Analysing Formula 1

It's well-known that the greatest number of Grand Prix winners in a single season is eleven, a number attained during the epic and tragic 1982 season. Did you know, however, that the second greatest number of winners in a single season is nine, from the 1975 season?

In fact, in the history of Formula 1, only six seasons have featured eight or more winners, and all but one of those occurred between 1975 and 1985, (the subsequent exception being 2003). On the basis of this fact alone, one might argue that the years between 1975 and 1985 define the most competitive era in the sport's history. It also begs all sorts of questions about the conditions which led to such a competitive environment, and why they have so rarely pertained since.

These facts are gleaned from Roger Smith's colourful 2008 tome on the statistics of Formula 1, which is now available in paperback. It's well worth a purchase, for this was clearly a labour of love for Smith. Of particular interest is the book's concluding chapter, where Smith expounds the results of a rating system which enables all the champion drivers to be ranked, irrespective of the eras in which they raced.

The basic performance indicator chosen by Smith is a driver's strike rate, the number of Grand Prix victories as a fraction of races contested. After ranking the drivers by strike rate, Smith then attempts to adjust the ranking to compensate for the superiority of the equipment at a driver's disposal, and the strength of the driving competition he faced. Sadly, Smith doesn't 'show his working' here, but he does explain that the superiority of a driver's equipment in any particular year can be estimated by factors such as: the absolute share of wins achieved by the driver's team; the number of 1-2s; the number of victories by the driver's team-mate; and the share of wins relative to the second most successful team that year. How Smith disentangles the strength of the driving competition from the strength of the equipment available to the competition is unclear, but the upshot is a rating system which places Fangio first, Clark second, and Schumacher third.

A first objection is that it's difficult to argue with Smith's reasoning without being able to see his detailed calculations. In addition, however, there is a serious omission which underlies Smith's ratings system, and it is an error which is committed by every published attempt to rank the all-time greats. It is the failure to adjust for the size of the competitive pool, and the fact that the pool has been steadily growing in size since the inception of the Formula 1 World Championship.

In the 1950s and 1960s, only a relatively small number of people were competing in single-seater motorsport, and there were only a small number of formulae. The number competing at all levels of the sport has increased massively from the 1970s and 1980s onwards. Developing hand-in-hand with this has been the proliferation of the different junior formulae, all arranged in a pyramidal structure, filtering out the best drivers at each stage (in theory!), and feeding them towards the world of Formula 1, located at the tip of the pyramid. At the base of the pyramid is the immensely competitive world of kart racing, into which thousands of children across the world every year, are now inducted at an early age, to begin learning the craft of racing driver.

As a general principle of performance statistics, all other things being equal, the best person from a large competitive pool is likely to be better than the best person from a small competitive pool. Aphoristically, it's easy to be a big fish in a small pond, and the best drivers of the 1950s and 1960s were essentially just that. The Fangios and the Clarks were the tips of very small pyramids, whilst the Sennas, Schumachers and Hamiltons are the tips of very large pyramids.

As a comparison, consider American single-seater racing. This is a much smaller competitive pool than the hierarchy of single-seater formulae in the rest of the world, which feeds into Grand Prix racing. Thus, it is easy for a driver such as Al Unser Jnr or Michael Andretti to look devastating in Indycar racing, but to fail badly when they attempt the transition to Grand Prix racing. The best Formula 1 drivers of the 1950s and 1960s are comparable to the best drivers in American single-seater racing. It's quite possible that the best driver ever could have raced in American single-seater racing, and it's still quite possible that Fangio or Clark was actually the best driver the world has ever seen, but on the basis of statistics alone, adjustment for the different sizes of the competitive pools mitigates against this conclusion.

The best drivers in the world effectively lie in the tail-end of the distribution of driver talent, and as a general statistical rule, unless you take very large sample sizes, you're unlikely to be sampling from the tail-end of a distribution. For example, if the frequency of great drivers (those in the tail-end of the talent distribution) is 1-in-100,000, then at a time when there are only, say, 100 drivers in the world, the chance that one of them will be a great driver will only be 1-in-1,000. Hence, it's highly unlikely (but not impossible), that the best driver the world has ever seen was part of the small sample of Grand Prix drivers found in the 1950s and 1960s.

All of which is maybe a way of saying that statistics alone cannot be used to support or refute the subjective appreciation of drivers, made by observers in the same era to which the drivers belong.

Are racing cars arthropods?

If a zoologist were forced to classify racing cars in biological terms, he might well be inclined to conclude that they match the body plan of those animals called arthropods. This so-called 'phylum' of the animal kingdom includes the insects and the crustaceans.

Arthropods are defined by the fact that they possess the following characteristics: an external skeleton ('exoskeleton'), which protects the more vulnerable organs within; bilateral mirror symmetry; a segmented body plan; and jointed attachments called appendages, which provide functions such as locomotion.
In racing cars, the monocoque, whether it is a modern carbonfibre shell, or a metal space-frame construction, plays the role of an exoskeleton, protecting the vulnerable parts within; most racing cars possess bilateral symmetry, with minor exceptions such as asymmetric radiator layouts, and the suspension 'stagger' used by Indycars on oval circuits; most racing cars have a segmented body plan, front-to-rear, with various bulkheads separating the modules; and the suspension and wheels on a racing car play the role of the appendages on an arthropod.

Arthropods, like all animals, possess a metabolism, in the sense that they burn a source of chemical potential energy, which has ultimately been created by photosynthesis in plants, in order to perform useful work. In so doing, they create waste heat, which has to be radiated to the environment. Similarly, a racing car equipped with an internal combustion engine, burns a source of chemical potential energy, which has ultimately been created by photosynthesis in plants, in order to perform useful work, and in so doing creates waste heat, which has to be radiated to the environment.

The analogy would be complete if the 'active-ride' technology, seen in Formula 1 in the late 1980s and early 1990s, had been permitted to develop to the point where racing cars became fully autonomous. The feedback loops involving sensors, computer control units, and actuators, would ultimately have allowed racing cars to drive around circuits on their own. As it is, racing cars play host to parasitic organisms belonging to the phylum of the chordates. These parasitic organisms provide the sensing, decision-making, and control inputs in a racing car. Such individual chordates possess various names, but are mostly called Mika.

A champion's drive?

Smoother than Roger Moore in his prime, and with the surgical precision of Professor Sid Watkins in-theatre, Jenson Button scythed through the field to win the Formula 1 World Championship on Sunday.

This was a fabulously exciting race, and reminded one that a Grand Prix conducted in dry weather conditions can still be exciting if the track favours a low-downforce set-up, and if a number of the faster cars start the race behind a number of the slower cars. Formula 1 racing has always been cerebral, but there was a time when the cerebral was entwined with the visceral, and the opening laps of the Brazilian Grand Prix were a reminder of those days.

In fact, many of the drivers almost appeared to have 'gone feral' in the opening laps, and the first incident occurred on the run down to turn 4 when Mark Webber veered across to block Raikkonen, and Kimi's front-wing was crumpled against the rear wheel of the Red Bull. Mark's move was later than a winning goal in extra-time at Old Trafford, and as such was an almost carbon-copy of the move Michael Schumacher pulled on Juan-Pablo Montoya on the first lap in 2002. Kimi almost got his fingers burnt here, but whilst Montoya's post-race criticism of Schumacher was vociferous, Kimi merely filed the incident away for future reference.

A minute or so later, as Raikkonen departed the pits with a new nosecone attached, he must have been at least somewhat startled to find Kovalainen's McLaren veering into his path like a V-bomber, its trailing proboscis offering the tantalising prospect of some air-to-air refuelling. The moment was brief, however, and after driving through the consequent conflagration, Kimi was left to reflect once more upon how close he had come to getting his fingers burnt.

The other major incident of note on the first lap occurred as an indirect result of the Webber/Raikkonen accident. As Kimi slowed through turn 4, Sutil's Force India had to get off the power momentarily, and Trulli opportunistically attempted to run around the outside of the Force India in turn 5. Sutil, however, ran the Toyota out wide, forcing Trulli over the kerbs, where he lost control and punted Sutil amidships, spitting the Force India into the infield and the Toyota into the wall. As the Force India hurtled over the grass, Fernando Alonso appeared to be strangely oblivious to its trajectory, and clobbered it as it returned to the track in the middle of Ferra Dura.

Trulli immediately remonstrated with Sutil by the trackside, and after the race accused the German of deliberately driving him off the road. Perhaps, however, this was something of an over-reaction, and one might recall that Robert Kubica's potentially fatal crash at Montreal in 2007 was also caused when he tried to run around the outside of a driver who didn't know that he was there. That driver was Jarno Trulli.

All of this mayhem eliminated four cars which would otherwise, most probably, have finished ahead of Jenson Button. Along with the retirement of Nico Rosberg, this was actually the primary reason that Jenson was able to seal the World Championship on Sunday. His overtaking moves were ballsy and brilliant, but they were executed to pass four rookie drivers (Grosjean, Nakajima, Kobayashi and Buemi). To put Jenson's drive into perspective, it's also worth noting that two of the established drivers who qualified behind him (Hamilton and Vettel), both finished ahead of him. In the case of Vettel, his Red Bull simply had superior pace to the Brawn, and in Hamilton's case his natural elan was combined with a superior one-stop strategy.

So, whilst it is indeed time to pay tribute to Jenson's triumph, with some irony it is actually Sebastien Vettel's team-mate, Mark Webber, to whom Jenson Button should owe the greatest thanks this week.

Maurice Hamilton meets Richard Dawkins

Cows can be modified to grow vast and ungainly udders, and these continue to yield copious quantities of milk indefinitely, long after the normal weaning period of a calf...And of course, the same would be true of dairy humans, if anyone wanted to try. All too many women...pay surgeons large sums of money to implant silicone...Does anyone doubt that, given enough generations, the same deformity could be achieved by selective breeding, after the manner of Friesian cows? (Richard Dawkins, The Greatest Show on Earth, p39).

Jacques [Villeneuve] was always a bit of a renegade - I'd say an odd character. He was a very appealing character but in many ways quite child-like. If he fancied a girl, he'd ask Ann Bradshaw [Williams Press Officer] to go out and chat her up for him. (Patrick Head, speaking to Maurice Hamilton in Williams, p296.)

Like most people, I've spent the past few weeks concurrently reading the latest books by Richard Dawkins and Maurice Hamilton. Both are absolute gems.

Dawkins's latest book is a fabulous exposition of the concepts and evidence to support evolution by natural selection. It's a gripping and eclectic work, whose range extends from embryology to plate tectonics, to artificial breeding. Did you know, for example, that cabbage, cauliflower, brussel sprouts and broccoli, have all been bred by horticulturalists from a single plant, the wild cabbage, over just a few centuries?

One of the big themes of Dawkins's book is the manner in which evolution yields modification over long periods of time, whilst retaining unmistakeable traces of what was in place at the outset. Such a description is also apposite for the history of the Williams Formula One team. As erstwhile Williams engineer John Russell points out, "Bernie Ecclestone once said that Frank and Patrick run the best grocery store in the business...It's no longer a grocer's shop; it's a supermarket,...but if you go behind the scenes the grocer's shop is still there."

In fact, in the early months of 1983, as a member of the nascent Williams Grand Prix supporters' club, I was one of the lucky few invited to visit the Williams factory at Station Road in Didcot. I remember being escorted around the factory by someone called Sheridan [Thynne], and being slightly underwhelmed to discover that it consisted of a small number of partitioned bays inside an industrial unit. As designer Neil Oatley recalls in Maurice Hamilton's book, the original Station Road factory was only about the size of two tennis courts.

Hamilton has sourced and written perhaps the best book ever on the Williams Formula One team. It has a somewhat unusual format in that it largely consists of personal recollections from the dramatis personae associated with Williams over the last four decades, but this plurality of subjective perspectives weaves into a fascinating and cohesive sporting and engineering history. In particular, the perspicacity of Patrick Head and Adrian Newey radiates from the page like a pair of Cobalt-60 sources.

The Funeral Games Grand Prix

Whilst the first Grand Prix motor race was held at Le Mans in 1906, the first de facto Grand Prix was actually the chariot race at the Funeral Games, held in honour of Patroclus, ca. 580 BC. Surprisingly, therefore, the first Grand Prix journalist was Homer, who published his race report in the Illiad 23.

Although the games were organised by Achilles (a chariot dealer from South Athens who had made good), the entry list for the first Grand Prix was disappointingly small, consisting of only five charioteers: Eumelus, Diomedes, Antilochus, Menelaus, and Meriones.

As the race approached its pivotal point. Eumelus was leading from Diomedes, but Diomedes was gaining ground, and making full use of what appeared to be an ancient Greek version of KERS. "Diomedes' stallions...seemed ever like to mount upon Eumelus' car, and with their breath his back waxed warm and his broad shoulders, for right over him did they lean their heads as they flew along."

Then, just as Diomedes was about to duck out of the slipstream and take the lead, he committed an error of concentration, lost his whip, and dropped behind as his horsepower plummeted. Eumelus now looked to be heading for a certain victory, but in chariot racing anything can happen, and usually does.

Without warning Eumelus' chariot suffered an axle failure, with disastrous consequences: "The mares swerved to this side and that of the course, and the pole was swung to the earth; and Eumelus himself was hurled from out the car beside the wheel, and from his elbows and his mouth and nose the skin was stripped, and his forehead above his brows was bruised; and both his eyes were filled with tears and the flow of his voice was checked."

As Eumelus licked his wounds, Diomedes raced to victory, and the crowd's attention switched to the battle for second. Menelaus was leading Antilochus, but the latter refused to accept defeat:

"Then quickly did Antilochus, staunch in fight, espy a narrow place in the hollow road. A rift there was in the ground, where the water, swollen by winter rains, had broken away a part of the road and had hollowed all the place. There drave Menelaus in hope that none other might drive abreast of him. But Antilochus turned aside his single-hooved horses, and drave on outside the track, and followed after him, a little at one side...the mares of [Menelaus] gave back, for of his own will he forbare to urge them, lest haply the single-hooved horses should clash together in the track, and overturn the well-plaited cars, and themselves be hurled in the dust in their eager haste for victory. Then fair-haired Menelaus chid Antilochus, and said: 'Antilochus, than thou is none other of mortals more malicious. Go, and perdition take thee, since falsely did we Achaeans deem thee wise. Howbeit even so shalt thou not bear off the prize without an oath'."

As the chariots crossed the finishing line, Diomedes was first, Antilochus second and Menelaus third. The result was then cast into doubt, however, when Menelaus lodged a protest with the organisers over Antilochus' driving. Summoned by the stewards, Antilochus was asked to demonstrate his innocence by swearing an oath to Zeus. Fearful of being on the receiving end of a thunderbolt which would prohibit his further participation in chariot racing, Antilochus accepted responsibility, but was permitted to retain his second place finish. Diomedes, for his part, was almost forgotten in the controversy, but as the winner he duly received a slave woman and a cauldron as his prize.

So, whilst the cauldrons are now made of Waterford crystal, it seems that little else has changed.

Bobsleighs humped by lawnmowers

Before his untimely death canonised him, Senna's realism was commonly called ruthlessness by everyone in the sport. To a certain extent it was: when he figured out that he would become champion if Prost could be removed from the track, he accomplished this by driving into Prost, thereby removing himself as well, but with the championship in the bag. He engineered the impact straight after doing the sums in his head, thus setting a bad precedent. Such behaviour brought formula one close to being a demolition derby, but it was a natural consequence of a team's readiness to back up its top man, even if his conscience-free behaviour was at the expense of its second man. More recently, tighter rules have made the deliberate shunt harder to pull off, but as with the professional foul in football, the spirit of the thing is hard to quench. (Clive James, The Guardian, 14th May 2002).

Sunday's Japanese Grand Prix was, by common consent, about as exciting as watching Astroturf grow. In fact, the curious placing of such synthetic turf beyond the kerbing at a number of Suzuka's corners transformed several small mistakes into several very large shunts, much like Lewis Hamilton's last-lap accident at Monza. Perhaps, then, Mr Ecclestone's next plan will be to introduce Astroturf at the apexes of the corners.

Such carnage is hardly a substitute for genuine racing, but as Clive James pointed out some years ago, "rarely does [Formula One] provide a thrilling spectacle. Apart from the occasional shunt, it mainly shows you a procession. But to the fan, the questions are endless, convoluted and enthralling."

It was James, of course, who unforgettably described Murray Walker as sounding, even in his calmest moments, like a man with his trousers on fire. Observing the World Championship finale at Estoril in 1984, James also noted that "the cars all look like a bobsleigh being humped by a lawnmower." It was an observation which sadly failed to appear in Autocourse 1984's Technical Analysis.

1984 did, however, possess the virtue of a Grand Prix calendar determined by the location of the great circuits, rather than one inspired by Marco Polo's travel itinerary. It was also a year free from deliberate shunts (?), but as James remarks, "the spirit of the thing is hard to quench."

There is currently a vacancy at the University of West England, Bristol, for a Senior Lecturer in Motorsport and Mechanical Engineering. Given the growing list of refugees from Formula One, the competition for this post may be stiff indeed.

The radiation dose to Formula One

Frequent business flyers, such as the team members and media representatives who fly to every Grand Prix in the Formula One World Championship, are subjected to a radiation dose which can exceed that received by workers in the nuclear industry. In fact, "estimates of potential exposures of aircrew and frequent travellers, validated by on-board measurements carried out by carriers in the United States, Canada and Europe, indicate that these groups sometimes receive exposures above the limits for the general public, and should be categorized as occupationally exposed." (Lawrence Townsend 2001, Radiation exposures of aircrew in high altitude flight, Journal of Radiological Protection, Vol. 21 pp5-8).

This dose to frequent flyers comes ultimately from galactic cosmic rays, 90% of which are high-energy protons, and 10% of which are alpha particles. These high-energy particles collide with the nuclei of atoms in the atmosphere, creating secondary particles. The dominant contribution to the radiation dose received by frequent flyers comes from the neutrons created in such reactions.

The dose varies with the 11-year solar cycle, and increases with latitude and altitude. Nevertheless, an estimate can be made of the dose to the Formula One community by taking some ballpark figures. Before launching into the calculation, it should be noted that radiation dose is estimated in units called Sieverts (Sv). The milli-sievert (mSv) is a thousandth of a Sievert, and the micro-sievert (μSv) is a millionth of a Sievert.

To estimate the average annual dose, we need to know the average dose-rate per hour, and the number of hours flown per year. Townsend reports that at typical commercial jet altitudes (30,000-40,000 ft), the dose-rate is 5-10 micro-sievert per hour. The US National Commission for Radiological Protection (NCRP) has produced a report in which this dose-rate is taken to be 20 micro-sievert per hour (NCRP Commentary No 12), but let us take 10 micro-sievert per hour as a more conservative figure.

In terms of the number of hours flown, there are 10 long-haul destinations on next year's 19-race Formula One calendar (Bahrain, Australia, Malaysia, China, Canada, Singapore, Japan, South Korea, Abu Dhabi, and Brazil). Taking round figures again, each one of these races involves a 10-hour flight each way (neglecting the fact that the Formula One community will fly direct from Singapore to Japan). That's approximately 200 flying hours to and from the long-haul races alone. Let's be conservative and neglect the additional flying hours to the European races. The total annual dose is still:

10 micro-sievert per hour * 200 = 2,000 μSv = 2 mSv.

Thus, over a 25-year career in Formula One, the cumulative dose will be 50 mSv. Now, the International Commission on Radiological Protection (ICRP) estimate that the risk of acquiring a fatal cancer from radiation exposure is per 5% per Sievert. That means that for a cumulative dose of 50 mSv = 0.05 Sv, the risk is only 0.25%. The background level of fatal cancer in the population is about 25%, so an individual pursuing a 25-year career in Formula One will experience only a one-in-a-hundred increase in the pre-existing risk of acquiring a fatal cancer.

However, there are a large number of people who fly to every Grand Prix, so we can also work out the collective dose. If we assume that there is a core of 1,000 people who fly to every race, then if the cumulative dose per person over a 25-year career is 0.05 Sv, the collective dose will be:

0.05 Sv * 1,000 = 50 Sv.

With the risk of a fatal cancer being 5% per Sievert, this entails that 2.5 people amongst the Formula One community can be expected to acquire a fatal cancer due to their exposure to cosmic radiation.

Dartington Hall

Devon sometimes seems like a region of unfulfilled geography; it is en route landspace, a zone to be traversed by holidaymakers going to and from Cornwall. Whilst inheriting some of the holiday lustre of its big brother, it is diminished by the fact that it is simply not the pointy bit at the end of the country.
Nevertheless, Devon has spectacular charms. Travelling by train, at Teignmouth one passes between garishly-hued sandstone cliffs, and the slate-grey expanse of the sea, stretching out towards horizon-hogging supertankers. Winding inland, the hillsides and valleys seem to be saturated with trees, luxuriantly limbed and leafed even in early Autumn.

Dartington Hall itself offers a tranquil escape from the hectic mundanity of life. The gardens are exorbitantly decked with a profuse variety of trees both tall and broad, vantaged by stairs and paths and outposts and green terraces, stimulating the eye from every angle.

All of which made the cancellation of the 17:27 from Totnes something of a comedown. Another train, it seems, had broken down, (tricky thing, this diesel technology), and as a consequence the train which goes from Paddington to Penzance, and back again, had been turned around at the next station up the line. A £20 taxi ride was therefore necessary to catch the 17:40 from Newton Abbot. Once aboard, an astonishingly generous (and monied) colleague sweet-talked the ticketman into two first-class upgrades for the cost of one. That'll be just the hundred pounds or so then.

And what do you get for your hundred pound first-class seat? You get a table, with an electric socket, and... (wait for it) a complimentary copy of The Times.

Night racing and Olbers's paradox

It would be easy to think that the existence of night is solely a consequence of the rotation of the Earth and its location relative to the Sun. But it is not. It is a consequence of the expansion of the Universe. If the Universe were not expanding then, wherever we looked into space, our line of sight would end at a star. The result would be like looking into a forest of trees. In a universe that didn't expand, the whole sky would resemble the surface of a star; we would be illuminated by perpetual starlight. What saves us from this everlasting light is the expansion of the Universe. It degrades the intensity of the light from distant stars and galaxies, and it leaves the night sky dark. (John Barrow, The Artful Universe, p45.)

As the Sun sets in Singapore, and European visitors fight the disruption to their circadian cycles, the aesthetics of Formula 1 undergoes a phase transition. A silver strip of metal halide light runs between the colonial palm trees, beneath the bejewelled post-modernist towers, underneath the concrete stanchions of the flyovers, alongside the armadillo-contoured concert hall and theatre, and beside the neoclassical and Palladian civic architecture. Specular reflections shimmer from the compound surfaces of the cars; the Ferraris become molten lava, and the McLarens dissolve into liquid metal.

Permitting this extravagant display of light and pattern is the dark night sky, a phenomenon whose very existence requires a cosmological explanation, as recognized by the 19th century German physician and amateur astronomer, Heinrich Olbers. Olbers realised that if our universe were a static universe, infinite in space and time, and homogeneously populated with stars (or galaxies or galaxy clusters etc.), then the sky should be bright throughout the day and night. The fact that the night sky is dark is therefore Olbers's paradox. Modern physics solves Olbers's paradox by virtue of the fact that light travels at a finite speed, and by virtue of the fact the Friedmann-Robertson-Walker models of relativistic cosmology represent our universe to be an expanding universe of finite age.

In a universe of finite age, in which light travels at a finite speed, there will be a finite cosmological horizon around every astronomically observant species; the universe is 14 billion years old, hence the light from stars more than 14 billion lights years away has not had time to reach us. Moreover, in an expanding universe, the expansion red-shifts distant starlight towards energies invisible to the naked eye, and reduces the brightness of the light.

There is a twist, however, for "it seems that the background of the sky is bright, even at night. Of course, it is not as bright as the surface of the Sun, nor does it shine at the same wavelengths. However, according to models of the big bang, the entire universe was so hot around 14 billion years ago that each of its points was a luminous as the surface of the Sun. Each direction leaving from our eye reaches a point of this past Universe. And by the same reasoning as that of Olbers, even in the absence of every star, we should be surrounded by this enormous bright object, the early Universe...We indeed receive this radiation, but it is shifted towards long wavelengths and weakened...Since it is very old, the shift is very strong: redshifted by a factor greater than 1000, it has transformed the light into microwaves. This electromagnetic fossil radiation, a vestige of the primitive epoch, was detected for the first time in 1964. Today it is being exhaustively observed under the name of cosmological background radiation." (Jean-Pierre Luminet, The Wraparound Universe, p159-160).

This weekend then, the Formula 1 cars will race at night through the streets of Singapore, and will do so oblivious to the omnipresent background radiation, and the cosmological expansion which permits this visual and kinetic cornucopia.

Singapore and interstellar gas clouds

Sitting at a country pub on the banks of the canal, basking in the Sun beside a weeping willow, the barges chugging languorously past, Singapore seems more than a world away. Swatting away an over-ambitious wasp, the skyline of this extraterrestrial plutopolis rises in the imagination like numerous glass, steel and concrete stalagmites, slowly precipitating from the steady drip of money over economic aeons.

Four and a half billion years ago, a cloud of interstellar gas and 'dust' (tiny grains of solid matter) contracted under the force of its own gravity, started to spin, and formed a rotating disk. The ball of material at the centre of the disk reached ever higher temperatures and pressures, until nuclear fusion ignited inside, and a star, our Sun, was born. The residual material in the surrounding disk then coalesced into an array of planets, our solar system.

The most abundant elements in the contracting cloud were hydrogen H, helium He, oxygen O, carbon C, nitrogen N, neon Ne, magnesium Mg, silicon Si, iron F and sulphur S. The silicon combined with oxygen to make silicates, and further combined with iron and magnesium to make what is colloquially known as rock. The remaining oxygen combined with hydrogen to make water. Within the outer reaches of the protoplanetary disk, the water was frozen, and the accretion of such icy masses gave the more distant planetesimals a head-start over the rocky masses which formed closer to the Sun. The more distant objects acquired sufficient mass to attract hydrogen, helium, and compounds such as methane CH₄ and ammonia NH₃, thereby creating the gas giants: Jupiter, Saturn, Uranus and Neptune.

On the surface of the third planet from the Sun, a rocky planet, oceans of liquid water formed. The oceans were populated by countless microscopic photosynthesizing organisms. Many had shells or skeletons made of calcium carbonate, and these shells and skeletons were continuously returned to the ocean floor, where they accumulated over the ages as layers of chalk or limestone. Over those same timespans, grains of silica SiO₂, or sand, were created from the weathering of silicate rock on the surface of the planetary crust. Then, eventually, an intelligent species emerged from the biosphere of the planet, delved into the planetary crust, and devised construction materials such as glass (silicon oxide which has been cooled sufficiently rapidly that the molecules are unable to form a regular crystal lattice), steel (iron judiciously doped with carbon atoms), and concrete (a coarse aggregate of limestone or gravel, combined with cement, water and sand, the cement itself a product of the heating and grinding of limestone and clay).

Four and a half billion years after the contraction of that cloud of interstellar gas and dust, Singapore rose as a glass, steel and concrete monument to the growth of complexity.

Branching and interfering parallel universes

"This universe is constantly splitting into a stupendous number of branches, all resulting from the measurementlike interactions between its myriads of components. Moreover, every quantum transition taking place on every star, in every galaxy, in every corner of the universe is splitting our local world into myriads of copies of itself." (Bryce De Witt, 1970).

The many-worlds interpretation of quantum mechanics famously holds that if a physical system is prepared into a state which is a superposition with respect to the possible values of a quantity A, then when the value of A is measured the universe splits into multiple branches, each of which realises one of the different possible definite values of quantity A. The superposed state Ψ is a sum

Ψ = c₁ ψ₁ + ⋅ ⋅ ⋅ + c_n ψ_n ,

where each ψ_i is a quantum state in which quantity A possesses a definite value. The many-worlds interpretation proposes that when the universe branches, it branches into all the different states ψ_i from the superposition.

Extending this interpretation to the macroscopic world of human experience, it is postulated that the universe branches every time a choice is made, and that all the different possible lives we may have led if only our choices had been different, do in fact exist as different branches of the quantum universe.

The naive manner of picturing this process is to represent the different histories at each branching point as if they were the leaves of a book, radiating outwards from the spine. The spine in this analogy corresponds to a spacelike three-dimensional hypersurface, and the leaves radiating from it correspond to different four-dimensional space-time histories. John Earman illustrates this concept in the first diagram reproduced here, for the simple case where only two histories radiate from each branching point.

However, the notion that the entire universe branches in this style every time there is a measurement-like interaction, renders such branching a highly non-local process, and tacitly supposes that there is a unique global time coordinate for the universe. Treating a measurement-like interaction as a point event in space-time, there will be many spacelike hypersurfaces which pass through that point; the selection of only one of these as the branching hypersurface requires one to accept that there is a preferential time coordinate for the universe.

To avoid these difficulties, one can suggest that the universe only branches locally as the result of a measurement-like interaction. To be specific, one can suggest that the future light-cone of the interaction event has multiple branches, one for each possible outcome of the interaction. If one imagines such a universe as a two-dimensional sheet, then the image is one in which there are numerous pockets in the sheet, formed by the multiple branches of the future light cones. Roger Penrose drew just such an image of a branching universe in 1979, reproduced as the second diagram here.

Returning to the many-worlds interpretation, it is important to note that the different branches ψ_i of the overall wave-function Ψ, do not themselves correspond to different classical universes. Whilst the states ψ_i do indeed bestow definite values upon the quantity A, they are still quantum states in their own right, and as such, they fail to assign definite values to all the quantities possessed by the physical system under consideration.

This is crucial, because advocates of the many-worlds interpretation can often be found claiming that a quantum universe is a universe whose basic ontological fabric consists of interfering classical universes. One could conceivably subscribe to the many-worlds interpretation of quantum measurement without endorsing this stronger ontological claim. The many-worlds interpretation of quantum measurement requires one to accept that the universe is continually branching into components of the quantum wave-function, whilst the stronger ontological claim requires one to accept that the entire quantum wave-function and its branches, consist of bifurcating, interfering and merging classical histories.

The stronger claim seems to be fuelled by the 'sum-over-histories', or path-integral formulation of quantum mechanics, in which each branch of the quantum wave-function corresponds to a different set of interfering classical histories, and in which the different branches of the wave-function interfere when there is interference between these different sets of classical histories.

For example, in the famous double-slit experiment, there are two possible sets of classical histories. If we label the slits as Slit A and Slit B, then one set consists of all the possible trajectories through Slit A, and the other consists of all the possible trajectories through Slit B. When both slits are open, the paths through the different slits duly interfere with each other to produce the overall wave-function on the distant screen.

(The interference between the different branches of the wave-function are purportedly removed by decoherence on macroscopic scales, thereby explaining why we never observe quantum superpositions on such length scales).

In this sense, the classical histories form the warp and weft of the quantum fabric of the universe. Whilst this is a stronger ontological claim than the basic many-worlds interpretation, in many ways this is a more coherent picture than one in which an overall quantum state branches by fiat into its component quantum states.