This anecdote is memorably presented as an application of modular arithmetic (along with the behavior of Enigma machine rotors) in Neal Stephenson’s Cryptonomicon. See the discussion in this review of the book: https://www.ams.org/notices/199911/rev-kasman.pdf
Your mention of modular arithmetic makes me draw a connection between this story and the way I think about ceiling fan pull chains.
Suppose the fan is clearly starting on high and you want to turn it off, but you don't know if the order of settings is "high-low-off", or "high-medium-low-off" (three states or four).
It's often difficult to tell if the fan is coasting off or just spinning down to low. If you get it wrong and pull one time too many, it goes back to high and you have to start over.
Therefore I always pull fan chains eleven (11) times to turn them off, so that I don't have to know if it's a three or four state fan. (Also works for on-off pull chains.) Pulling the chain 59 times would extend this technique to cover five-state fans, but I've never encountered one.
I have a love/hate relationship with his writing – he adores going off on tangents, and action often gets put on pause while he savors them (REAMDE has several long paragraphs of a character pulling themselves onto a boat, admiring the hull, praising the usefulness of the tires lashed to it, etc).
But it also introduces me to so many lovely things in the process. That moment in Cryptonomicon was quite cool.
Interestingly the bike repair bible that most of the experts defer to, i.e. https://www.sheldonbrown.com/ is written by a man who claims he was strongly influenced by the writings of Bertrand Russell, and he is married to a mathematician and has two children who are both mathematicians.
The webpage contains several simple maths formulas for calculating things such as gear ratios.
It's kept upto date by employees of the bike shop he worked for (Harris Cyclery), his widow, and his friend John Harris (a nationally recognized bicycle expert).
The most recent post was about 3 weeks ago: https://sheldonbrown.com/blog/
This was the website that made me fall in love with the World Wide Web and the “old Internet”. In the mid to late nineties, I was big into biking and liked to do my own bike maintenance. I came across Sheldon’s wonderful site (probably via Alta Vista) some time after a friend brought me to an Internet café and introduced me to the web. I still have a folder with all the pages I printed out so I could read it at home.
From occasionally repairing modern bicycles the way the chain is falling off "regularly" doesn't click with me: If one spoke is bent so much (more like a serious wheel wobble) that - at some point - it touches a "damaged" chain link; this is quite a spectacular failure.
I have to assume back then those bicycles were built quite differently that something like that could happen in such a predicatble fashion. Or along the way some information was lost and something different added in that anectode.
Curious to see other versions I have quickly traced back the ancetode to a Nature article "Are mathematician logical" (1987) by Ian Stewart[0]; the version/wording unfortunately here is quite the same.
Bikes back then weren't much different. Some from that time are still around and are not technically anything special.
I don't buy the story. It might be what he told himself, but what fixed it was just the maintenance in general. Probs moved the wheel and tightened the chain. In any case if the bump of a crocked wheel could throw his chain, it was waaaay to loose.
That's a charming story. It reminds me of skid-patch calculations on fixed gear bikes.
If you have ride a brakeless track bike and "stop" or slowdown the bike by skidding, there's certain gear ratios that you'll want to avoid. Basically, in skidding, you unweight the rear wheel by moving your body forward and lock the position of your cranks. The rear wheel will then skid and you can end up with a flat spot of wear on your tire (assuming your don't face-plant yourself when you do it wrong). The fewer the number of skid patches for your gear-ratio, the more rapid the wear-down on the tire will be.
If you're over 35, please, just use brakes and keep the track bike on the track! (stupid dangerous examples of skidding https://www.youtube.com/watch?v=W9xQcMniV84, don't do it).
Even if you don't do skiddy tricks (but do a lot of miles) knowing the arithmetic leads into all sorts of interesting thinking about prime numbers - and, indeed, figuring out optimum gears ratios for (personal perference for) particular gear inches. Added to which, to minimise chain wear how big can we go at the front without hitting the chainstays/still making kerb jumps plus we want the number of links in the chain to be relatively prime as well.
Agreed re brakes, on a bike having a decent front brake is all that matters (irrespective of age)- if stopping quickly, the back wheel is generally waving about in the air anyway so whether its leg-braked or other makes no difference (except fixed riders will still be pedalling, force of habit and all that).
That's funny, I'm 41 and planning on setting up my brakeless track bike for the road again this afternoon (changing the gearing to 42-17 and switching from drops to a wide riser bar). If it goes badly I'll think of your warning.
> And in Turing’s case, it was further complicated by the fact that he always wore a gas mask as he rode, to prevent triggering his allergies. But the alarm clock he was known to wear around his waist might have helped.
What? Is that for real?
Edit: It is!
> He loved to ride his bicycle through the countryside. To time himself, he would simply tie an alarm clock around his waist. During the war, according to I.J. Good, a Cambridge mathematician, Turing suffered horribly from hay fever during the first week of June every year. So to keep the pollen off while riding, he wore a military gas mask.
I wonder which part of this (admittedly odd fashion statement) seemed surprising to you? Gas masks were of course ubiquitous during the war, so this was probably only about as eccentric as wearing an N95 mask would seem now. And the alarm clock seems like a fairly practical if whimsical hack in an age when chronometers would have been mechanical, and therefore quite expensive.
I feel like the average person absolutely was not wearing a gas mask at all times. Having myself used an N95 in my neverending war on pollen I totally understand though.
No, of course people didn't routinely wear them. But every citizen was issued one. Children took them to school every day, and were drilled in their use.
It's still an eccentric thing to do, of course. But, like the alarm clock tied around his waist, it was an instance of solving a problem by grabbing something close to hand.
I bought one at a swap meet in Amsterdam a few years ago. They were piled up high and cheap. It made a fun gift for a friend back home, but we got really odd looks on the train when wearing it :).
I guess if you have a hammer, everything looks like a nail, even if you are Alan Turing!
Tangentially related, I have recently become aware of a sound similar to a bag of potato chips being crushed coming from the rear wheel of my mountain bike. Closer inspection showed that the rear wheel hub exhibited an unwanted additional degree of freedom with respect to the rest of the wheel. The hub was able to be moved slightly back and forth radially. "An excuse to get my hands dirty, and an easy fix", I thought foolishly -- it is only mechanics, after all, nothing that will not surrender immediately to the agile mind and nimble hands of a computer programmer! Fast forward 300$ in tools of and 4 weeks of attempts to fix this, which typically resulted in the purchase of a new specialized tool, and the bike runs again. I learned more than I had hoped about the vast number of very small metal balls that reside in a modern freewheel hub, and it was a good lecture in humility.
When I was a kid, I had a similar noise (and resistance to turning). I took apart my bike with whatever tools I had and at some point found the ball bearings. Took them out, wiped off "all the dirty grease" and put them back in and tightened it all up. Needless to say, it got worse. It was a good lesson, though- only take apart your spare bike, and bring the prod bike to a real mechanic until you know what you're doing.
You could make some of the money back from those tools, and help others in a similar situation, by renting them out on https://fatllama.com (or some other sharing website/app such as https://olioex.com/).
Sharing & hiring saves money and environmental impact.
Kind of reminded me of the traditional wisdom that you should select chains and sprocket to avoid patterned wear. I don't know if there is an effect (especially with bikes and their low torque),but I always think about it when changing my chain. I suppose this only applies to single sprocket bike.
This is called "hunting tooth". It applies to any system of gears or timing belts.
In gears, if you make sure the number of teeth of mating gears is coprime then you wear the teeth of one gear evenly against the other gear. If there are low factors, then each tooth of one gear only engages with a small number of teeth on the other gear, exacerbating wear.
With belts, if the number of teeth on the belt shares low factors with the number of teeth on either of the pulleys then you get the same effect: any given tooth on the belt only ever meshes with a small number of teeth on the pulley.
Isn't this only relevant for fixed gear bikes, where you can skid the rear wheel to brake, and you want maximize the number of possible "skid patches", or orientations of the rear wheel while the pedals are in a fixed position?
It could also apply to the wear on the chainring and cog if their teeth are in a perfect proportion to one another, but without thinking too hard about it, I'm guessing that's actually rare. For instance a 2:1 ratio would be awkward for most cyclists riding on pavement (too low) and 3:1 too high, if they had to choose exactly one ratio.
I ride 46:19, because those are the parts that were in my bin when I built the bike, and it's a pretty good all-round ratio for city riding unless you're a lot more athletic than I am.
Also, those components wear out soon enough anyway -- a chain lasts 2 to 3 thousand miles, and a cog lasts a few chains.
It applies to any bike. Ideally the number of links in the chain would be coprime to the number of teeth on every sprocket. The easiest way to achieve this is to make the number of links in the chain prime.
It'll still work just fine if you ignore this idea, but it might wear out more quickly. If you're a hobbyist just trying to make something work, you can safely ignore it and do whatever is most convenient. If you're a bicycle engineer trying to make things reliable and long-lasting, then there's no downside to making the number of links prime if you can arrange it.
I don't know whether bike companies actually do choose prime-numbered chains, maybe they have other constraints that are more important.
On a bike with derailleur gears, every time you change gears the derailleur will add some slippage so you won't get this effect.
> The easiest way to achieve this is to make the number of links in the chain prime
The chainring is fixed, but you might need to add or remove a link in the chain. In practice it seems more common to make the chainring have a prime number of teeth (53 or 47).
> then there's no downside to making the number of links prime if you can arrange it.
Bike chains always have to have an even number of links because they come in inner and outer pairs. But this has an effect on chainrings as well. When the tooth on a chainring or sprocket is in between two inner plates it's in a narrow gap. When the tooth is in between outer plates that's a wide gap. If you have a chainring with an even number of teeth then you can have the teeth match the narrow and wide profiles (called a narrow-wide chainring) which is supposed to make it less likely that you drop your chain off the chainrings. I'm not sure if it works, tbh. It seems to only be a thing in mountain or gravel bikes with a single chainring. I can't find any track chainrings that have the profile but I only looked for a second.
For chains themselves there's usually a pretty narrow number of links that work on a road drivetrain. I think I can live with one fewer or more pair of links on mine.
Track chainrings don't need a narrow-wide profile because there's very little slack in the system for the chain to come off. Saint Sheldon warns of the possibility of losing a finger also for this reason.
Yeah, chains and cassettes are considered wear items, so I doubt manufacturers put much thought into patterned wear. Avoiding chain drop and crisp shifting are higher priorities.
Thanks for the fun story. I began appreciating mathematics and sincerely loving it due to the biographies of mathematicians like Sir Isaac Newton in Veritasium channel.
This reminds me of when I unstuck a drawer using thermodynamics.
The drawer was getting locked after I pulled it out X or so inches, where Y is the full length the drawer can come out and X < Y.
I thought, "this is an implicit restriction on the degrees of freedom the drawer should have". Then I thought, well, if this system has fewer degrees of freedom than it should have, then I need to add degrees of freedom to it so that it may come unstuck.
How to add degrees of freedom? Well, I could attempt to add entropy to the system, so that becoming unstuck could asymptotically become a part of its configuration space, and it may come free!
