100 Essential Things You Didn't Know You Didn't Know (8 page)

20

Emergence

A politician needs the ability to foretell what is going to happen tomorrow, next week, next month, and next year. And to have the ability afterwards to explain why it didn’t happen.

Winston Churchill

One of the buzz words in the sciences that study complicated things is ‘emergence’. As you build up a complex situation step by step, it appears that thresholds of complexity can be reached that herald the appearance of new structures and new types of behaviour which were not present in the building blocks of that complexity. The world-wide web or the stock market or human consciousness seem to be phenomena of this sort. They exhibit collective behaviour which is more than the sum of their parts. If you reduce them to their elementary components, then the essence of the complex behaviour disappears. Such phenomena are common in physics too. A collective property of a liquid, like viscosity, which describes its resistance to flowing, emerges when a large number of molecules combine. It is real but you won’t find a little bit of viscosity on each atom of hydrogen and oxygen in your cup of tea.

Emergence is itself a complex, and occasionally controversial, subject. Philosophers and scientists attempt to define and distinguish between different types of emergence, while a few even dispute whether it really exists. One of the problems is that the most interesting scientific examples, like consciousness or ‘life’,
are
not understood and so there is an unfortunate extra layer of uncertainty attached to the cases used as exemplars. Here, mathematics can help. It gives rise to many interesting emergent structures that are well defined and suggest ways in which to create whole families of new examples.

Take finite collections of positive numbers like [1,2,3,6,7,9]. Then, no matter how large they are, they will not possess the properties that ‘emerge’ when a collection of numbers becomes infinite. As Georg Cantor first showed clearly in the nineteenth century, in finite collections of numbers possess properties not shared by any finite subset of them, no matter how large they are. Infinity is
not
just a big number. Add one to it and it stays the same; subtract infinity from it and it stays the same. The whole is not only bigger than its parts, it also possesses qualitatively different ‘emergent’ features from any of its parts.

A Möbius strip

Many other examples can be found in topology, where the overall structure of an object can be strikingly different from its local structure. The most familiar is the Möbius strip. We make one by taking a thin rectangular strip of paper and gluing the ends together after adding a single twist to the paper. It is possible to make up that strip of paper by sticking together small rectangles of paper in a patchwork. The creation of the Möbius strip then looks like a type of emergent structure. All rectangles that were put together to make the strip have two faces. But when the ends are twisted and stuck together the Möbius strip that results has only one face. Again, the whole has a property not shared by its parts.

21

How to Push a Car

There are only two classes of pedestrian in these days of reckless motor traffic – the quick and the dead.

Lord Dewar

There was an old joke that asked, ‘Why does a Lada car have a heater on its rear windscreen?’ The answer: ‘So that your hands don’t get cold while you are pushing it.’ But pushing cars presents an interesting problem. Suppose you have to push your car into the garage and bring it to a stop before it hits the back wall. How should you push it and pull it so as to get into the garage and stopped as quickly as possible?

The answer is that you should accelerate the car by pushing as hard as you can for half the distance to be covered and then decelerate it by pulling as hard as you can for the other half. The car will begin stationary and finish stationary and take the least possible time to do so.
²

This type of problem is an example of an area of mathematics called ‘control theory’. Typically you might want to regulate or guide some type of movement by applying a force. The solution for the car-parking problem is an example of what is called ‘bang-bang’ control. You have just two responses: push, then pull. Domestic temperature thermostats often work like that. When the temperature gets too high they turn on cooling; when the temperature gets too low they turn on heating. Over
a
long period you get temperature changes that zigzag up and down between the two boundaries you have set. This is not always the best way to control a situation. Suppose you want to control your car on the road by using the steering wheel. A robot driver programmed with the bang-bang control approach would let the car run into the left-hand lane line, then correct to head to the right until it crossed the right-hand lane line, and so on, back and forth. You would soon end up being stopped and invited to blow into a plastic tube before being detained in the local police cells if you followed this bang-bang driving strategy. A better approach is to apply corrections that are proportional to the degree of deviation from the medium position. A swing seat is like this. If it is pushed just a little way from the vertical then it will swing back more slowly than if it is given a big push away from the vertical.

Another interesting application of control theory has been to the study of middle- and long-distance running – and presumably it would work in the same way for horse racing as well as human racing. Given that there is only a certain amount of oxygen available to the runner’s muscles, and a limit to how much it can be replenished by breathing, what is the best way to run so as to minimise the time taken to complete a given distance? A control theory solution of bang-bang type specifies that for races longer than about 300 metres (which we know is where anaerobic exercise is beginning and oxygen debt begins) you should first apply maximum acceleration for a short period, then run at constant speed before decelerating at the end for the same short period that you accelerated for initially. Of course, while this may tell you how best to run a time-trial, it is not necessarily the best way to win a race where runners are competing against you. If you have a fast finish or have trained to cope with very severe changes of pace you may well have an advantage over others by adopting different tactics. It will be a brave competitor who sticks to the optimal solution while others
are
racing into a long lead. Sitting in behind the optimal strategist, sheltering from the wind and getting a free ride before sprinting to victory in the final straight is a very good alternative plan.

22

Positive Feedback

Accentuate the positive. Eliminate the negative. Latch on to the affirmative. Don’t mess with Mr In-between.

Johnny Mercer and Harold Arlen, ‘Ac-cent-tchu-ate the Positive’

Earlier this year I had an odd experience while staying in a new hotel in Liverpool during an unexpected snowfall. The hotel was a new ‘boutique’ style of establishment in greatly altered premises dating back to the nineteenth-century commercial heyday of the city. In the early morning I had had a tortuous journey through heavy snow from Manchester, by a slow train that was eventually halted for a significant time after an announcement by the driver that the signal cabling along the next stretch of track had been stolen overnight – another market pointer to the steadily growing value of copper, I noted. Eventually, by means of coordinated mobile phone calls, the train moved slowly past all the signals set by default at red and trundled safely into Lime Street Station.

My hotel room was cold and the temperature outside the snow-covered skylight windows comfortably below zero. The heating was underfloor and so rather slow to respond, and it was difficult to determine whether it was answering to changing the thermostat. Despite assurances from the staff that the temperature would soon rise, it seemed to get colder and eventually a fan heater was brought to help out. Someone at reception suggested that because the heating was new it was important not to turn it up too high.

Much later in the afternoon the building engineer called by, much concerned by the ‘old wives’ tale’ about not turning the heating up too much because it was newly installed and, like me, confused by the fact that the heating was working just fine in the corridors outside – so much so that I left my door open. Fortunately, the master panel for all the room heating was just opposite my door and so we looked together at what it revealed before the engineer investigated the temperature in the room next door where the guest had left for the day. It was very warm next door.

Suddenly, the engineer realised what was the cause of our problem. The heating in the room next door had been wired to the thermostat in my room and my heating to next door’s thermostat. The result was a beautiful example of what engineers would call a ‘thermal instability’. When my neighbours felt too warm they turned their thermostat down. As a result my room got colder, so I turned the temperature up on my thermostat, which made them feel warmer still, so they turned their thermostat down even further, making me colder still, so I turned my heat up even more . . . Fortunately, they gave up and went out.

This type of instability feeds upon the isolated self-interest of two players. Far more serious environmental problems can arise because of the same sort of problem. If you run lots of fans and air-conditioning to keep cool, you will increase the carbon dioxide levels in your atmosphere, which will retain more of the sun’s heat around the Earth, which will increase your demand for cooling. But this problem can’t be solved by a simple piece of rewiring.

23

The Drunkard’s Walk

Show me the way to go home.

I’m tired and I want to go to bed.

I had a little drink about an hour ago

And it went right to my head.

Irving King

One of the tests of sobriety that is employed by police forces all over the world is the ability to walk in a straight line. Under normal circumstances it is a (literally) straightforward task for an able-bodied person to perform. And if you know how long your stride length is, then you will know exactly how far you will have walked after any number of steps. If your stride is one metre long then after S steps you will have gone S metres from your starting point. But, imagine that, for one reason or another, you are unable to walk in a straight line. In fact, let’s suppose you really don’t know what you are doing at all. Before you take your next step, choose its direction at random so that you have an equal chance of picking any direction around you and then step one metre in that direction. Now pick a new direction at random and step in that direction. Keep on picking the direction of your next step in this way and you will find that your path wiggles about in a rather unpredictable fashion that has been called the Drunkard’s Walk.

An interesting question to ask about the Drunkard’s Walk is
how
far it will have gone when measured in a straight line from the starting point after the drunkard has taken S steps. We know that it takes the sober walker S one-metre steps to go a straight-line distance S, but it will typically take the drunkard S
²
steps
³
to reach the same distance from the starting point. So,100 sober steps will get you a distance of 100 metres as the crow flies but the drunkard will usually need 10,000 steps to achieve the same distance.

Fortunately, there is rather more to this numerical insight than the example of the inebriated stroller suggests. The picture of a succession of steps in random directions is an excellent model of what is going on when molecules spread from one place to another because they are hotter than those nearby, and so move faster on the average. They scatter off other molecules one after another in a random way and spread out from the starting point just like the shambling drunkard, taking about N
²
scatterings to go a distance equal to N of the distances that the molecules fly in between scatterings. This is why it takes a significant time to feel the effects of turning on a radiator across a room. The more energetic (‘hotter’) molecules are ‘drunkenly’ meandering across the room, whereas the sound waves from the clunking of the air locks in the pipes fly straight across at the speed of sound.

24

Faking It

Richard Curtis and Ben Elton

One of the most interesting general statistical misconceptions is what a random distribution is like. Suppose that you had to tell whether a particular sequence of events was random or not. You might hope to tell if it was not random by finding a pattern or some other predictable feature. Let’s try to invent some lists of ‘heads’ (H) and ‘tails’ (T) that are supposed to be the results of a sequence of tosses of a coin so that no one would be able to distinguish them from real coin tossings. Here are three possible fake sequences of 32 coin tosses: