Mom, Zoë, and I had fun riding the blue-wheeled trike along the catenary curves:
It really is a smooth ride and one does not feel the bumpiness of a washboarded dirt road as the length of the squares' sides roll perfectly into the endpoints of the catenaries. It is intriguing that quarter circles were used to move large square blocks of marble around the pyramids.
So all those jokes about square wheels really aren't quite as funny any more. . .
Jan's questions about washboarding on dirt roads dovetails into our square wheel discussion:
This article discusses why wash board ripples or corrigations form whenever a vehicle travels more than 5 miles (8 km) per hour over a gravelly or sandy surface. My best guess as to why they extend over the whole road is that drivers try to avoid existing ripples, thereby inevitably creating more right next to them.
The most mysterious part of the researchers' results is that the ripples appear even when the springy suspension of the car and the rolling shape of the wheel are eliminated.
Letting some air out of tires when going over washboarded roads approaches traveling over catenary-like bumps with a square-ish tire.
What other questions spring from square wheels, catenaries (which makes me think of a cat who ate the canary) and washboarded roads?
Sponge Steph, Square Pants
Interesting that the catenary curve under your square-wheeled bike doesn't appear at all on Wikipedia's page on the topic. We may have to edit that. The term is from the Latin for "chain", and describes the shape assumed by a chain suspended from both ends. I'm familiar with the term being applied to the overhead catenary wires that supply electric power to railroad locomotives. The power is transferred to the train by another beautifully named device, the pantograph, which is a particular form of a wonderfully multipurpose tool.
ReplyDeleteYes, it ought to be added to the Wiki. Have you edited Wiki before.
ReplyDeleteThe pantograph was a fun childhood tool. Great to see how it is used to transfer power to trains.
Pantographs reminded me of another great toy, the spirograph. I actually made one of these out of wood when I was a kid, sawing out triangular nubs (cogs) that I nailed onto the circumference of circular wooden gears. It worked great but the cogs were the weak link (catenary!) as they split apart. Pounding a nail through tiny pieces of wood is not a good woodworking strategy.
DeleteHow’s that for drifting off-topic?
LegoSpiro Graft
I've edited a few Wikipedia pages, but not as many as you might suppose, given how often I mouth off elsewhere. I even created a new page once, for Roxana Saberi, when she was arrested in Iran in 2009.
DeleteSpirographs were another favorite toy. I never made one, though.
DeleteDrift away!
Did you know the Jersey-born but brought up in North Dakota, Roxana Saberi, jan?
Never met Saberi. (My wife doesn't let me hang with beauty queens.) (As if!) I just happened to see a brief piece on the AP wire about her possibly being detained in Iran, and her name was familiar from her occasional NPR reports, so I went to Wikipedia to read more about her, and there was no entry. So, I Googled together enough info to start a page on her.
DeleteI am not as well versed in geometry and mathematics as many of you who follow and contribute to this wonderful PEOTS blog. But I am still curious about the “two-sided wheel” (a tire-tread-clad plank, basically) that I posited on last week’s thread. In theory, I thought it would “roll” smoothly over a surface consisting of a series of adjacent semicylinders (quonset-hut-shaped).
ReplyDeleteFirst, Is this correct? Second, would this actually work in practice? Or would there be slippage along the surface, or some other glitch?
Also, what surface works for the equilateral-triangular wheel?
LegoLambDuPont:BetterWheelsForBetterRollingThroughGeometry
A triangle wouldn't work (nor would a plank). Here's a nice animated demonstration of how all regular polygons with 4 or more sides work.
DeleteThat's a great animation!
DeleteLego, I never considered a plank or triangle as a wheel; that's thinking outside the box! The DuPont commercial was a hoot!
DeleteYes, jan, wonderful animation.
DeleteSo the smooth rolling works if the interior angles of the regular polygon are greater than or equal to 90 degrees, with the limit being 180 degrees, as the number of sides approach infinity and a the polygon approaches circle status?
But regular polygons with interior angles of 60 degrees (triangle) and zero degrees (plank!) just go “ker-plunk-plunk-plunk when they try to “roll?”
I still want the triangle and plank to roll… somehow. I noticed that the square in the animation, after rolling to the right and back to the left, briefly morphs into a triangle before morphing back into a square and then polygons of increasingly more sides.
And, at least mathematically and x-axis/y-axis-graphically if not in practice, wouldn’t the midpoint of a plank of length =2 (that is “rolling” across a series of semicircles of radius =1) retain a y-value of 1 for every value of x?
Water freezes at zero degrees. Apparently so do waterwheels. And they even freeze up at 60 degrees!
Letterman’s“WillItFloat?”IsDead!LongLiveLego’s“WillItRoll?”
Lego, I have been thinking about this all day, whilst trying to get Zoe in to see the dentist last minute, get her some good shoes, etc.
DeleteI think you need at least 4 equal sides. The plank has two long ones and two short so that won't work. The triangle won't work even though it momentarily appears to. . .up and over one catenary requires two equal sides per rotation. I hope that makes sense.
How's the spelling bee going?
Perhaps they can work in "soi-disant," a so-called new word for me today. ;-)
DeleteYou can tell Macalester that I'd be generously willing to donate all of my flat tires for their square-wheeled bike...
ReplyDeleteI will pass that generous offer along, jan. . .
DeleteMac won't take your tires, jan. They are only flat on one side (!).
DeleteLegoTellsJokesThatFallFlat
G(rim)shot, eh?
DeleteOr (G)rim shot if you prefer. .
DeleteI assume that the glut of sand in the Southwest can't somehow be used to address the worldwide sand shortage we spoke of here last year. Just as this week's floods in Texas can't do California's drought any good. Pity. Wrong place, wrong time?
ReplyDeleteIndeed. . .There's lots of natural sand around the shores of Lake McConaughey. Shifting it is the key. . .
DeleteTopologist's sine curve
ReplyDeleteYou know how many calories there are in one of those little yellow birds?
DeleteEvelyn Lamb once more. . .
DeleteThe vigilante copy editors who are correcting Ecuador’s graffiti grammar
ReplyDeleteThat's an EGG!
ReplyDeleteEcuador Graffiti Grammar (EGG). . .as in throwing eggs, writing on property.
DeleteThe article on washboarding that you link to above implies that it doesn't matter whether the road surface contains a variety of gravel sizes. That is contradicted by this article from the South Carolina Transportation Technology Transfer Service and this one from the South Dakota Local Transportation Assistance Program, which say that "surface gravel should be a blend of stone, sand and fines that will compact into a dense, tight mass with an almost impervious surface."
ReplyDeleteI suspect that with non-uniform gravel sizes, the math becomes much more complicated that with your article's model of a layer of uniform small disks. Probably, all the pieces of gravel just do what I did, and throw their tiny hands in the air and give up.
I wonder if there's any relation between washboard roads and ripples in sand dunes? Does the angle of repose play a role in washboarding?
I had similar thoughts about ripples forming in flumes. . .Must be some connections there!
DeleteA whole year and the new kindergarten grads (see above). One wants to be a paleontologist. Might you guess which one?
Can't guess the future rock hound, but cute pic! My daughter-in-law once coached kids that age in soccer. She ran a drill: drew a circle on the ground, all the kids inside the circle. When she blew a whistle, the kids were supposed to stay inside the circle. Last one still inside wins.
DeleteIf you mean fifth from left, the shorter boy in blue, you are uncannily correct, Paul!
DeleteUncannily, I miscounted!
DeleteWait a minute, that wouldn't be, oh shit ...Guess?!
DeleteNo, not Guess. . .It starts with an M and ends with an e. . .
DeleteIf it's Maizie I'll eat the most disgusting thing you can think of.
DeleteNope. Think mountain wild life. . .
DeleteI'm only guessing 'Moose' because I used to work with someone who called her nephew that, so it doesn't count ... unless it's right.
DeleteYou are right, Paul! (You did have a pretty big hint, though). "Moose" fits for Moose; it's so wrong, it's right. . .
DeleteHis name is really Moose? His first name? I swear, we need tougher licensing standards for parents. I guess it's better than Moon Unit or Blue Ivy, or Southwest (the name chosen by a woman who gave birth to the kid on that airline). (I suggested that he could grow up to marry Kim and Kanye's daughter. A friend said they could form a band named More Than One Direction.)
DeleteYes, his first name really is Moose. Better than Guess, I suppose.
DeleteSouthwest Northwest--that really is more than One Direction.
Why did the cows return to the marijuana field?
ReplyDeleteIt was the pot calling the cattle back.
Had enough chicken jokes, eh?
DeleteYes, having a beef with chicken jokes ;-).
DeleteNew post on "Cherry Blossom Stones: I knew I Muscovite Away" is up.
ReplyDelete