Proof: Gwyneth Paltrow and Fermat's last theorem

A few months ago the movie Proof, in which Gwyneth Paltrow plays a mathematics genius’ daughter, was released. The film is an adaptation of a very successful play, written by David Auburn, that caught the eye of both audiences and critics. It dealt with the demonstration of a famous theorem that had driven mathematicians crazy for a long time. The theorem’s name wasn’t actually mentioned but somewhere during the play the main character explained in a brief way its statement, which resembled very much that of the well known Fermat’s last theorem.

Fermat was a French mathematician from the XVIIth century. He observed several cases of two numbers being perfect squares and its sum being another perfect square. However, he couldn’t find anything similar happening with cubes, fourth powers or any other exponent. For example, the addition of 9, which is the square of 3, and 16, which is the square of 4, is 25, which is the square of 5. Another case is 64 (square of 8) + 36 (square of 6) = 100 (square of 10). Nevertheless, no matter how hard we try, we won’t find any sum of perfect cubes the result of which is a perfect cube as well.

Maths is a perfect science instead of an empirical one, though. Seeing that a theorem works many times isn’t enough: it must be proved that it has to be always true, with no exceptions. Fermat claimed to have found a very simple solution to this conjecture. It was never found and that left the mathematicians clueless for over three centuries, a time in which they were trying to find that easy solution. The theorem was finally proved a short time ago, in 1993, with a very complex proof over 200 sheets long. It makes us believe that the brief explanation Fermat stated to have reached had to be mistaken. Anyway, both the play Proof and the movie based on it avoid any kind of mathematical complexity and focus on the effort of a wise man (Anthony Hopkins) who gives all his time to science and the consequences it brings to his family. But … what if there is a great woman behind the great man, his daughter in this case? The title this movie has been given in Spain (La verdad oculta, which means the hidden truth) shows what this might be about ….

Mission Impossible: Tom Cruise against photoresistors

Those who have seen the successful film Mission: Impossible by Brian de Palma (1996) will undoubtedly remember the scene in which Tom Cruise has to perpetrate a burglary inside a room provided with a sophisticated alarm system. Cruise sees himself forced to hang upside down from the ceiling, go down easily and avoid by all means getting below a certain level in which the alarm will buzz.

This sequence, apart from being homage or maybe plagiarism to Topkapi, a Jules Dassin film from the 60s, is one of the best examples of a movie making the most of everyday technology. We are so used to photovoltaic cells opening the bank’s or the mall’s doors for us that we find it usual to link light and electricity. Nevertheless, the discovery of the photoelectric effect (electric current produced by light, which means a wave that spreads through a vacuum acting over matter) was one of the highlights in later 19^th / early 20^th century science. It got to solve a several centuries long controversy among physicians about the nature of light (wave or matter) and meant a Nobel Prize for Albert Einstein (he actually got it for this reason, not for his Theory of Relativity).

Once the link between light and electricity was found, photoresistors (known by the acronym LDR, light-dependent resistor) didn’t take long to be launched. They are electric resistors whose resistance changes with light: in the absence of light they have a big resistance to electricity and therefore they block an electric circuit; in presence of light, though, they don’t offer any resistance and allow electricity to flow. We can see an image of a LDR here.

Thus, when the ray of light received by the LDR gets interrupted by the body of someone passing by, the LDR doesn’t allow current to flow anymore. The electronic circuit avoiding the opening of an automatic door, or the buzz of an alarm, or whatever, gets blocked. This is the simple mystery hidden by a lot of automatic systems like water taps opening and closing and lamps switching on and off without a manual switch. LDRs aren’t expensive and could be used for light and water automatic regulation at home, but the experiences which took place up to the present produced a backlash from users. Like Tom Cruise in the movie, we feel restless about the sensation of being controlled by invisible forces.

Jodie Foster's contacts

Contact (1997) is an unusual extraterrestrial movie for several reasons. First, it doesn't judge the main character, played by Jodie Foster, for being a feisty single woman with no kids. She doesn’t have a romantic affair in the movie either, and what is even worse, she is an atheist. Apart from that, we’re talking about the adaptation of a best-seller from Carl Sagan, one of the most important writers in the popularization of science. Therefore, the subject is dealt in quite serious terms, far away from good men from Mars or bad men from Mars invading Earth movies.

The contact with extraterrestrials about which this Robert Zemeckis movie talks about is inspired by a real event that took place in 1967: a young Irish woman working on her doctorate in Physics noticed a sign on her radiotelescope. It was repeated in very short and extraordinarily regular time laps, a sort of heartbeat coming from somewhere in space. Any possible terrestrial source for that sign, such as radio hams, radiotaxis, etc. was studied until certainty of the fact that it came from out of our planet was reached. The young woman, named Jocelyn Bell, believed for some time that she had got in touch with extraterrestrial civilizations, until she found out another sign, identical to the previous one, which had its origin somewhere else in the galaxy. Were two extraterrestrials civilizations sending the same sign at the same moment in the same direction?

The Carl Sagan book and the Zemeckis film dealt with the attractive possibility of Jocelyn actually getting in touch with intelligent beings from out of the Earth, but prosaic scientific logic stated that they were pulsating stars or pulsars, something unknown until that moment. This discovery meant a Nobel prize, not for Jocelyn but, as it is usual, for her work’s supervisor.

What are these pulsars and how are they created? Stars, like the sun, stay alive because fusion of hydrogen nuclei into helium is constantly taking place inside them. The heat coming from the sun and allowing life on the Earth comes from the energy generated in these reactions. What happens when combustible material runs out and fusion doesn’t take place anymore? Nothing can thwart the star’s gravitational force then, so it collapses and implodes into its centre: this phenomenon is known as a supernova. The gravitational force reduces all the star’s volume into a tiny hiperdense nucleus spinning at great speed. Its huge gravitational field produces a magnetic field causing radiation noticed by Jocelyn: this is a pulsar. If a pulsar’s density gets even higher, then a black hole is created: this is a subject it would be worth discussing in greater detail on another occasion.

Total recall: Schwarzenegger under pressure

Living on another planet is quite complicated for any animal on Earth, man included. Evolution has provided us with bodies fully adapted to the Earth, actually to very definate and restricted habitats (the sea, rivers, jungles, the pole, …). The idea of human beings living in another planet is like thinking of a fish living on the mainland or a polar bear living on the tropics.

Apart from the gravity problem we dealt with a couple of posts ago, which must be taken into account seriously, we have the atmosphere matter. Our bodies are machines which work by burning food with oxygen in a chemical reaction of combustion, quite similar to the one that takes place inside a car’s engine. We need an atmosphere that allows us to breathe oxygen in order to live. In addition to that, it protects us from ultraviolet rays coming from the sun, and, in the third place, it allows blood to run through our veins.

As with any other fluid moving, blood flows through our body under a certain pressure, and the same is true for the air on our lungs. If we were out on the void with no atmosphere, blood and air pressure would swell our skin and take our eyes out of their orbits, among other things. It is thought, though, that our skin is strong enough to take this without breaking, or without bursting at least. In normal life the atmospheric pressure on the outside makes up for pressure inside our bodies; not always in a perfect way, though, and that’s why some people have high or low blood pressure. On the void, nevertheless, the very low pressure would evaporate the water inside the cells, causing death in less than one minute.

This phenomenon was shown in a rather harsh way in the 1990 movie Total recall. Harsh but probably exaggerate, as we have said the body is believed not to burst on the void. This film, though, hits the nail on the head at explaining what a human colony on other planet would be: a bunker or a bubble with an artificial atmosphere inside with the same quantity of oxygen and the same pressure that there are on the Earth.

Lack of oxygen can cause malformations on pregnant women’s fetus, which produced all sort of mutants in the movie. On the other hand, too much oxygen would raise the risk of fire and turn the planet into a powder magazine. And in case someone pierces the artificial atmosphere bubble, like Arnold Schwarzenegger and his girlfriend do at the beginning of the movie (in a premonitory dream) and also in the end, he would find himself exposed to radiation and with his head on the verge of exploding. Therefore, thinking of human beings walking peacefully over the surface or remote planets with no need of a spacesuit is nothing but a beautiful dream.

Cube: caught among prime numbers

Cube was a work of independent science-fiction cinema directed by Vincenzo Natali in 1997. It dealt with a bunch of people locked inside a huge cube divided into a lot of cube-shaped rooms, some of which were safe while the others were full of mortal traps. One of the persons locked inside, a young mathematician, achieves the key to pass from one room to the next in this dangerous game: if the numbers written below each window of the room are prime numbers, it means that the next room is safe.

Knowing if a number with many digits is a prime or not (which means if it has a divisor different from 1 and the number itself) is quite complex and turns into one big headache for the authority in maths. The only way to prove it is trying to divide it into all the smaller prime numbers, which is quite a lot of work ... Nevertheless, there is a quite simple rule: only odd numbers can be prime, for the simple reason that even numbers are divisible by two. Neither can be numbers ending in 5 be primes, as they are always divisible by 5. Therefore, only numbers which have 1, 3, 7 or 9 as their last digit can be primes. The screenplay’s writer forgot this and made the mathematician girl rack her brains to find out if a number ending in 2 was or wasn’t a prime number ...

Mathematicians have been thinking of prime numbers since the beginning of time, and they actually couldn’t come to any other conclusion about the subject. Distribution among those ending in 1, 3, 7 or 9 is quite uniform, and it is also well known that prime numbers are infinite, even if they get more and more distant from each other as we get into bigger and bigger amounts (the percentage of prime numbers is greater between 1 and 1000 than the percentage between 1.000.001 and 1.001.000). The fact is, though, that there can’t be a last prime number. Why is that? If we multiply that huge prime number by all the smaller prime numbers and we add 1 to the result, we would find another prime number. We actually would never get an exact result dividing that number by any prime number at all, as we would always get 1 as remainder.

But going back to Cube, numbers came to be coordinates defining the position of each room inside the cube, from which the people locked inside reached the conclusion that the cube was spinning. Then, if they stood still inside one cell, there would come a moment in which the cell would become an outer one: that would allow them to reach the cube’s exit … That’s a funny conclusion, as in a movement of rotation (spinning) the inner sides of the rotating object always stay inner, while the outer ones are always outer … otherwise we would find ourselves in the very core of the Earth one day, as our planet is turning round and round all the time. Nevertheless, years go by and we are still on its surface. And that is a property shared by both spheres and cubes.

Star wars and others: The unbearable lightness of being

Everybody probably remembers the image of North American spacemen stepping over the Moon. Their steps over our satellite were light, because we weigh less on the Moon than we do on the Earth. How can we slim so suddenly? It’s because in our ordinary lives we give the same meaning to two different physical concepts: mass and weight. Mass is an inner property of an object, while weight is the strength with which it is attracted by a greater object (our planet in this case). Therefore, a person having a mass of 70 Kg will always have the same mass in the outer space, but he or she should no longer say ‘I weigh 70 Kg’ out of the Earth.

The force of gravity on a planet or satellite depends on its mass and its radius. The Moon is quite lighter than the Earth, that’s why a man with a mass of 60 kg (60 kg weight on the Earth) weighs only 10 Kg on the Moon, even though he still has the same 60 Kg mass. This has an effect on many things: you can jump higher, raise objects that would be too heavy on the Earth, etc. It has its disadvantages, though: the frictional force against the ground gets tiny too. This force is important for walking, as we would slide over the ground without it. With such little gravity, the Moon resembles a skating rink .... Nevertheless, gravity is twice or three times bigger on Jupiter than it is on Earth: our 60 Kg man would weight over 150 Kg there. This would cause problems such as broken joints.

All these considerations are rarely taken into account in the movies: the Star Wars series is one of the most remarkable examples. Humans there are able to walk on the surface of several planets, never feeling lighter, nor heavier. Gravity has an influence on the appearance of extraterrestrials as well: Moon inhabitants would be tall and slim, while Jupiter men would be stocky in order to adapt themselves to their environment.

Chaotic cinema II: Jurassic Park

In the previous post we talked about The butterfly effect and A sound of thunder, but chaos theory had actually been already explained ten years before in Jurassic Park (Steven Spielberg, 1993). In this movie, a mathematician played by Jeff Goldblum tells a biologist (Laura Dern) that it is impossible to predict the trace of a waterdrop falling in the back of your hand. Invisible imperfections of the skin may alter the result and make the drop fall either to one side or the other.

It should be mentioned that this very didactic dialogue is shocking and worthy of admiration in a Hollywood blockbuster mainly, even if not only, aimed to a child audience. In a video as pedagogical as poor when it comes to storytelling, the film explained as well what DNA was and how a dinosaur could be rebuilt from his DNA chain got from the blood found in a fossil mosquito.

What could actually lead to a certain confusion is the mathematician saying rather enigmatically that nature finds its own way when he is told that the dinosaur population is under control in jurassic park, since all the animals are female. We find out later that nature did find its own way so that some female dinosaurs went through genetic changes becoming hermaphrodites capable of procreating as males. The belief in this concrete way of understanding the wisdom of nature is the mathematician’s personal point of view. It isn’t related to the chaos theory he is an authority of, but to a tradition that can be traced through science-fiction novels, like Frankenstein, according to which dreadful disasters come when human beings dare their Creator assuming his job of giving life.

The butterfly effect and A sound of thunder: chaotic cinema

Aston Kutcher had the chance of going back to the past and changing little things there to alter the future in the movie The butterfly effect. A sound of thunder, on the other hand, dealt with millionaires who travelled to the past to amuse themselves hunting dinosaurs. They were seriously warned, though, not to modify anything they could find or see in prehistory, since the consequences of their actions were unpredictable. Putting aside what Einstein’s theory of relativity has to say about time travelling, which is something we will talk about later, the question is: can a little change in the past really modify the future in a significant way?

We mentioned, when dealing with the problem of scale in King Kong, that the human mind thinks in a linear way but reality doesn’t always work like that. Machines usually do, as they are human inventions: if we get out of Madrid by car and set our speed at a hundred kilometres per hour, after six hours we will have travelled 600 km, so that we could be reaching Barcelona. What if we don’t start at Madrid exactly, but at Alcorcon, which is a few kilometres away? This is what the mathematicians call a change in initial conditions. Human logic would say that a slight change in initial conditions should result in a slight change in the final result: instead of Barcelona, we would reach some nearby town. It works this way with cars because they are human inventions and they are therefore linear, but in nature we could find that planning a trip Madrid – Barcelona and starting at Alcorcon instead of Madrid can lead us to Sevilla, Rome, Paris or Moscow.

It seems illogical but this is what a researcher named Lorentz found. Trying to predict the weather for the next few days, he measured all the variables he thought could have an influence, like temperature, air pressure, wind speed, etc., and he found out that tiny errors of measurement altered the final result completely. That’s why it is considered impossible at present to predict the weather, even in a rough way, with more than three days in advance. Measuring instruments are never perfectly precise. Measuring a 12 ºC temperature instead of a precise temperature of 12,00000000000034 ºC seems to be an insignificant error, but it can make our calculations predict rain instead of sun for tomorrow. These systems in which the results of changes in initial conditions are unpredictable, instead of proportional, are called chaotic systems, and chaos theory studies them.

Lorentz explained his theory with his famous metaphore that a butterfly shaking his wings could produce a hurricane. He was proving something that had come to other people’s minds before, the well-known science-fiction writer Ray Bradbury being one of them: he wrote the short story which inspired the film A sound of thunder. The butterfly doesn’t actually produce a hurricane by itself, but, in order to study a chaotic system, even the least important variables must be taken into account: ignoring the little butterfly can lead us to serious calculation errors.

How can nature be so anarchic? The answer is because systems are more stable this way, as paradoxical as it might sound. Any natural system, as a river, a mountain or an ocean, is under the influence of countless little changes at every moment: stones falling, rain, animal action, etc. If the river or the mountain reacted in a linear way to every single disturbance, the river would change course and the mountain would fall down. But chaos provides systems with flexibility: a flexible framework may bend easier than a rigid one, but it is nevertheless much more difficult to break.

The Simpsons, Jane Fonda and Meryl Streep against nuclear power

The sight of Springfield town is eclipsed by the huge nuclear power station where Homer Simpson works. It has a worrying look due to its large cooling towers. These, though, are completely harmless in spite of its enormous size. Smoke coming from them is made of non-polluting water vapor. Its function consists of cooling the steam coming out of the nuclear reactor so that it can start its cycle all over again: coming into the reactor with a low temperature, getting warm and moving a turbine to produce electricity. The steam is cooled either by water coming from a river or, in a more simple way, by atmospheric air. The size of the cooling towers is huge, and bigger at the base than at the uppert part, so that air flows cooling the water, just like in a chimney: the higher it is and the more its section gets reduced from bottom to top, the better it works.

Of course, even if these dreadfully looking towers (that exist in no nuclear power stations as well) are harmless, that doesn’t mean nuclear power isn’t dangerous. It doesn’t pollute atmospheric air directly, and that’s the reason why it was introduced at first as a ‘clean’ energy compared to toxic gases produced in power plants getting electricity from burning coal or oil. Nevertheless, the process taking place at a nuclear reactor is actually a controlled atomic explosion. If, because of a failure in the security system, the chain reaction couldn’t be stopped, the catastrophe produced would be much worse than that of Chernobyl, which consisted ‘only’ of a leak of radioactive material with the very serious consequences we all know well.

Above all this, even with the process taking place in ‘secure’ conditions, the problem of dealing with radioactive waste produced in the nuclear reaction still remains. This waste has to be securely stored, as it will produce for hundreds, sometimes thousands, of years, radiation harmful for all living things. Nuclear repositories where this waste is kept are already reaching their capacity limit.

In 1983, Meryl Streep, Kurt Russell and Cher starred in Silkwood, a drama about a nuclear power station worker found dead in strange circumstances after having reported irregularities related to security in the station she worked in. Jane Fonda, Michael Douglas and Jack Lemmon also reported the dangers surrounding nuclear power in The China syndrome (1977), a more didactic movie, but a plainer and poorer one as well.

Nowadays, movies about nuclear power are out of date because nuclear power itself is: it’s been a long time already without new stations being built in most countries. Alternative and more ecological energies, like solar or aeolic, are being studied instead. These aren’t still enough developed, though, for allowing the closing-down of all the nuclear stations still active (five of which are located here in Spain, by the way). Few people doubt that is the destiny meant for all those stations either in the short or in the long term, though. Nuclear power was absolutely the biggest failure of engineering in the 20th century.

2001: The silent space

When Stanley Kubrick released 2001: A space odyssey in 1968, he proved that science-fiction gender didn’t have to consist only in B – movies, but it could be a gender as good as any other to tell more serious stories. Along with this more elevated artistic aim came a greater scientific rigour. Apart from introducing one of the first electronic cinema characters, the famous computer HAL, the movie tried to make space trips look believable, in an age in which these were becoming a reality.

The mute takes of outer space were a shocking component of the movie. Space is full of light coming from the stars but it is quiet; sound can’t be transmitted through it. Light is able to spread through a vacuum, just like heat, but sound is a pressure wave instead: air particles collide and transmit a vibration which is turned into sound when reaching our eardrums. Therefore, there can be no sound without any atmospheric air, so when we hear the spacecrafts’ collisions and explosions in a countless number of movies, we must think that we are dealing with one of those well known poetic licenses of movie makers.