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
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We need to talk about the online radicalisation of young, white women

Alt-right women are less visible than their tiki torch-carrying male counterparts - but they still exist. 

In November 2016, the writer and TED speaker Siyanda Mohutsiwa tweeted a ground-breaking observation. “When we talk about online radicalisation we always talk about Muslims. But the radicalisation of white men online is at astronomical levels,” she wrote, inspiring a series of mainstream articles on the topic (“We need to talk about the online radicalisation of young, white men,” wrote Abi Wilkinson in The Guardian). It is now commonly accepted that online radicalisation is not limited to the work of Isis, which uses social media to spread propaganda and recruit new members. Young, white men frequently form alt-right and neo-Nazi beliefs online.

But this narrative, too, is missing something. When it comes to online radicalisation into extreme right-wing, white supremacist, or racist views, women are far from immune.

“It’s a really slow process to be brainwashed really,” says Alexandra*, a 22-year-old former-racist who adopted extreme views during the United States presidential election of 2016. In particular, she believed white people to be more intelligent than people of colour. “It definitely felt like being indoctrinated into a cult.”

Alexandra was “indoctrinated” on 4Chan, the imageboard site where openly racist views flourish, especially on boards such as /pol/. It is a common misconception that 4Chan is only used by loser, basement-dwelling men. In actuality, 4Chan’s official figures acknowledge 30 percent of its users are female. More women may frequent 4Chan and /pol/ than it first appears, as many do not announce their gender on the site because of its “Tits or GTFO” culture. Even when women do reveal themselves, they are often believed to be men who are lying for attention.

“There are actually a lot of females on 4chan, they just don't really say. Most of the time it just isn't relevant,” says Alexandra. Her experiences on the site are similar to male users who are radicalised by /pol/’s far-right rhetoric. “They sowed the seeds of doubt with memes,” she laughs apprehensively. “Dumb memes and stuff and jokes…

“[Then] I was shown really bullshit studies that stated that some races were inferior to others like… I know now that that’s bogus science, it was bad statistics, but I never bothered to actually look into the truth myself, I just believed what was told to me.”

To be clear, online alt-right radicalisation still skews majority male (and men make up most of the extreme far-right, though women have always played a role in white supremacist movements). The alt-right frequently recruits from misogynistic forums where they prey on sexually-frustrated males and feed them increasingly extreme beliefs. But Alexandra’s story reveals that more women are part of radical right-wing online spaces than might first be apparent.

“You’d think that it would never happen to you, that you would never hold such horrible views," says Alexandra. "But it just happened really slowly and I didn't even notice it until too late."


We are less inclined to talk about radical alt-right and neo-Nazi women because they are less inclined to carry out radical acts. Photographs that emerged from the white nationalist rally in Charlottesville this weekend revealed that it was mostly polo shirt-wearing young, white men picking up tiki torches, shouting racial slurs, and fighting with counter-protestors. The white supremacist and alt-right terror attacks of the last year have also been committed by men, not women. But just because women aren’t as visible doesn’t mean they are not culpable.  

“Even when people are alt-right or sympathisers with Isis, it’s a tiny percentage of people who are willing or eager to die for those reasons and those people typically have significant personal problems and mental health issues, or suicidal motives,” explains Adam Lankford, author of The Myth of Martyrdom: What Really Drives Suicide Bombers, Rampage Shooters, and Other Self-Destructive Killers.

“Both men and women can play a huge role in terms of shaping the radicalised rhetoric that then influences those rare people who commit a crime.”

Prominent alt-right women often publicly admit that their role is more behind-the-scenes. Ayla Stewart runs the blog Wife With a Purpose, where she writes about “white culture” and traditional values. She was scheduled to speak at the Charlottesville “Unite the Right” rally before dropping out due to safety concerns. In a blog post entitled “#Charlottesville May Have Redefined Women’s Roles in the Alt Right”, she writes:

“I’ve decided that the growth of the movement has necessitated that I pick and choose my involvement as a woman more carefully and that I’m more mindful to chose [sic] women’s roles only.”

These roles include public speaking (only when her husband is present), gaining medical skills, and “listening to our men” in order to provide moral support. Stewart declined to be interviewed for this piece.

It is clear, therefore, that alt-right women do not have to carry out violence to be radical or radicalised. In some cases, they are complicit in the violence that does occur. Lankford gives the example of the Camp Chapman attack, committed by a male Jordanian suicide bomber against a CIA base in Afghanistan.

“What the research suggests in that case was the guy who ultimately committed the suicide bombing may have been less radical than his wife,” he explains. “His wife was actually pushing him to be more radical and shaming him for his lack of courage.” 


Just because women are less likely to be violent doesn’t mean they are incapable of it.

Angela King is a former neo-Nazi who went to prison for her part in the armed robbery and assault of a Jewish shop owner. She now runs Life After Hate, a non-profit that aims to help former right-wing extremists. While part of a skinhead gang, it was her job to recruit other women to the cause.

“I was well known for the violence I was willing to inflict on others… often times the men would come up to me and say we don’t want to physically hurt a woman so can you take care of this,” King explains. “When I brought other women in I looked for the same qualities in them that I thought I had in myself.”

King's 1999 mugshot


These traits, King explains, were anger and a previous history of violence. She was 15 when she became involved with neo-Nazis, and explains that struggles with her sexuality and bullying had made her into a violent teenager.

“I was bullied verbally for years. I didn't fit in, I was socially awkward,” she says. One incident in particular stands out. Aged 12, King was physically bullied for the first time.

“I was humiliated in a way that even today I still am humiliated by this experience,” she says. One day, King made the mistake of sitting at a desk that “belonged” to a bully. “She started a fight with me in front of the entire class… I’ve always struggled with weight so I was a little bit pudgy, I had my little training bra on, and during the fight she ripped my shirt open in front of the entire class.

“At that age, having absolutely no self-confidence, I made the decision that if I became the bully, and took her place, I could never be humiliated like that again.”

Angela King, aged 18

King’s story is important because when it comes to online radicalisation, the cliché is that bullied, “loser” men are drawn to these alt-right and neo-Nazi communities. The most prominent women in the far-right (such as Stewart, and Lauren Southern, a YouTuber) are traditionally attractive and successful, with long blonde hair and flashing smiles. In actuality, women that are drawn to the movement online might be struggling, like King, to be socially accepted. This in no way justifies or excuses extreme behaviour, but can go some way to explaining how and why certain young women are radicalised. 

“At the age of 15 I had been bullied, raped. I had started down a negative path you know, experimenting with drugs, drinking, theft. And I was dealing with what I would call an acute identity crisis and essentially I was a very, very angry young woman who was socially awkward who did not feel like I had a place in the world, that I fit in anywhere. And I had no self-confidence or self-esteem. I hated everything about myself.”

King explains that Life After Hate’s research reveals that there are often non-ideological based precursors that lead people to far right groups. “Individuals don’t go to hate groups because they already hate everyone, they go seeking something. They go to fill some type of void in their lives that they’re not getting.”

None of this, of course, excuses the actions and beliefs of far-right extremists, but it does go some way to explaining how “normal” young people can be radicalised online. I ask Alexandra, the former 4Chan racist, if anything else was going on in her life when she was drawn towards extreme beliefs.

“Yes, I was lonely,” she admits.                                                       


That lonely men and women can both be radicalised in the insidious corners of the internet shouldn’t be surprising. For years, Isis has recruited vulnerable young women online, with children as young as 15 becoming "jihadi brides". We have now acknowledged that the cliché of virginal, spotty men being driven to far-right hate excludes the college-educated, clean-cut white men who made up much of the Unite the Right rally last weekend. We now must realise that right-wing women, too, are radicalised online, and they, too, are culpable for radical acts.  

It is often assumed that extremist women are radicalised by their husbands or fathers, which is aided by statements by far-right women themselves. The YouTuber, Southern, for example, once said:  

“Anytime they [the left] talk about the alt-right, they make it sound like it’s just about a bunch of guys in basements. They don’t mention that these guys have wives – supportive wives, who go to these meet-ups and these conferences – who are there – so I think it’s great for right-wing women to show themselves. We are here. You’re wrong.”

Although there is truth in this statement, women don’t have to have far-right husbands, brothers, or fathers in order to be drawn to white supremacist or alt-right movements. Although it doesn’t seem the alt-right are actively preying on young white women the same way they prey on young white men, many women are involved in online spaces that we wrongly assume are male-only. There are other spaces, such as Reddit's r/Hawtschwitz, where neo-Nazi women upload nude and naked selfies, carving a specific space for themselves in the online far-right. 

When we speak of women radicalised by husbands and fathers, we misallocate blame. Alexandra deeply regrets her choices, but she accepts they were her own. “I’m not going to deny that what I did was bad because I have to take responsibility for my actions,” she says.

Alexandra, who was “historically left-wing”, was first drawn to 4Chan when she became frustrated with the “self-righteousness” of the website Tumblr, favoured by liberal teens. Although she frequented the site's board for talking about anime, /a/, not /pol/, she found neo-Nazi and white supremacist beliefs were spread there too. 

“I was just like really fed up with the far left,” she says, “There was a lot of stuff I didn't like, like blaming males for everything.” From this, Alexandra became anti-feminist and this is how she was incrementally exposed to anti-Semitic and racist beliefs. This parallels the story of many radicalised males on 4Chan, who turn to the site from hatred of feminists or indeed, all women. 

 “What I was doing was racist, like I – deep down I didn't really fully believe it in my heart, but the seeds of doubt were sowed again and it was a way to fit in. Like, if you don't regurgitate their opinions exactly they’ll just bully you and run you off.”

King’s life changed in prison, where Jamaican inmates befriended her and she was forced to reassess her worldview. Alexandra now considers herself “basically” free from prejudices, but says trying to rid herself of extreme beliefs is like “detoxing from drugs”. She began questioning 4Chan when she first realised that they genuinely wanted Donald Trump to become president. “I thought that supporting Trump was just a dumb meme on the internet,” she says.

Nowadays, King dedicates her life to helping young people escape from far-right extremism. "Those of us who were involved a few decades ago we did not have this type of technology, cell phones were not the slim white phones we have today, they were giant boxes," she says. "With the younger individuals who contact us who grew up with this technology, we're definitely seeing people who initially stumbled across the violent far-right online and the same holds for men and women.

"Instead of having to be out in public in a giant rally or Klan meeting, individuals find hate online."

* Name has been changed

Amelia Tait is a technology and digital culture writer at the New Statesman.

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