Cryptonomicon

The chain of Turing's bicycle has one weak link. The rear wheel has one bent spoke. When the link and the spoke come into contact with each other, the chain will part and fall onto the road. This does not happen at every revolution of the wheel--otherwise the bicycle would be completely useless. It only happens when the chain and the wheel are in a certain position with respect to each other.

Based upon reasonable assumptions about the velocity that can be maintained by Dr. Turing, an energetic bicyclist (let us say 25 km/hr) and the radius of his bicycle's rear wheel (a third of a meter), if the chain's weak link hit the bent spoke on every revolution, the chain would fall off every one-third of a second.

In fact, the chain doesn't fall off unless the bent spoke and the weak link happen to coincide. Now, suppose that you describe the position of the rear wheel by the traditional [theta]. Just for the sake of simplicity, say that when the wheel starts in the position where the bent spoke is capable of hitting the weak link (albeit only if the weak link happens to be there to be hit) then [theta] = 0. If you're using degrees as your unit, then, during a single revolution of the wheel, [theta] will climb all the way up to 359 degrees before cycling back around to 0, at which point the bent spoke will be back in position to knock the chain off And now suppose that you describe the position of the chain with the variable C, in the following very simple way: you assign a number to each link on the chain. The weak link is numbered 0, the next is 1, and so on, up to l – 1 where l is the total number of links in the chain. And again, for simplicity's sake, say that when the chain is in the position where its weak link is capable of being hit by the bent spoke (albeit only if the bent spoke happens to be there to hit it) then C = 0.

For purposes of figuring out when the chain is going to fall off of Dr. Turing's bicycle, then, everything we need to know about the bicycle is contained in the values of [theta] and of C. That pair of numbers defines the bicycle's state. The bicycle has as many possible states as there can be different values of ([theta], C) but only one of those states, namely (0, 0), is the one that will cause the chain to fall off onto the road.

Suppose we start off in that state; i.e., with ([theta] = 0, C = 0), but that the chain has not fallen off because Dr. Turing (knowing full well his bicycle's state at any given time) has paused in the middle of road (nearly precipitating a collision with his friend and colleague Lawrence Pritchard Waterhouse, because his gas mask blocks his peripheral vision). Dr. Turing has tugged sideways on the chain while moving it forward slightly, preventing it from being hit by the bent spoke. Now he gets on the bicycle again and begins to pedal forward. The circumference of his rear wheel is about two meters, and so when he has moved a distance of two meters down the road, the wheel has performed a complete revolution and reached the position [theta] = 0 again--that being the position, remember, when its bent spoke is in position to hit the weak link.

What of the chain? Its position, defined by C, begins at 0 and reaches 1 when its next link moves forward to the fatal position, then 2 and so on. The chain must move in synch with the teeth on the sprocket at the center of the rear wheel, and that sprocket has n teeth, and so after a complete revolution of the rear wheel, when [theta] = 0 again, C = n. After a second complete revolution of the rear wheel, once again [theta] = 0 but now C = 2n. The next time it's C = 3n and so on. But remember that the chain is not an infinite linear thing, but a loop having only l positions; at C = l it loops back around to C = 0 and repeats the cycle. So when calculating the value of C it is necessary to do modular arithmetic--that is, if the chain has a hundred links (l = 100) and the total number of links that have moved by is 135, then the value of C is not 135 but 35. Whenever you get a number greater than or equal to l you just repeatedly subtract l until you get a number less than 1. This operation is written, by mathematicians, as mod I. So the successive values of C, each time the rear wheel spins around to [theta] = 0, are

[C sub i] = n mod l, 2n mod l, 3n mod l,...,in mod l

where i = (1, 2, 3, ... [infinity]) more or less, depending on how close to infinitely long Turing wants to keep riding his bicycle. After a while, it seems infinitely long to Waterhouse.