all day thinking, "We're all misfits!" and....and the song...

I'm sticking with the wheel

[MojoRilla]

How many roads do you have in where you live that has perfect bumps in it?

Also, steering with this would be impossible. Basically, this goes in a straight line only.

And, this is good food for thought. Perhaps this priciple can be applied to other things.

Web design with Mathematica?!?

[Just Some Guy]

If you follow the link to the designer's own web page [stanwagon.com], and scroll to the bottom, you see:

Created by Mathematica (February 3, 2004)

I just realized that any geek cred I thought I had was just an illusion. I don't ever want to hear jokes about Emacs again. Understand?

Re:Web design with Mathematica?!?

[vanza]

Dude, we're talking about *square wheels*. The guy surely is not a big fan of using the right tool for the job (in any situation, it seems).

Re:Web design with Mathematica?!?

[Just Some Guy]

I'd say you're right. When all you have is a square, everything begins to look like a fractalated surface?

If he uses Mathematica for his "real" website, I wonder if he blogs with Octave?

Re:Web design with Mathematica?!?

[passion]

Actually, it appears that he's not alone...

http://www.google.com/search?q=%22Converted+by+Mat hematica%22 [google.com]

google returns 10,000+ results for that phrase...

*BOOM*

[H3lldr0p]

It was at this point that my brain attempted to explode:

"So far, no one has found a road-and wheel combination in which the road has the same shape as the wheel."

Re:*BOOM*

[derkaas]

It seems like a flat road on a spherical earth would be the same shape.

Tricycle sounds like the Dymaxion Car

[G4from128k]

That backwards tricycle sounds like the Buckminster Fuler's Dymaxian Car [washedashore.com]. That beast was designed for minimum air resistance. Also having the two wheels in front provides better stability when cornering during hard braking. Still, tricycles do have some roll-over stability problems because the CG is closer to the sides of the wheelbase.

Re:Tricycle sounds like the Dymaxion Car

[YrWrstNtmr]

This 'backwards' tricycle design (commonly known as a 'tadpole' shape) is very common among recumbent bikes.

Greenspeed [greenspeed.com.au]
Windcheetah [windcheetah.co.uk]
Catrike [catrike.com]
Build your own [ihpva.org]
Many others [ihpva.org]

The Dymaxion suffered from having rear wheel steering. Tends to be very, very unstable at anything over walking speed. Too easy to overcorrect, swinging the back end wide.

Re:Tricycle sounds like the Dymaxion Car

[CastrTroy]

Except for the fact that the post is messed up, and the bike is actually an normal tricycle, with 1 front, and 2 rear wheels

Now I know

[frodo from middle ea]

Why those people , who drive like crazy on the roads, drive like so....it's the road stupid, not the driver.

Old News!

[back_pages]

I've seen the South Park kids travel to French Canada. They have square wheels on their bicycles as well as their cars. I really don't see what the big excitement is all about.

Re:Old News!

[ehiris]

Actually all of Canada has square wheels. And this project would even have use in the South Parkian Canada since they only had one road in Canada so you don't have to turn.

old joke

[Al Al Cool J]

They should have used triangular wheels. One less bump.

Elegant solution-

[baudilus]

While a mathematical solution is technically perfect, I can think of an easy way to determine the requisite road shape: use a square wooden block, cut a hole in teh center so you can roll it, then do so over a reasonably soft surface. You can even observe how the shape of the catenaries elongates as the rotational speed stays constant but the horizontal velocity increases. Would be fun for downhill rides. :)

Duh, physics class 101

[192939495969798999]

Every college physics class has one day where they talk about this,where the road is lumpy in a specific way, and then the bicycle with square wheels can drive. You know what else has a smooth ride? the space shuttle crawler. If you weigh enough, you just crush anything that would otherwise be a bump. I'll be happy when I see a vehicle besides a tank whose method of ground contact changes shape to accommodate for the road (i.e. tank tread on a bicycle). That would be sweet!

http://www.fulcrumgallery.com

Re:Duh, physics class 101

[el-spectre]

Check this out [mattracks.com]. So far as I know, you still use the original shocks and stuff, so the ride would be fairly smooth...

Finally we get some improvements!

[comedian23]

I was wondering when someone was going to get around to improving the wheel. The current version is so impractical, inefficient, and has such a limited range of applications it has been screaming for a face-lift. Someone get this guy a $250 million research grant ASAP!!!

Re:Finally we get some improvements!

[BrookHarty]

I was wondering when someone was going to get around to improving the wheel. The current version is so impractical, inefficient, and has such a limited range of applications it has been screaming for a face-lift. Someone get this guy a $250 million research grant ASAP!!!

He did, this was the solution to Seattles Washingtons Pothole problem.

On the bandwagon

[polyp2000]

Stan Wagon, a mathematician at Macalester College in St. Paul, Minn., has a bicycle with square wheels. It's a weird contraption, but he can ride it perfectly smoothly. His secret is the shape of the road over which the wheels roll.

Is it me or do others find it amusing that a chap researching vehicles with square wheels has a surname "Wagon" ?

nick ...

Meow/Chirp, Meow/Chirp

[malia8888]

From the article: A square wheel can roll smoothly, keeping the axle moving in a straight line and at a constant velocity, if it travels over evenly spaced bumps of just the right shape. This special shape is called an inverted catenary.

Dear Esteemed Committee: I would like a million dollar grant. As a good geneticist I am going to see if I can cross a cat with a canary. I will call it "cantenary"! (Since you refused my grant for the monkey with four asses research) Part bird and part cat--that is something useful. Regards, Dr. Mephisto...

wrong description of the trike

[gosand]

The Math Trek section of Science News Online tells us more about this strange bicycle -- actually a tricycle with two front wheels and one back wheel.

It actually has 1 front wheel and two rear wheels.

A Lesson about Inventions

[Esion Modnar]

The best ones conform the invention's design to fit the environment, not the other way round.

Reminds me of the british 20p coin

[Funkitup]

This coin has 7 sides so you wouldn't expect it to roll smoothly.

However, they are cleverly made so that the diameter is equal right the way around the coin. Therefore, since the center of mass doesn't move, the coin will roll smoothly in slot machines etc. Try it!

I'm not sure whether the 50p is the same or not. I don't have one in my wallet to test as I used it to buy a packet of wine gums...

MMmmmm wine gums...

Stupid American Question

[The Ape With No Name]

I am intrigued. What the hell are 'wine gums?'

Re:Stupid American Question

[SnappleMaster]

Wine gums! [cadburyschweppes.com]

They're a little bit like ju-jubes but you probably don't know what those are either, heathen!

Farming..

[bigattichouse]

wow, looks like the furrows in a field, if you had a large rectangular solid, you could move across a field without disurbing the furrows.. hmm, wonder if there is something useful in that idea.

Reuleaux Triangle

[ortholattice]

Also neat is the Reuleaux Triangle [whistleralley.com] that is not round but even so has a constant width as it rotates. If it is used as a roller between two planks, it will roll smoothly and the distance between the planks will remain constant. This java applet [whistleralley.com] demonstrates it.

Gear and rack

[Animats]

This is a gear and rack [bostongear.com] assembly. It's a funny shaped one, but it's a gear and rack.

Standard gear and rack interaction is well understood. Racks are usually straight-sided, while gear teeth are involute curves. [howstuffworks.com] Two gears which will mesh with the same straight-sided rack will mesh properly with each other. This fact reduces the size of simple gear inventories from O(N^2) to O(N).

"Mesh properly" has a specific meaning. There has to be contact on both sides of each gear tooth when the axes of the meshing gears are a constant distance apart. Getting this right improves gear life by orders of magnitude.

There's a nice little section in the back of every Boston Gear catalog which explains all this. Available online [bostongear.com], too.

Nonstandard rack shapes are rare, but not unheard of. The drive system on the IBM RS-1 electrohydraulic gantry robot used a curved-sided rack.

Let me be the one to point out the obvious,

[pair-a-noyd]

way

too

much

time

on

his

hands.

Square Wheel?

[Himring]

Ok, I'm risk asking this, but by definition, a "wheel" cannot be "square...."

wheel [reference.com]

n.

1. A solid disk or a rigid circular ring connected by spokes to a hub, designed to turn around an axle passed through the center.

And, without pasting it too, a disk must be circular....

So, whatever those things are on that bicycle frame, they are not wheels

Re:Square Wheel?

[black mariah]

I suggest human interaction, and a lot of it. FAST.

Errant pedant prompts Lakatos reference

[melquiades]

OK, so the parent post was kind of silly, but it gives me a chance to mention Imre Lakatos [st-and.ac.uk], my favorite mathematical philosopher. (Yes, I have a favorite favorite mathematical philosopher. Don't you?)

He wrote a marvelous little book called Proofs and Refutations -- here's a very brief bit of summary and context [wikipedia.org] -- which present a very interesting very of the process of mathematical discovery: instead of accumulating an ever-increasing series of perfect truths, he argues, mathematicians are constantly shif

Never understood...

[Comatose51]

Do mathematicians have to justify the purpose of their paper/research in the papers they publish? If so, I would be interested in reading it for this project because he would have to be a damn good English professor as well to pull that one over the eyes of the committee. :-)

Finally!

[freeze128]

YES! Finally a way to take the speedbumps as fast as I want!

Lets get wild with the sides.

[acarrig]

This whole 4 sided wheel thing is great. But lets keep going.

If 4 wheels needs small hills to run on.... lets add a side so we have 5 sides. 5 sides will need smaller hills saving material in the rebiuld the road project.

And if 5 saves materal lets keep adding sides... 6, 8, 20, 100, 1000. Imagine how small the hills will be... we don't need to redo the roads as much.

Infact if we keep adding sides... we'll get.... a circular wheel... with no need to change the roads.

Well. That was easy.

Re:Lets get wild with the sides - fractals

[SmackCrackandPot]

But if we use a fractal patterned tread, we'll need an infinite amount of road surface!

Wheel / Road Same Shape?

[SlipJig]

So far, no one has found a road-and wheel combination in which the road has the same shape as the wheel. That's an intriguing challenge for mathematicians.

Seems to me you could ride a bike with circular wheels on the inside (or outside) of a circular track (2001 space odyssey-style). The wheels and road would have the same shape then, right?

Re:Wheel / Road Same Shape?

[ahem]

Yeah Baby! I want to ride on the wall of death, one more time!!

Speedbumps

[gouldtj]

Hmmm, with the number of speedbumps there are today I'm not sure what would be the smoother ride on the road.

And...

[Bluesman]

...to improve his "ride," Stan Wagon will be adding a "spoiler" (shaped like a rectangle) and a cylindrical exhaust "muffler" to make the vehicle more appealing to "the bitches."

Re:Good for elementary schoolers

[Defender2000]

Being a bit of a bike nut I notice this bike would have some issues with turning and fixing flats.

Don't you mean, fixing rounds?

Re:Groklaw?

[kgarcia]

they are both radio blog themes, based on Bryan Bell's Woodland's Theme [bryanbell.com]

Re:WORTHLESS HORSESHIT!

[Mr. Darl McBride]

what's with you fucktarded slashdot people? what moron would make a fucking bike with fucking square wheels?

If you've ever tried to bicycle across giant corrugated steel planes like I have, you would recognize the value of this contribution.

Now if only the train to Chicago didn't run 1/3 as fast as the train to New York and leave 2 hours earlier.

Re:brakes

[Comatose51]

Not a problem if you disc brakes, which is pretty much the way to go for mountain bikes becauses they can grip harder without damaging the rim and will not heat up the tires through friction. Also, since they are center mounted, they stay clean from dirt accumulated by the wheels.