Geometry and the Imagination Based on materials from the course taught at the University of Minnesota Geometry Center in June 1991 by John Conway, Peter Doyle, Jane Gilman, and Bill Thurston Version 0.93, Fall 1999 Derived from works Copyright (C) 1991 John Conway, Peter Doyle, Jane Gilman, Bill Thurston Contents 1 Bicycle tracks 3 2 Pulling back on a pedal 6 3 Bicycle pedals 7 4 Bicycle chains 9 5 Push left, go left 13 6 Knots 13 7 Reidemeister moves 15 8 Notation for some knots 15 9 Knots diagrams and maps 15 1 10 Unicursal curves and knot diagrams 18 11 Exercises in imagining 19 12 Pizza 21 13 Ideas for projects 22 14 Polyhedra 24 15 Maps 26 16 Euler numbers 26 17 Gas, water, electricity 29 18 Topology 30 18.1 Letters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 18.2 Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 19 Surfaces 31 20 How to knit a Mo¨bius Band 33 21 Classi(cid:12)cation of surfaces 34 22 Mirrors 35 23 More paper-cutting patterns 35 24 Symmetry and orbifolds 36 25 Names for features of symmetrical patterns 37 25.1 Mirrors and mirror strings . . . . . . . . . . . . . . . . . . . . 37 25.2 Mirror boundaries . . . . . . . . . . . . . . . . . . . . . . . . . 47 25.3 Gyration points . . . . . . . . . . . . . . . . . . . . . . . . . . 47 25.4 Cone points . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 26 Names for symmetry groups and orbifolds 47 2 27 The orbifold shop 48 28 The Euler characteristic of an orbifold 49 29 Positive and negative Euler characteristic 53 30 A (cid:12)eld guide to the orbifolds 54 30.1 Conway’s names . . . . . . . . . . . . . . . . . . . . . . . . . . 55 30.1.1 The pre(cid:12)x . . . . . . . . . . . . . . . . . . . . . . . . . 55 30.1.2 The descriptor . . . . . . . . . . . . . . . . . . . . . . . 56 30.2 How to learn to recognize the patterns . . . . . . . . . . . . . 57 30.3 The manuscript . . . . . . . . . . . . . . . . . . . . . . . . . . 58 31 Geometry on the sphere 58 31.1 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 32 The angle defect of a polyhedron 64 33 Descartes’s Formula. 64 33.1 First proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 33.2 Second proof . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 34 The celestial image of a polyhedron 66 35 Curvature of surfaces 67 36 Clocks and curvature 70 36.1 Clocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 36.2 Curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 36.3 Where’s the beef? . . . . . . . . . . . . . . . . . . . . . . . . . 73 1 Bicycle tracks C. Dennis Thron has called attention to the following passage from The Adventure of the Priory School, by Sir Arthur Conan Doyle: ‘This track, as you perceive, was made by a rider who was going from the direction of the school.’ 3 ‘Or towards it?’ ‘No, no, my dear Watson. The more deeply sunk impression is, of course, the hind wheel, upon which the weight rests. You perceive several places where it has passed across and obliterated the more shallow mark of the front one. It was undoubtedly heading away from the school.’ Problems 1. Discuss this passage. Does Holmes know what he’s talking about? 2. Try to come up with a method for telling which way a bike has gone by looking at the track it has left. There are all kinds of possibilities here. Which methods do you honestly think will work, and under what conditions? For example, does your method only work if the bike has passed through a patch of wet cement? Would it work for tracks on the beach? Tracks on a patch of dry sidewalk between puddles? Tracks through short, dewy grass? Tracks along a thirty-foot length of brown package-wrappingpaper,madebyabikewhosetireshavebeencarefully coated with mud, and which has been just ridden long enough before reaching the paper so that the tracks are not appreciably darker at one end of the paper than the other? 3. Try to determine the direction of travel for the idealized bike tracks in Figure 1. 4. Tryto sketch some idealized bicycle tracks ofyour own. Youdon’t need a computer for this; just an idea of what the relationship is between the track of the front wheel and the track of the back wheel. How good do you think your simulated tracks are? 5. Go out and observe some bicycle tracks in the wild. Can you tell what way the bike was going? Keepyour eye out forbike tracks, andpractice until you can determine the direction of travel quickly and accurately. 6. Tell your friends about this problem. Don’t give them the answers; just the questions. Let them think for themselves. Let them stew a little. See what they come up with. You can give them a little help, if need be, once they’ve spent enough time thinking about it for themselves to 4 Figure 1: Which way did the bicycle go? 5 appreciate the problem. Make sure that they, too, (cid:12)nally master this useful trick of bike direction-(cid:12)nding. 2 Pulling back on a pedal ImaginethatIamsteadying abicycle tokeepitfromfallingover, butwithout preventing it from moving forward or back if it decides that it wants to. The reason it might want to move is that there is a string tied to the right-hand pedal (which is to say, the right-foot pedal), which is at its lowest point, so that the right-hand crank is vertical. You are squatting behind the bike, a couple of feet back, holding the string so that it runs (nearly) horizontally from your hand forward to where it is tied to the pedal. Problems 1. Suppose you now pull gently but (cid:12)rmly back on the string. Does the bicycle go forward, or backward? Remember that I am only steadying it, so that it can move if it has a mind to. No, this isn’t a trick; the bike really does move one way or the other. Can you reason it out? Can you imagine it clearly enough so that you can feel the answer intuitively? 2. Try it and see. John Conway makes the following outrageous claim. Say that you have a group of six or more people, none of whom have thought about this problem before. You tell them the problem, and get them all to agree to the following proposal. They will each take out a dollar bill, and announce which way they think the bike will go. They will be allowed to change their minds as often as they like. When everyone has stopped wa(cid:15)ing, you will take the dollars from those who were wrong, give some of the dollars to those who were right, and pocket the rest of the dollars yourself. You might worry that you stand to lose money if there are more right answers than wrong answers, but Conway claims that in his experience this never happens. There are always more wrong answers than right answers, and this despite the fact that you tell them in advance that there are going to be more wrong answers than right answers, and allow them to bear this in mind during the wa(cid:15)ing process. 6 (Or is it because you tell them that there will be more wrong answers than right answers?) Problem (cid:15) Try this trick for yourself, and see if it works. Of course the fact that this trick invariably works for Conway doesn’t mean it will necessarily work for you; mileage may vary. In other words, do not expect anyone to cover your losses if you mess this trick up. 3 Bicycle pedals It is fairly commonly known, at least among bicyclists, that there is some- thing funny about the way that the pedals of a bicycle screw into the cranks. One of the pedals has a normal ‘right-hand thread’, so that you screw it in clockwise|the usual way|like a normal screw or lightbulb, and you un- screw 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 un- screw 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 notion on the part of bike manufacturers. If the pedals both had normal threads, one of them would fall out before you got to the end of the block. If you’re assembling a new bike out of the box, the fact that one of the pedals screws in the wrong way may cause some momentary confusion, but you can easily (cid:12)gure out what to do by looking at the threads. The real confusion comes when for one reason or another you need to unscrew one of your pedals, and you can’t remember whether this pedal is the normal one or the screwy one, and the pedal is on so tightly that a modest torque in either direction fails to budge it. You get set to give it a major twist, only which way do you turn it? You worry that if you to turn it the wrong way you’ll get it on so tight that you’ll never get it o(cid:11). Ifyoutryto(cid:12)gureoutwhich pedalisthenormaloneusingcommonsense, the chances are overwhelming that you will (cid:12)gure it out exactly wrong. If you remember this, then you’re all set: Just (cid:12)gure it out by common sense, and then go for the opposite answer. Another good strategy is to remember 7 that ‘right is right; left is wrong.’ Problems 1. What is the di(cid:11)erence between a screw and a bolt? 2. Do all barber poles spiral the same way? What about candy canes? What other things spiral? Do they always spiral the same way? 3. Take two identical bolts or screws or lightbulbs or barber poles, and place them tip to tip. Describe how the two spirals meet. 4. Take a bolt or a screw or a lightbulb or a barberpole and hold it per- pendicular to a mirror so that its tip appears to touch the tip of its mirror image. Describe how the two spirals meet. 5. When you hold something up to an ordinary mirror you can’t quite get it to appear to touch its mirror image. Why not? How close can you come? What if you use a di(cid:11)erent kind of mirror? 6. Why is a right-hand thread called a ‘right-hand thread’? What is the ‘right-hand rule’? 7. Which way do tornados and hurricanes rotate in the northern hemi- sphere? Why? 8. Which way does water spiral down the drain in the southern hemi- sphere, and how do you know? 9. Use common sense to (cid:12)gure out which pedal on a bike has the normal, right-hand thread. If you come up with the correct answer that ‘right is right; left is wrong’ then we o(cid:11)er you our humblest apologies. 10. Now see if you can (cid:12)gure out the correct explanation. 11. You can simulate what is going on here by curling your (cid:12)ngers loosely around the eraser end of a nice long pencil (a long thin stick works even better), so that there’s a little extra room for the pencil to roll around inside your grip. Get someone else to press down gently on the business end of the pencil, to simulate the weight of the rider’s foot on 8 the pedal, and see what happens when you rotate your arm like the crank of a bicycle. 12. The best thing is to make a wooden model. Drill a block through a block of wood to represent the hole in the crank that the pedal screws into, and use a dowel just a little smaller in diameter than the hole to represent the pedal. 13. If your pedal is on really, really tight, you might be tempted to use a ‘cheater’, which is a pipe that you slip over the end of a wrench to increase the e(cid:11)ective length of the handle. If it takes a force of 150 pounds on the end of a 9-inch long adjustable wrench to loosen the pedal, how much force will be required on the end of an 18-inch long cheater? 14. Wrenchmanufacturers,pipemanufacturers,bicyclemanufacturers,your insurance underwriter, your family doctor, and your geometry teacher all maintain that using a cheater is a bad idea. Do you understand why? Discuss some of the dangers to which you might expose yourself by using a cheater. If despite all this well-meaning advice you should go ahead and use a cheater, and some harm should come to you, who will be to blame? 4 Bicycle chains Sometimes, when you come to put the rear wheel back on your bike after (cid:12)xing a flat, or when you are fooling around trying to get the chain back onto the sprockets after it has slipped o(cid:11) (what? you don’t always keep your gears adjusted perfectly?), you may (cid:12)nd that the chain is in the peculiar kinked con(cid:12)guration shown in Figure 2. Problems 1. Since you haven’t removed a link of the chain or anything like that, you know it must be possible to get the chain unkinked, but how? Play around with a bike chain (a pair of rubber gloves is handy), and (cid:12)gure out how to introduce and remove kinks of this kind. 9 Figure 2: Kinked bicycle chain. 10