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GEOMETRY AND THE IMAGINATION

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Geometry and the ImaginationJohn Conway, Peter Doyle, Jane Gilm an, and Bill ThurstonVersio n 0.9 41, Winter 2010∗1 Bicycle tracksC. Dennis Thron has called attention to the following passage from TheAdventure of the Priory School, by Sir Arthur Conan Doyle:‘This track, as you perceive, was made by a rider who wa sgoing from the direction of the school.’‘Or towards it?’‘No, no, my dear Watson. The more deeply sunk impressionis, of course, the hind wheel, upon which the weight rests. Youperceive several places where it has passed across a nd obliteratedthe more shallow mark of the front one. It was undoubtedlyheading away from the school.’Problems1. D iscuss this pa ssage. Does Holmes know what he’s talking about?2. Try to determine the direction of travel for the idealized bike tracks inFigure 1.∗Based on materials from the course taught at the University of Minnesota GeometryCenter in June 1991 by John Conway, Peter Doyle, Jane Gilman, and Bill Thurston.Derived from works Copyright (C) 1991 John Conway, Peter Doyle, Jane Gilman, BillThurston.1Figure 1: Which way did the bicycle go?23. Try to sketch some idealized bicycle tracks of yo ur own. You don’t needa computer for this; just an idea of what the relationship is betweenthe track of the front wheel and the track of the back wheel. How g ooddo yo u think your simulated tracks are?4. G o out and observe some bicycle tracks in the wild. Can you tell whatway the bike was going? Keep your eye out for bike tra cks, and practiceunt il you can determine the direction of travel quickly and accurately.2 Pulling back on a pedalImagine that I am steadying a bicycle to keep it from falling over, but withoutpreventing it from moving forward or back if it decides that it wants to. Thereason it might want to move is that there is a string tied to the right-handpedal (which is to say, the right-foot pedal), which is at its lowest point, sothat the right-hand crank is vertical. Yo u are squatting behind the bike, acouple of feet back, holding the string so that it runs (nearly) horizontallyfrom your hand forward to where it is tied to the pedal.Problems1. Suppose you now pull gently but firmly back on the string. Does thebicycle go forward, or backward? Remember that I am only steadyingit, so that it can move if it has a mind to. No, this isn’t a trick; the bikereally does move one way or the other. Can you reason it out? Can youimagine it clearly enough so that you can feel the answer intuitively?2. Try it and see.John Conway makes the following outrageous claim. Say t hat you have agroup of six or more people, none of whom have thought about this problembefore. You tell them the problem, and get them all to agree to the followingproposal. They will each take out a dollar bill, and announce which way theythink the bike will go. They will be allowed t o change their minds as o ften asthey like. When everyone has stopped waffling, you will take the dollars fromthose who were wrong, give some of the dollars to those who were right, andpocket the rest o f the dollars yourself. You might worry that you stand to3lose money if there are more right answers than wrong answers, but Conwayclaims that in his experience this never happens. There are always morewrong answers than right answers, and this despite the fact that you tellthem in advance that there are going t o be more wrong answers than rightanswers, and allow them to bear this in mind during the waffling process.(Or is it beca use you tell them that there will be more wrong answers thanright answers?)3 Bicycle pe dalsThere is something funny about the way that the pedals of a bicycle screwinto the cranks. One of the pedals has a normal ‘right-hand thread’, sothat you screw it in clockwise—the usual way—like a normal screw or light-bulb, and you unscrew it counter-clockwise. The other pedal has a ‘left-hand thread’, so that it works exactly backwards: You screw it in counter-clockwise, and you unscrew it clockwise.This ‘asymmetry’ between the two pedals—actually it’s a surfeit of sym-metry we have here, rather than a dearth—is not just some whimsical notionon the part of bike manufacturers. If the pedals both had normal threads,one of them would fall out befo r e you got to the end o f the block.If you try to figure out which pedal is the normal one using common sense,the chances are overwhelming that yo u will figure it out exactly wrong. Ifyo u remember this, then you’re all set: Just figure it out by common sense,and then go for the opposite answer. Another good strategy is to rememberthat ‘right is right; left is wrong.’Problems1. Take a screw or a bolt (what’s the difference?) or a candy cane, andsight along it, observing the twist. Compare this with what you seewhen you sight along it the other way.2. Take two identical bo lts or screws or candy canes (or lightbulbs orbarber poles), and place them tip to tip. D escribe how the two spiralsmeet. Now take one of them and hold it perpendicular to a mirror sothat its tip appears to touch the tip of its mirror image. Describe howthe two spirals meet.43. Why is a right-hand thread called a ‘right-hand thread’? What is the‘right-hand rule’?4. Use common sense to figure out which pedal on a bike has the normal,right-hand thread. Did yo u come up with the correct answer that ‘rightis right; left is wrong’?5. You can simulate what is going on here by curling your fingers looselyaround the eraser end of a nice long pencil (a long thin stick works evenbetter), so that there’s a little extra room for the pencil to roll aro undinside your grip. Press down gently on the business end of the pencil,to simulate the weight of the rider’s foot on the pedal, and see whathappens when you rotate your arm like the crank of a bicycle.6. The best thing is to make a wooden model. Drill a block through ablock of wood t o represent the hole in the crank that t he pedal screwsinto, and use a dowel just a little smaller in diameter t han the hole torepresent the pedal.7. D o all candy canes spiral the same way? What about barber poles?What other things spiral? Do they always spiral the same way?8. Which way do tornados and hurricanes rotate in the northern hemi-sphere? Why?9. Which way does water spiral down the drain in the southern hemi-sphere, and how do you know?10. When you hold something up to an ordinary mirror you can’t quite getit to appear to touch its mirror image. Why not? How close can


GEOMETRY AND THE IMAGINATION

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