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Odds'n ends

Postscript fractals -- Bathroom physics -- Bathroom maths -- The chaotic butterfly -- Schrödinger's cat -- Statistics -- Oracles -- The earth is flat

PostScript fractals

Many easy-to-reproduce fractals are point sets invariant under a set of transformations and their combinations. These transformations are typically rotations and changes of scale.

The PostScript language, otherwise known mainly for being understandable by some printers without the help of a computer, makes such transformations easy to implement. Rotations and rescalings of coordinate systems are a basic ingredient of this language.

Here is a PostScript "program" of a star made up of three Koch curves. The small size of the file (580 Bytes) reflects the fact that such fractals are most briefly described by the transformations with respect to which they are invariant. Print it on a PostScript-capable printer and become one of few persons who have made a printer compute a fractal!

Physics in the bathroom

If you have some knowledge about the physics of electric currents, you may have heard that an electric signal propagates along a wire at the speed of light (in the respective medium) while the electrons which make up the current move much more slowly.

This may seem counterintuitive: How can a peak in the current arrive much before the electrons, which after all make up that current? There is an easy way to see that. You can try it out yourself.

Not for electrons, actually ;). We will substitute water molecules for the electrons and their velocity for the electric current (which is after all nothing more than the flow of electric charge). Set the tap of your bathroom shower to cold and let the water run until it really is freezing. Then you set it to a nice cosy warm (but not too hot) temperature. Step under the shower and turn on the tap full blast.

Yowl!! You are hit by a gush of cold water. Isn't physics great? After you have hurled the shower away from you, towelled yourself off, noticed that the shower is still running and that now it is warm, hopped under the warm shower again, towelled yourself off again, mentally quartered me and cursed all physicists to hell, let's analyse this.

The change in the velocity of the water resulting from turning on the tap arrived instantly at the end of the shower (in fact, at the speed of sound in water). The actual warm water molecules took some more time, which gave you this invigorating experience. Why is obvious: there were a lot of cold molecules in front of them which had to be pushed out first. The same goes for electrons: They just push each other via electric fields rather than knocking against each other. Electric fields propagate at the speed of light, so electric signals move at that speed.

Mathematics in the bathroom

This example is more complex than the previous one because you need two bathrooms for it, or at least two different types of water taps. With some taps one has to adjust hot and cold water volumes separately while others allow to select both total volume and temperature by moving one lever in two directions.

This nicely illustrates what is called a change of base in linear algebra. The settings for hot and cold water at one sort of tap and for volume and temperature at the other represent two different coordinate systems slanted relative to each other:

Acquittal of the butterfly

"A butterfly flapping its wings in Japan could cause a hurricane a month later in North America."

Nice headline, isn't it? Probably that is why it pops up so often in articles about "Chaos" Theory (another word that makes good copy). Has anyone given the butterfly a chance to defend itself against such a baseless accusation? It seems not, and it's about time someone does.

Let's recapitulate the argument of the prosecution: Aerodynamics exhibits a property called "sensitive dependence on initial conditions". This means that if a few air molecules are knocked about by a butterfly's wings in Japan, that may well make the difference between a hurricane and no hurricane in North America some time later.

Well, that's true, in fact. So could a slight change in the position or momentum of many other air molecules anywhere else in the world. If you fine-tune the momenta of all particles in the atmosphere in a way that a butterfly in Tokyo causes a hurricane in Florida, you might just as well fine-tune them so that that happens anyway. Claiming that the few cubic centimetres of air shifted by the butterfly are the cause of the hurricane while all other cubic kilometres just happen to be around creating the right conditions stretches the notion of cause and effect beyond breaking point. It's rubbish.

So the verdict is: not guilty. The butterfly is free to flap its wings anywhere, including Tokyo. It has been framed by sensation-hungry journalists and scientists who ought to know better.

Post scriptum: If you go whitewater kayaking one day and have an involuntary swim, don't blame the water fleas. It's your fault!

Further physicozoology: Schrödinger's cat

Another creature from the physicists' zoo frequently invoked by people whose grip on physics is hazy at best is Schrödinger's cat. It is supposed to be locked up in a box together with some radioactive material and a device which releases some poison once the radioactive stuff emits a particle of radiation, thus poisoning the cat. Radioactive decay is a quantum process and the cat's life obviously depends on it. So the story goes, since quantum systems can be in different superimposed states so long as nobody measures them, the cat is both dead and alive until someone opens the box (which can conveniently double as a coffin) to look.

It is less well known that Schrödinger invented this thought experiment to show that quantum mechanics goes against the grain. Maybe it does, but in fact the thought experiment goes against quantum mechanics. The cat is a macroscopic object and therefore extremely unlikely to be in two superimposed states.

Far from explaining how quantum mechanics works, Schrödinger's cat is an example of what the measuring process in quantum theory is not supposed to be. In the story, the measuring process is the opening of the box, which puts the cat into one of the two possible states, alive or dead. But the measuring process in quantum mechanics has nothing whatsoever to do with a human observer. Rather, it is defined as an interaction with a macroscopic object. In the case of Schrödinger's cat, this happens in the device which releases the poison (a macroscopic process) when an atom decays.

Unsurprisingly, this device is never specified. To implement it, one would have to build a particle detector and link it up to a device which opens a valve to release the poison. But explaining that would obviously render the argument spurious.

Lies, damned lies and statistics

Statistics describe somebody's ignorance of something. If an extraterrestrial comes to earth who knows nothing about traffic-lights, he will assume it to be equally likely for them to be red, green, blue, ultraviolet or any other colour. After he has seen one, he will assume that they are equally likely to be red, green or amber.

A person living on earth, on the other hand, (or an extraterrestial with some experience of local life forms ;)) knows that traffic lights are red or green most of the time, and would ascribe a much smaller probability to amber. To someone watching a traffic light, it will clearly be red, green or amber with 100% probability (give or take a few percent to account for poor eyesight).

So the probability of a traffic light showing one or the other colour tells us rather more about the person (or furry creature, or martian, or...) contemplating it than about traffic lights themselves. That's why I'm not interested in statistical physics.

The quandary of the oracle

An oracle which can see into the future seems a useful thing to have. One could leave the umbrella at home unless it really is going to rain, avoid traffic jams, to say nothing of exam questions and stock-market investments. But unfortunately oracles are not only thin on the ground these days, but history records them as being quite frequently ambiguous or misleading. Out of sheer malevolence.

Really? Take one example: The ancient king Croesus considers going to war and asks the Delphi oracle's advice. It tells him that he will destroy a great empire, whereupon he starts his war and loses. The destroyed empire turns out to be his own.

This is not decent behaviour from the point of view of the oracle's customers. But an oracle's primary duty is not to serve its customers but to tell the truth. What if the king in question happened to be of a difficult temperament which made him throw caution in the wind after being told he will win but which would otherwise have boosted his fighting grit so much that he would have won? Faced with a petitioner whose actions are going to prove it wrong one way or the other, what is a self-respecting oracle to do? There is only one solution: to say one thing and to mean another.

The earth is flat

Haven't you also learnt at school that the earth is flat? No? You can't have been paying proper attention. Have you not heard that an object thrown on earth has a parabolic trajectory? Ah, there we are. This is because gravity points in the same direction everywhere — the component of the object's velocity parallel to gravity changes, while the component perpendicular to it remains the same. Of course, this precludes gravity pointing to a fixed point from everywhere, such as to the putative centre of the earth — every schoolchild knows that two different rays extending from the same point are never parallel. So the earth is flat.

Faced with this observation, people (those who have a response at all) tend to mutter something about approximations, and that the typical distance an object can be thrown is clearly much smaller than the earth's diametre. Fair enough — so it does not matter for the purpose of things for which it does not matter whether the earth is flat or round. But that does not solve the underlying problem, that an inference from a terribly good approximation can lead to an utterly wrong world view. A miss can really be as bad as a mile.