Philae comes in to land on 67P/Churyumov-Gerasimenko. It reached the comet using carefully calculated forces of attraction. Image: 2014 European Space Agency/Getty
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Wandering in the heavens: how mathematics explains Saturn’s rings

Ian Stewart shows how maths is changing cosmology, and explains why the best way to reach a comet near Mars is to go round the back of the sun.

The Enūma Anu Enlil, a series of 70 clay tablets, was found in the ruins of King Ashurbanipal’s library in Nineveh (on the eastern bank of the River Tigris, opposite modern-day Mosul in Iraq). The name means “in the days of Anu and Enlil”; Anu was the sky god, Enlil the wind god. The tablets, which date as far back as 1950BC, list 7,000 omens from Babylonian astrology: “If the moon can be seen on the first day, the land will be happy.” But tablet 63 is different: it gives the times when Venus first became visible, or disappeared, over a 21-year period. This Venus tablet of Ammisaduqa is the earliest known record of planetary observations.

The Babylonians were expert astronomers who produced star catalogues and tables of eclipses, planetary motion and changes in the length of day. They were also capable mathematicians, with a number system much like ours, but using base 60 rather than ten. They could solve quadratic equations and calculate the diagonal of a square with precision, and they applied their mathematical skills to the heavens. In those days, mathematics and astronomy were part and parcel of astrology and religion, and the whole package was intimately bound up with agriculture through the progression of the seasons.

The torch of astronomy passed by way of ancient Greece to India. In 6th-century India, mathematics was a sub-branch of astronomy, and astronomy still played second fiddle to reading omens in the stars. The Arab world made further advances in our understanding of the cosmos, and kept the ancient knowledge alive until Europe once more turned its attentions to the science of the heavens.

In 1601 Johannes Kepler became imperial mathematician to the Holy Roman emperor Rudolf II. Casting the emperor’s horoscope paid the bills, and it also left time for serious mathematics and astronomy. Kepler had inherited accurate observations of Mars from his former master Tycho Brahe, and from these he extracted three mathematical patterns, his laws of planetary motion. By then, thanks to Nicolaus Copernicus, it was known – though still controversial, to say the least – that the planets revolve round the sun, not the Earth. Their orbits were thought to be combinations of circles, but Kepler’s calculations showed that planets move in ellipses. His other two laws govern how quickly the planet moves and how long it takes to go round the sun.

In his epic Mathematical Principles of Natural Philosophy of 1687, Isaac Newton built on Kepler’s laws and deduced his law of universal gravitation: every body in the universe attracts every other body with a force that obeys a specific mathematical rule. These forces determine how moons, planets and stars move. Newton’s book paved the way to a rational scientific understanding of nature based on precise mathematical laws, and opened up the metaphor of the clockwork universe.

One of the great tests of Newtonian gravitation was Edmond Halley’s prediction about a comet. In ancient times comets, bright bodies with long curved tails that seemed to appear from nowhere, were seen as omens of disaster. From old records, Halley realised one particular comet was a repeat visitor, with an elliptical orbit that took it near the Earth every 76 years. He predicted its next return in 1758. By then Halley was dead, but his prediction proved correct.

Even today, Newton’s law remains vital to astronomy and space exploration; Einstein’s later refinements are seldom needed. A topical example concerns another comet, rejoicing in the name 67P/Churyumov-Gerasimenko, which takes about six and a half years to orbit the sun. In 2004 the European Space Agency (ESA) launched the Rosetta probe to visit the comet and find out what it looked like and what it was made of. Famously, it resembled a rubber duck: two round lumps joined by a narrow neck. On 12 November 2014 a small capsule, Philae, landed on the head of the duck, which was 480 million kilometres from Earth and travelling at over 50,000 kilometres per hour. Unfortunately Philae bounced and ended up on its side, but even so it had sent back vital and unprecedented data.

It’s worth visiting the ESA’s “Where is Rosetta?” web page to see an animation of the astonishing route the probe took. It wasn’t direct. The probe began by moving towards the sun, even though the comet was far outside the orbit of Mars, and moving away. Rosetta’s orbit swung past the sun, returned close to the Earth, and was flung outwards to an encounter with Mars. It then swung back to meet the Earth for a second time, then back beyond Mars’s orbit. By now the comet was on the far side of the sun and closer to it than Rosetta was. A third encounter with Earth flung the probe outwards again, chasing the comet as it now sped away from the sun. Finally, Rosetta made its rendezvous with destiny.

Why such a complicated route? The ESA didn’t just point its rocket at the comet and blast off. That would have required far too much fuel, and by the time it got there the comet would have been somewhere else. Instead, Rosetta performed a carefully choreographed cosmic dance, tugged by the combined gravitational forces of the sun, the Earth, Mars and other relevant bodies. Its route was designed for fuel efficiency; the price paid was that it took Rosetta ten years to get to its destination. Each close fly-by with Earth and Mars gave the probe a free boost as it borrowed energy from the planet. An occasional small burst from four thrusters kept the craft on track. And every kilometre of the trip was governed by Newton’s law of gravity.

Complex trajectories such as this one have now become standard in many unmanned space missions. They originated in mathematical studies of chaotic dynamics in the motion of three gravitating bodies, and go back to pioneering work by Edward Belbruno at the Jet Propulsion Laboratory in California in 1990. He realised that these techniques could put a Japanese probe, Hiten, into lunar orbit after a failure of its parent craft, even though there was hardly any fuel available.

Mathematics has always enjoyed a close relationship with astronomy; not just in the technology of space missions but in understanding planets, stars, galaxies – even the entire universe. How, for example, did the solar system form? We can’t go back to take a look, so we have to do some celestial archaeology, inferring what happened from the evidence that remains. Our main tool is mathematical modelling, which lets us test whether hypothetical scenarios make sense.

When Galileo first spied Saturn in 1610, he took it to be a trinity of planets. Image: Nasa/Eyevine

Observations and theoretical astrophysics tell us that the sun came into being about 4.8 billion years ago, and the planets of the solar system formed at much the same time. Everything condensed out of the solar nebula, a huge cloud of gas – mainly hydrogen and helium, the two commonest elements in the universe. The cloud was about 65 light years across, 15 times the distance to the nearest star today. One fragment, about four light years across, gave rise to the solar system; other fragments became other stars – many of which, we now know, have their own planets. As our fragment collapsed under its own gravitational field, most of the gas collected at the centre, where enormous pressures ignited nuclear reactions to create the sun. Much of the remaining gas clumped into smaller, but still gigantic, bodies: the planets. The rest either got swept away or remains as various items of clutter – asteroids; centaurs (small bodies with characteristics of both comets and asteroids); Kuiper Belt objects, in the debris field beyond Neptune; comets in the Oort Cloud, which is a quarter of the way to the next-nearest star.

This scenario, minus the nuclear physics, was first proposed in the 18th century, but fell out of favour in the 20th because it seemed not to account for the sun’s low angular momentum (a measure of how much rotation it has, taking into account its mass and speed) compared to that of the planets. But in the 1980s astronomers observed gas clouds round young stars, and mathematical modelling of the collapsing clouds showed plausible, and very dramatic, mechanisms that fitted the observations.

According to these ideas, the early solar system was very different from the sedate one we see today. The planets formed not as single clumps, but by a chaotic process of accretion. For the first 100,000 years, slowly growing “planetesimals” swept up gas and dust, and created circular rings in the nebula by clearing out gaps between them. Each gap was littered with millions of these tiny bodies. At that point the planetesimals ran out of new matter to sweep up, but there were so many of them that they kept bumping into each other. Some broke up, but others merged; the mergers won and planets built up, piece by tiny piece.

Late in 2014 dramatic evidence for this process was found: an image of a proto-planetary disc around the young star HL Tau, 450 light years away in the Taurus
constellation. This image showed concentric bright rings of gas, with dark rings in between. The dark rings are almost cer­tainly caused by nascent planets sweeping up dust and gas.

Until very recently, astronomers thought that once the solar system came into being it was very stable: the planets trundled ponderously along preordained orbits and nothing much changed. No longer: it is now thought that the larger worlds – the gas giants Jupiter and Saturn and the ice giants Uranus and Neptune – first appeared outside the “frost line” where water freezes, but subsequently reorganised each other in a lengthy gravitational tug of war.

In the early solar system, the giants were closer together and millions of planetesimals roamed the outer regions. Today the order of the giants, outwards from the sun, is Jupiter, Saturn, Uranus, Neptune. But in one likely scenario it was originally Jupiter, Neptune, Uranus, Saturn. When the solar system was about 600 million years old, this cosy arrangement came to an end. All of the planets’ orbital periods were slowly changing, and Jupiter and Saturn wandered into a 2:1 resonance – Saturn’s “year” became twice that of Jupiter. Repeated alignments of these two worlds then pushed Neptune and Uranus outwards, with Neptune overtaking Uranus. This disturbed the planetesimals, making them fall towards the sun. Chaos erupted in the solar system as planetesimals played celestial pinball among the planets. The giant planets moved out, and the planetesimals moved in. Eventually the planetesimals took on Jupiter, whose huge mass was decisive. Some were flung out of the solar system altogether, while the rest went into long, thin orbits stretching out to vast distances. After that, it mostly settled down.

These theories are not idle speculation. They are supported by huge computer calculations of the solar system’s dynamics over billions of years, carried out in particular by the research groups of Jack Wisdom of the Massachusetts Institute of Technology and Jacques Laskar of CNRS, the French national centre for scientific research. Some cunning mathematics is required even to set up these simulations: the deep structure of the laws of motion must not be disturbed by the unavoidable numerical approximations that occur. This structure includes the laws of conservation of energy and angular momentum, whose totals cannot change. Amazingly, the planetary migrations not only keep these quantities in balance, but happen because they balance.

Another playground for mathematicians and astronomers investigating Newtonian gravitation is the rings of Saturn. The most distant of the planets known to the ancients, Saturn is about 1.3 billion kilometres from Earth. In 1610, when Galileo looked at Saturn through his telescope, he sent his fellows a Latin anagram: smaismrmilmepoetaleumibunenugttauiras. This was a standard way to preannounce a discovery without giving it away. Kepler deciphered it as reading – in translation – “Be greeted, double knob, offspring of Mars,” and thought Galileo was claiming Mars had two moons (as Kepler had predicted, and rightly so). But Galileo later explained that his anagram actually meant: “I have observed the most distant of planets to have a triple form.” That is, Saturn consists of three bodies.

So much for anagrams.

Galileo’s image of the planet was blurred. Using a better telescope, the Dutch mathematician Christiaan Huygens realised that the middle body was the planet and the others were parts of a gigantic system of rings. Mathematics proves – contrary to an early suggestion by the French scholar Pierre-Simon Laplace – that the rings cannot be solid. In fact, they are made up of ice particles, ranging in size from fine dust to lumps ten metres across. There are several current theories for the rings’ formation: the break-up of a moon, or perhaps leftovers from Saturn’s own primordial nebula. Mathematics is being used to try to find out which explanation, if any, is correct.

Mathematical studies also explain many puzzling features of Saturn’s rings. For one thing, the rings are dense in some regions, but so thin in others that at first sight there seem to be gaps. Some of these gaps come from resonances between the rings and the periods of Saturn’s 62 moons, which can systematically disturb gas in orbits related to that of the moon itself. Other gaps are organised by “shepherd moons” that hustle out any sheepish moonlet that strays into the gap. When the spacecraft Voyager 1 flew past in 1980, some rings appeared to be braided. We now know that they are kinked and lumpy, another subtle consequence of Newtonian gravity in this complex system.

Mathematics has illuminated many other cosmic puzzles: the formation of Earth’s moon, the future of the solar system, the formation and dynamics of galaxies – even the origin of the universe itself in the Big Bang. In ancient India, mathematics was a sub-branch of astronomy. Today, if anything, it is the other way round. Mathematicians are making discoveries and inventing methods; astronomers and cosmologists are making ever greater use of the latest mathematical tools and concepts to advance this utterly fascinating subject. Mathematical thinking teaches us more about humanity’s place in the universe. And it helps us to seek out new places.

Ian Stewart is an emeritus professor of mathematics at the University of Warwick

This article first appeared in the 19 December 2014 issue of the New Statesman, Christmas Issue 2014

Andre Carhillo
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The decline of the Fifth Republic

With the far right and far left surging in the run-up to a defining presidential election, the French seem intent on blowing up the political establishment.

On a cold Saturday evening in late February, cycling back to my flat in southern Paris, I accidentally ran into a pack of lads on a rampage. They were turning over bins, kicking over expensive motorbikes parked on the street, and obviously looking for someone to fight.

It wasn’t the first time that I’d seen this sort of thing, even in this relatively gentrified part of the city. Usually the best course of action is to stop, let them swarm past and allow the police to do their job. But on this particular night, although I could hear the buzz of a police helicopter above us, there were no officers on the ground. As I nervously became aware of this, one of the lads, no more than five yards away, looked at me and screamed: “T’es qui toi?” (“Who the f*** are you?”). His mates turned and gathered round. Now panicking, I saw that he was pointing a screwdriver at me.

I pelted down the street, heart racing as the young men followed me, so shocked that when I reached my apartment building I twice tapped in the wrong entry code. It was only once indoors, now safe but genuinely scared and sweating, that I understood what had happened.

This was a gang from one of the local ­cités – council estates – that border this part of Paris. They had been flushed out of their normal dens, where they deal in weed and mess about, by police using helicopters and unmarked cars, and were now taking their revenge on these unfamiliar surroundings. When they saw me, a tall, white, male figure, watching in the dark on my bike (stupidly the same dark blue as a police bike), they assumed I could only be one thing: a police spotter. In other words, their most hated enemy.

In the past few weeks, in Paris and across France, there has been a new and special danger in being identified by such gangs as a lone policeman. This is because the ever-present tensions between police and the youth of the cités have become particularly acute following the so-called Affaire Théo. On 2 February in Seine-Saint-Denis, north-east of Paris, four police officers violently attacked an innocent black man, identified only as Théo. The assault was caught on camera and allegedly involved the man’s “rape” with a telescopic baton.

The details of the case caused widespread outrage, right up to the highest level of ­government. In the banlieue, the suburbs where many young people feel excluded from mainstream French life, some felt a desire for revenge. And though their anger related to a specific incident, it was in keeping with the emotions sweeping across France, at all levels of society, in the lead-up to the first round of this year’s presidential election on 23 April.

***

France is in a state of political disarray. This much was obvious during the first live “great debate” on 20 March, organised by the television channel TF1, featuring five front-runners for the presidency.

Probably the greatest loser on the night was François Fillon of the centre-right party les Républicains, who served as prime minister from 2007 to 2012. Fillon has gone from being a sure favourite to outsider in the presidential contest, following allegations of dodgy financial dealings. Most damagingly, a formal judicial investigation has been launched into reports that he paid upwards of €800,000 of taxpayers’ money to his wife and other family members for jobs they didn’t actually do. Fillon, who denies any wrongdoing, has also been accused of failing to declare a €50,000 loan from a French businessman in 2013 (which he has since repaid). He held himself in check during the debate, trying to look dignified and presidential, but he has become the object of scorn from all sides, including his own.

Benoît Hamon, the candidate for the Parti Socialiste (PS), the party of the outgoing and discredited president, François Hollande, did not perform much better in the debate. Hamon identifies with the far left and green wings of the PS and favours a basic income, the legalisation of cannabis, and euthanasia. He resigned from Hollande’s government in 2014 claiming that the president had abandoned socialist values. But at every public appearance Hamon still looks surprised to be in the race. Although he has positioned himself as the “anti-Hollande” candidate – no surprise, as Hollande has the lowest polls ratings of any French president – even Hamon’s supporters concede that he has no reach outside the party faithful, and his dismal poll ratings reflect this.

In recent weeks, Jean-Luc Mélenchon, a veteran left-winger and now leader of his own party, France Insoumise (“Unsubmissive France”), has surged in the polls. He has been compared to Jeremy Corbyn but is more like George Galloway, in that he can be trenchant and biting and speaks fluently without notes. Some of his views – anti-EU, anti-Nato, pro-Russia – are close to those of Marine Le Pen, the leader of the far-right Front National (FN). The candidate of the centre or centre-left is Emmanuel Macron, a 39-year-old former investment banker and protégé of Hollande, under whom he served as minister of the economy, industry and digital data. Macron broke with the PS in 2016 to set himself up as an independent candidate with his new movement, En Marche! (“onward”). He presents himself as a voice of moderation and common sense. He defends the EU and the eurozone and is an unashamed liberal globaliser. But Macron is also hard to love: his enemies claim that he is self-serving, an opportunist who cannot be trusted, and, worse, that he lacks experience of high office. On television he can be vain and testy – as was the case when he came under attack from Marine Le Pen, during the TF1 debate.

In many ways, Macron was a gift to Le Pen. She accused him of being out of touch and of not knowing what he was talking about. Even non-FN supporters, who didn’t necessarily agree with her views on security and immigration, conceded that Le Pen was the most convincing speaker. As I was told by a neighbour with an impeccable PS background, it was as if she was the only politician on the night of the debate in charge of what she believed. Le Pen’s popularity increased as a consequence.

So is it now possible to think the unthinkable: that Marine Le Pen could triumph not only in the first round of the presidential election but in the second as well? If that happens, not only would she become the first female president of France but she would transform French politics and further destabilise the European Union.

***

When I put this to Jean-Pierre Legrand, the leader of the Front National in Roubaix, a town of 90,000 inhabitants in the north of France, he shook his head. He wishes Le Pen well but fears that in the second round the mainstream parties will gang up and back whoever her opponent is. “This is what always happens,” he told me. “This is why so-called French democracy is actually a form of dictatorship. You can never really get your hands on power. It belongs to an elite, people like Emmanuel Macron.”

Legrand, 69, has been a supporter of the FN for decades. He smiles a lot and can be witty, but he also likes talking tough, like the hard-headed factory boss he used to be. He admires the way Le Pen has reinvented the party, shedding some of the old-school neo-Nazi trappings. But he is also faithful to, maybe even nostalgic for, the old FN of her father, Jean-Marie Le Pen, who reached the second round of the 2002 presidential election (he lost to the centre-right Jacques Chirac). So I asked him if he was not really a democrat but, like Le Pen père, basically a fascist. “I am not afraid of being called a fascist, or even a Gaullist,” he said. “But all I really believe in is order and authority. And that is what France needs now.”

I had come to Roubaix because it is officially the poorest town in France. It is also, according to most media reports, one of the most troubled. It’s not far from Paris – just over 90 minutes on a fast train – but when you get there it feels like a different, distant place. The train station is scruffy and there is little sense of the usual Gallic civic pride; the stroll down the main boulevard to the Grand Place is drab and quiet, unlike in most French towns.

Roubaix has a large immigrant population, mainly from North Africa but comprising more than 60 nationalities. It has a reputation as a refuge for illegal migrants making for Calais and then the UK, and as a hotbed of Islamist radicalisation. In May last year the conservative news weekly Valeurs actuelles described Roubaix as “le Molenbeek français”. The magazine was referring to the suburb of Brussels where several of the terrorists and sympathisers involved in the November 2015 attacks on Paris, which killed 130 people, including 89 at the Bataclan concert hall, grew up.

Legrand and his FN colleague Astrid Leplat offered to show me around the town, just as they had done with the writer from Valeurs actuelles. The article was criticised by the local newspaper La Voix du Nord as depicting a fantasy version of France conjured up by the FN. I was aware of this argument, but also keen to take up the offer of a tour: it was a rare chance to see an ordinary French town through the eyes of the FN.

I quite liked Roubaix. With its sooty terraced houses, empty textile mills, iron bridges and dirty canals, it reminded me of Salford in the 1970s. The town is neatly laid out even if the streets are scruffy. It is also busy with small businesses – Arabic-language bookshops, kebab houses and tea shops, as well as traditional French cafés and bistros. It looked no more menacing than Bradford or Rusholme in Manchester.

Legrand is proud of Roubaix, or at least of what Roubaix used to be, and has chosen to live here rather than in nearby Lille. Having been a blue-collar worker, too, he admires the noble ambitions and graft of the people who built the town. These were the original indépendants – the aspiring working class, much cherished by the FN, who believe in the values of hard work and public service. But Legrand told me that when he looks at the streets today he sees not the cluttered life of 21st-century, multicultural France but what he called “conquered territory”.

There are problems in Roubaix: 45 per cent of the town’s residents live below the official French poverty line of €977 a month. Describing the local poverty, Legrand used the term “misère”, a word that also translates as “wretchedness”. The unemployment rate is high (40 per cent in parts of town) and on a typical weekday afternoon there are many young men sitting around with nothing to do.

As we drove through some of the tougher areas, Legrand pointed out so-called Salafist mosques, most of them shielded from the streets by the high walls of disused factories. It is these places, unknown and unvisited by outsiders, which have given Roubaix its reputation for radicalism.

It is true that in the recent past Roubaix has produced many extremists. The most notorious is Lionel Dumont, a former soldier who is white and working class, and is viewed as the leader of radical Islam in the French prison system, where he is serving a 25-year sentence for terrorism offences that include trying to set off a car bomb during a G7 meeting in Lille in 1996. Islamists such as Dumont are, in effect, beyond the control of the penal authorities because French laws forbid the monitoring of prisoners on grounds of race or religion. One frustrated director of prisons in the Paris region complained to me that the French penal system was “the real engine room of radicalisation”.

The main reason why Roubaix has produced so many terrorists – including Mehdi Nemmouche, the gunman who fired the shots at the Jewish Museum in Brussels in May 2014 that killed four people – is not immigration, as the Front National would have it, but geography. This part of France is depicted in the media as “a security black hole”, partly because of its proximity to the Belgian border. You can drive into Belgium from Roubaix in ten minutes, as I did with Legrand; the border is just a roundabout and unmonitored. The French and Belgian intelligence services are minutes away from each other but do not share information or collaborate properly. This allowed some of the terrorists who led the 2015 Paris attacks to escape after the killing spree.

***

Crossing the border to Belgium, you notice that the roads are lined with gleaming new warehouses belonging to Amazon and other technology companies. ­Roubaix suddenly seems like a ruin from the early 20th century. It must be difficult for its people not to feel trapped and abandoned – by the French elite to the south and the new economy to the north.

“If you live in Roubaix it is hard to feel connected to the rest of France,” said Hélène Robillard, a junior civil servant. I had come across her in the centre of town. She was leading a group of young women, merrily banging tambourines, blowing whistles and chanting slogans outside one of the
offices of the local council. They were striking against work conditions at the council, but having a laugh, too, in the best Made in Dagenham style.

I asked the women about the film Chez nous (This Is Our Land), which had been released only a few weeks earlier and was playing to packed houses across France. Set in a fictionalised town much like Roubaix, it tells the story of a young woman, Pauline Duhez, a nurse who is seduced into joining the FN and standing for a seat on the council. As she learns the party’s true positions, she becomes disillusioned and angry. The film ends with Pauline returning to the socialist values of her unemployed father, a former steelworker, culminating in a family trip to watch a game featuring the local football team Lens.

The women protesting with Robillard were all determinedly anti-FN. Those who had seen the film were full of enthusiasm. “It is our real life,” said one of them, laughing. “It shows our true values – not fascism, but football, beer and chips.”

Like Pauline in the film, the FN’s Astrid Leplat is a nurse. Jean-Pierre Legrand explained to me that this was why she had been hand-picked by Marine Le Pen to stand
as a regional councillor. The party has adopted a policy of recruiting fonctionnaires (civil servants), especially those who work in the health and support services. This is partly to demonstrate that the FN has left behind its neo-Nazi origins and is now the party of everyday folk, but also to undermine PS dominance of the public services.

When I asked Leplat why she supported the FN, she said that she had witnessed the disastrous effects of repeated budget cuts on hospitals, with overstretched departments and increasingly run-down facilities. “The Front National are there to protect us,” she said.

Leplat told me she hadn’t seen Chez nous and that she probably wouldn’t, because it would upset her. There were also political reasons why she didn’t want to see it: it had been financed with public money from Hauts-de-France, the northern region that covers Roubaix, as well as the television companies France 2 and France 3. When I pointed out that most French cinema relies on public subsidy, she argued that the film’s release had been deliberately timed to undermine the February launch of the FN’s presidential campaign.

“How else can this be explained?” she said. “The Front National is always persecuted by the establishment elites in culture and politics.”

***

Back in Paris, as part of a documentary I was making for BBC Radio 4, I interviewed Émilie Dequenne, the actress who plays Pauline in Chez nous, and the film’s director, Lucas Belvaux. We met at the production company’s office just off the rue du Faubourg Saint-Honoré in the swish heart of Paris – a corner of the city that couldn’t be further removed from the streets of Roubaix. But both Dequenne and Belvaux are intimately connected with the region and the northern working-class life, because they grew up near the Franco-Belgian border and still have family ties there. I asked them whether the FN had a point about the film.

“The film is not ambiguous,” Dequenne said. “It is clearly a warning about being ­seduced by the far right. But it also has lots of [different] ambiguities. The main character, Pauline, is a good person, and not stupid. She wants to help people. She thinks that this is not the case with the main pol­itical parties. So she is attracted by a party that seems to care.”

“I agree it is a warning,” Belvaux said. “We are not yet a fascist country, but I do fear that this could happen.

“There are big social and cultural divisions in France. Not everybody who will vote for the Front National is a bad person, but there are many angry people in this country who feel hurt and damaged. When this is the case, fascism can arrive much more quickly than you think.”

Until now, voting for the FN has been a sign of protest, historically a safety valve for releasing discontent. Whenever the FN has got near to victory, right and left have come together as a bloc to exclude it from power. This is what happened in 2002, of course, when Jean-Marie Le Pen, the then leader of the FN, made it through to the second round of the presidential elections. Jacques Chirac won the run-off with 82 per cent of the vote, despite accusations of corruption. The rallying cry across all non-FN political lines was: “Vote for the crook, not the fascist!” Yet there is no guarantee that this will happen again, because Marine Le Pen has successfully reinvented and rebranded the FN, making it more acceptable to mainstream voters.

Even if Marine loses, there is another danger. If those French parties of the left and right which historically have been strongest continue to implode, there will be a new constituency of voters who in future will be “homeless”. Even if Macron wins – having blurred the lines between right and left – he will disappoint at some stage. When this happens, those who supported him may not find their way back to the established parties, thus opening up an avenue to power for the far right. Sylvain Bourmeau, an associate professor at the École des Hautes Études en Sciences Sociales in Paris, told me that this was part of the Front National’s long-term strategy.

The withering of a historically strong party has already happened in the UK, where voters’ movement to Ukip and the SNP has undermined, if not destroyed, Labour as a national force. Marine Le Pen has already voiced her admiration for Ukip for “breaking the mould”. However, it is important to remember that the FN is not “populist” in the way that Ukip, or indeed Donald Trump, is. Nor are Roubaix and the north of France the same as the “rust belt” of the United States.

Rather, the present conflicts in France are ideological, with roots in the antagonisms and turmoil of French history. The FN’s ultimate goal is to get rid of the present French Republic – the result of the “mistake” of the “liberal revolution” of 1789. In other words, the promise of liberté, égalité, fraternité is to be replaced by an “awakening”, which would lead to a “national movement”: that is, the rebirth of the French nation. The FN is not just about racism, immigration or identity: it wants to send French history into reverse gear.

That is how high the stakes are, and why the coming elections are the most important in France since the Second World War. There is a generalised tension right now – the tension that I encountered on my bike on my own street in southern Paris – which sometimes finds expression in gang violence, anti-police riots and even terrorism, all fuelling the rise of the FN.

For all the polls, signs and omens, it is ­impossible to predict the election result. Whatever happens in the coming weeks and months, with the old political certainties melting away, it seems more than ever that France is set on a long and unstoppable journey into darkness. L

Andrew Hussey is the author of “The French Intifada” (Granta Books). He lives in Paris. His documentary “Culture, Class and Le Pen” will be broadcast on BBC Radio 4 on 24 April (8pm)

This article first appeared in the 20 April 2017 issue of the New Statesman, May's gamble

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