Category Archives: Mathematics

Maths and murals: Leiden’s wall formulae

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One of Einstein's field equations - part of the Leiden wall formulae project.

One of Einstein’s field equations – part of the Leiden wall formulae project.{credit}Ivo van Vulpen and Sense Jan van der Molen. Photograph by Hielco Kuipers.{/credit}

Albert Einstein’s field equations from his theory of general relativity combine wonderful scientific intuition with the honed concision of poetry. Yet relatively few of the culturally inclined marvel at the shape of a mathematical equation in the way they might at a line from Shakespeare. Now, however, the Dutch university town of Leiden is giving its citizens a chance to try, through iconic formulae by physicists and astronomers painted on walls throughout the city.

The formulae join 100-plus murals of poems, painted by artists over more than two decades as a way of highlighting Leiden’s long connection with the arts, not least as Rembrandt’s birthplace. These celebratory artworks, some in delicate Japanese calligraphy, have become part of an urban aesthetic. But the city is also a crucible for discoveries such as superconductivity, by Heike Kamerlingh Onnes, in 1911.

lorentzkracht

The Lorentz force formula.{credit}Ivo van Vulpen and Sense Jan van der Molen. Photograph by Hielco Kuipers.{/credit}

The idea of ‘wall formulae’ arose a few years ago, when physicists Ivo van Vulpen and Sense Jan van der Molen convinced municipal authorities (and house-owners) to embrace the scheme as a way of celebrating science in the city. Dutch artists Jan Willem Bruins and Ben Walenkamp were first in, painting Willebrord Snellius’s law of refraction (Snell’s law), Hendrik Lorentz’s force formula, and Einstein’s field equations. These were unveiled in 2016. Three more – the Oort constants, the Lorentz contraction and electron spin (discovered by Lorentz’s students Samuel Goudsmit and George Uhlenbeck) – are officially unveiled today.

Oort constants.

Oort constants.{credit}Ivo van Vulpen and Sense Jan van der Molen. Photograph by Hielco Kuipers.{/credit}

Van der Molen notes that the equations, like poems, distil realities and are “beautiful to behold and inspiring”. To help convey their meaning to non-mathematicians, the artists add a simple graphical representation of the physical phenomenon described. Thus the Lorentz contraction, which expresses how objects shrink to an observer travelling near speed of light, is illustrated by a circle and a series of ‘squeezed’ ellipses. The Oort constants, which refer to the angular velocity of the Sun around the centre of the Milky Way, are symbolized by a spiral galaxy (with a dot showing the Sun’s position). And to picture Einstein’s field equations – which describe how space is deformed by big objects – we see a ray of starlight’s curved path around a heavy mass, known as gravitational lensing.

By inviting comparison between these and more familiar lines of beauty, Leiden is leading the way in inspiring its citizens about physics and maths on the hoof.

Quirin Schiermeier is a senior reporter for Nature based in Munich.

 The Leiden wall formulae feature on city-centre buildings including the Boerhaave science history museum. Tourists can visit the sites on a leisurely 90-minute walk. Guided tours and an app for smartphones, developed by Leiden physics students involved in a science communication project, will be available by the start of 2018.

 

For Nature’s full coverage of science in culture, visit www.nature.com/news/booksandarts.

Fibonacci’s real mathematical legacy

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Statue of Leonardo Pisano (Fibonacci) in Pisa.

Monument of Leonardo Pisano (Fibonacci) by Giovanni Paganucci (1863) in the Camposanto di Pisa.{credit}Hans-Peter Postel, Wikimedia Commons{/credit}

For hundreds of years until the ebb of the Italian Renaissance, one name was synonymous with arithmetic. This was Leonardo — not the polymath from Vinci, but Leonardo Pisano (ca. 1170-1250), now popularly known as Fibonacci.

Yet we know little of Fibonacci’s life beyond the nickname and his Pisan roots: most details come from a 160-word autobiographical sketch written in 1202. He is often assumed to have discovered the so-called ‘Fibonacci sequence’, which starts with zero and 1 and is thereafter the sum of the two previous numbers (so 1, 2, 3, 5 and so on). The sequence shows up with astonishing frequency in natural spiral structures such as shells and plant tendrils.

Fibonacci did not, however, discover the sequence – it was recorded in Sanskrit at least as far back as 200 BC. Nor does the sequence explain anything about artistic beauty via the so-called ‘golden section’, as Keith Devlin reminds us in his new book Finding Fibonacci. The Pisan’s greatest legacy was to help Europe dump the ancient system of Roman numerals and switch to Hindu-Arabic numbers from 1 to 9 and, perhaps most importantly, 0, which Fibonacci called zephirum after the Arabic ṣifr. (Finding Fibonacci repeats some of Devlin’s arguments in his 2011 The Man of Numbers, and indeed is in large part a meta-narrative exploring the making of that earlier book.)

A page of Fibonacci's Liber Abaci from the Biblioteca Nazionale di Firenze showing (in box on right) the Fibonacci sequence with the position in the sequence labeled in Roman numerals and the value in Hindu-Arabic numerals.

A page of Fibonacci’s Liber Abaci from the Biblioteca Nazionale di Firenze showing (in box on right) the Fibonacci sequence with the position in the sequence labeled in Roman numerals and the value in Hindu-Arabic numerals.{credit}National Library of Florence, Wikimedia Commons.{/credit}

During Fibonacci’s lifetime, much of Italy was part of the Holy Roman Empire, yet many Italian cities were in practice independent city-states. Pisa, Genoa, Amalfi and Venice had been gaining prominence as maritime powers, establishing trade routes across the Mediterranean. As commerce boomed, Italian merchants needed to keep track of finances efficiently. Roman numerals made multiplication and division extremely cumbersome (try dividing MXCI by LIII); they were no match for the 10-digit positional system invented by the Hindus some time before 700 AD and common in the Arab world. And compared to using, say, an abacus, calculations in Hindu-Arabic numbers also allowed an “audit trail”, as Devlin points out: “An individual sitting in Pisa controlling a network of traders needed to be able to review the financial books on a regular basis.”

To fill that need, in 1202 Fibonacci (the son of a notary working for Pisan traders) published Liber Abaci, a compendium of Hindu-Arabic arithmetic and its practical applications to trade. The 600-page book introduces the numerals and explains how to use them for basic calculations. Like every good maths textbook, it also features many practical problems, such as how to convert currencies (Italy alone had 28 at the time, Devlin notes), or puzzles such as this:

It is proposed that 7 rolls of pepper are worth 4 bezants and 9
pounds of saffron are worth 11 bezants, and it is sought how
much saffron will be had for 23 rolls of pepper.

Such problems may seem trivial to someone trained in modern elementary-school algebra, but the symbolic notation for equations with x’s and y’s had not yet been invented at the time, so all solutions had to be spelled out in words. As mathematician John Hannah wrote in his 2011 review of The Man of Numbers,“It is awe-inspiring to see how far medieval mathematicians could progress using such primitive tools.”

Liber Abaci was published in Latin, as was the norm for learned texts. But soon, ‘popular arithmetic’ books in local vernacular, many citing Fibonacci as their source, began to appear. These ‘abacus books’ became standard in schools; at least 600 were written over the next few centuries. Through these texts Italy, and later Europe, learned to do maths.

In Finding Fibonacci Devlin tells us (22 times) that Liber Abaci “changed the world”, comparing the medieval mathematician to tech giant Steve Jobs. He even contends that the book made Western science and technology possible. But although Liber Abaci seems to predate the vernacular abacus books, did it actually inspire them?

Devlin points out that Fibonacci had also written a shorter, simpler abacus book in the vernacular, intended for merchants. That is now generally considered to be lost. If this book could be found, he argues, it might turn out to be the “missing link” between Liber Abaci and the spread of popularized arithmetic texts that came later.

Medieval whodunit

In 2003, historian of mathematics Raffaella Franci discovered such a vernacular text, Livero de l’abbecho, from the late 1200s. Devlin centres both his books on the assertion that Franci concluded that this text was a copy of Fibonacci’s lost book; Devlin avers that it is a “slavish” copy.  He states that thanks to Franci and subsequent studies by other researchers, “we can now say with historical certainty” that Livero de l’abbecho is indeed Fibonacci’s missing link.

But is this as certain as Devlin claims? Franci wrote to me: “I do not believe and I have never claimed that Livero de l’abbecho should be attributed to Leonardo Pisano.” She found evidence that Livero de l’abbecho was based on Fibonacci’s lost book — not that it was a word-for-word copy. Another historian of mathematics, Elisabetta Ulivi, adds that Livero cannot be an exact copy as it’s written in an Umbrian dialect, not Fibonacci’s Tuscan. And historian Jens Høyrup even disputes the importance of Livero and Fibonacci to the importation of Hindu-Arabic arithmetic.

Devlin emailed me that Livero “can be taken to be a fairly close copy” (in Finding Fibonacci he describes it as “a medieval equivalent of a photocopy”) of Leonardo’s lost book. “My duty as a writer of history is not to list the ‘facts’,” he added. “It is to present the best account I can.” Devlin did not respond to follow-up questions about why, in both his books, he describes his attribution of Livero to Fibonacci as “Franci’s conclusion”.

Still, Finding Fibonacci showcases Devlin’s writerly flair. My favourite passages are the incredible story of how Liber Abaci (or at least, the edition he wrote in 1228, the sole surviving one) became available in English for the first time – to this day the only modern-language translation. Mathematician Laurence Sigler had made it his mission to translate the book, rushing to complete the task right before he died of lymphocytic leukemia in 1997. But his editor moved on, and the manuscript languished on floppy disks for years. For a while Sigler’s widow Judith Sigler Fell, fearing the project would be killed, took the extraordinary step of impersonating her husband in communiqués.

By the time Fell found a new publisher, Springer Verlag (now part of the same publisher as Nature), floppy disks had been superseded and she had to hire a hacker to extract the files. Fell then discovered that Springer only accepted submissions in TEX format, the technical standard for physics and mathematics texts. She learned it and spent six months retyping the text. Fibonacci’s Liber Abaci was finally published in 2002 — the 800th anniversary of the book’s first appearance.

Davide Castelvecchi is senior physical sciences reporter at Nature. He tweets at @dcastelvecchi.

 

For Nature’s full coverage of science in culture, visit www.nature.com/news/booksandarts.

Hidden Figures: the movie

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Taraji P. Henson as NASA ‘human computer’ Katherine Johnson. Over the course of her career, Johnson calculated the trajectories and launch windows for flights including the early missions of John Glenn and the Apollo 11 flight to the Moon, and did early work on the Mars mission.{credit}Courtesy of Twentieth Century Fox{/credit}

High-profile protests dominated the media during the civil rights era in 1960s America. At NASA, a quieter struggle was already underway. From the 1940s, African-American women had been chipping away at perceptions and making incursions into the early space programme — that otherwise very white, male world.

The stories of three of these scientific whizzes – Dorothy Vaughan, Katherine Johnson and Mary Jackson – are now told in Hidden Figures, a film directed by Theodore Melfi and based on a book of the same name by Margot Lee Shetterly (reviewed here for Nature by Alexandra Witze).

This sharp, witty triple biopic captures the focused frenzy of the United States’ space race with the Soviet Union, when NASA was trying to figure out how to achieve the remarkable feat of launching a man into orbit atop a rocket and returning him safely. That all-out effort meant opening the doors to the best people — which in turn created an opportunity for these pioneering African-American women to take on roles previously barred to them.

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The Langley band of ‘human computers’ led by Dorothy Vaughan (played by Octavia Spencer).{credit}Courtesy of Twentieth Century Fox{/credit}

The movie recreates NASA’s Langley Research Center in Virginia, a state that in the early 1960s remained segregated. Vaughan, Johnson and Jackson are among Langley’s human “computers”: women hired to do the mathematics behind space flight, in the days just before the room-sized first IBM machine did it for them. This smart, passionate band, who made up the West Computing group, spend their days calculating launch and landing trajectories and air flow around capsules, armed only with pencils and reams of paper.

The trio were truly extraordinary. Vaughan, played by Academy Award-winner Octavia Spencer, is the matriarch. Although head of the computing group, she is not initially recognised as such for racist reasons. The film shows her initiative over the years in becoming an expert programmer of computing machines as the march of technology sees electronic counterparts to human computers emerge. Meanwhile Jackson, played with spirit by singer Janelle Monáe, wants to be an engineer. She struggles to reach ever-moving goalposts, including segregation laws that prevent her from attending the only school where she could get the necessary qualifications. Monáe’s vivacity earns her most of the film’s best lines.

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Octavia Spencer as ‘human computer’ supervisor Dorothy Vaughan.{credit}Courtesy of Twentieth Century Fox{/credit}

The main focus in on Johnson, perhaps the most remarkable of the three. Her work stands at the very heart of US success in space. The film opens with her as a child prodigy, then zips past degrees in mathematics and French, and graduate school at West Virginia University — where she was one of the first black students to attend. At NASA she was soon picked to join the Space Task Force, who needed her talents in calculating the geometries of parabolic and, later, orbital flight. So indispensable was she that astronaut John Glenn asked for her to personally check the calculations of his trajectory by hand, ahead of the first US orbital flight in 1962.

Johnson is portrayed by Taraji P. Henson as quiet, tenacious and warm-hearted. The character could not be more different from Henson’s role as gangster Cookie Lyon in the music-industry television drama Empire. Johnson is a whizz with the chalk, often seen up a ladder scrawling calculations on a giant blackboard. She carves out her own position in the team, and in colourful outfits and heels offers a human face as often the only woman in a sea of white-shirted, pencil-tied men. (Among many excellent supporting actors, such as The Big Bang Theory’s Jim Parsons, Kevin Costner as a fictional amalgamation of several real NASA leaders deserves special mention. Gum-chewing and hard-nosed, he insists on referring to his team as “gentlemen” despite Johnson’s presence; but his desire to reach the heavens is what gives her her chance.)

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Janelle Monáe as Mary Jackson, who later became a NASA engineer.{credit}Courtesy of Twentieth Century Fox{/credit}

Hidden Figures succeeds in revealing the institutionalised racism faced by the women and their families. Bathrooms, drinking fountains, schools, libraries — all were segregated. One of the best exchanges is between Vaughan and computing pool supervisor Vivian Mitchell (Kirsten Dunst), who insists, “I have nothing against y’all”. To this, Vaughan kindly replies: “I know you probably believe that.” And the women’s status as invisible engines driving the space programme contrasts clearly with the pomp surrounding the astronauts, who as the faces of NASA seem constantly showered with red, white and blue confetti.

Yet the upbeat film can sometimes come across as sanitised. There are no real baddies: even the racist characters, flawed with conscious or unconscious bias, seem ultimately good. A touch more anger wouldn’t have detracted from the enjoyable feel-goodness, epitomised by a bouncing soundtrack  by co-producer Pharrell Williams (composer of mega-hit Happy).

On another level, this may be an effort to avoid the film being solely about race. Rather, it is about women and their love of science. Vaughan, Johnson and Jackson had families to support and could not risk everything in the political fight for equality. In chasing their passions, these three chose to foment change from the inside. Hidden Figures fleshes its characters out into real human beings, and tells their cracking story with grace.

Elizabeth Gibney is a reporter on physics for Nature based in London. She tweets at @LizzieGibney. Hidden Figures’ US premiere is 25 December 2016; general release is on 6 January. The film’s UK premiere is 10 February 2017; general release is on 17 February.  

 

For Nature’s full coverage of science in culture, visit www.nature.com/news/booksandarts.

Top 20 books: a year that made waves

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beach-1836366_960_720This was a year that made waves — some so steep that I found myself reaching for a psychological surfboard. I skimmed along the discovery of gravitational waves (featured in Janna Levin’s Black Hole Blues and Other Songs of Outer Space), and rode the CRISPR tsunami. The political turbulence stateside, in Britain and beyond had me scrabbling for balance — and historical precedents. Yet amid all the Sturm und Drang, it has been a terrific year for science and culture.

In Nature’s first sci-fi special, we celebrated two anniversaries that stand as reminders of profound — and much-needed — humanistic vision. One was the 150th of the birth of H.G. Wells, ‘Shakespeare of science fiction’, prolific author and frequent Nature contributor; the other, the 50th of Gene Roddenberry’s pioneering franchise Star Trek. And as ever I was able to trace bright currents in the bookish deeps.

Oncologist and writer Siddhartha Mukherjee plunged into the genetics riptide with The Gene — fortuitously, in a year when Richard Dawkins’s name-making classic The Selfish Gene hit 40 and a pod of genome-editing studies surfaced. There was a glut of big physics, notably Roger Penrose’s trenchant Fashion, Faith, and Fantasy in the New Physics of the Universe. And forests, earthquakes, biomechanics and military technology were all ‘trending’. But in trawling hundreds of books for my top 20, one of the more astonishing confluences was in the history of women in science — specifically, the ‘computers’ or number-crunchers behind key astronomical discoveries and space missions. (I’ve cheated here by counting three books on this phenomenon as one — as they are both important self-contained stories and part of a great historical trajectory.) The rest are pretty wonderful too. Enjoy.

The Glass Universe: How the Ladies of the Harvard Observatory Took the Measure of the Stars, Dava Sobel. Viking. The science writer traces the stories of pioneering women ‘computers’ who, from the late nineteenth century, made astronomical history at Harvard College Observatory. (Reviewed here.) 
Hidden Figures: The American Dream and the Untold Story of the Black Women Mathematicians Who Helped Win the Space Race, Margot Lee Shetterly. William Morrow. A historian extols the brilliant African-American women mathematicians at NASA’s Langley Research Center who helped propel postwar America to the Moon and beyond. (Reviewed here.)
Rise of the Rocket Girls: The Women Who Propelled Us, from Missiles to the Moon to Mars, Nathalia Holt. Little, Brown. The HIV researcher on the women at NASA’s Jet Propulsion Lab who from the 1940s number-crunched in near-secrecy to launch missiles and the first US satellite. (Reviewed here.)

Lab Girl, Hope Jahren. Knopf. A palaeobiologist reveals the joy (and strangeness) of field and lab life through the lens of a woman in science. (Reviewed here.)

Half-Earth: Our Planet’s Fight for Life, Edward O. Wilson. Liveright. The eminent biologist issues a compelling call to commit half the planet to the rest of nature. (Reviewed here.)

Reality Is Not What It Seems, Carlo Rovelli. Allen Lane. The theoretical physicist invites us to gaze through a window at a world where space is granular and time does not exist. (Reviewed here.)

The Serengeti Rules: The Quest to Discover How Life Works and Why It Matters, Sean B. Carroll. Princeton University Press. An evolutionary biologist distils a vast body of biological research into six rules of regulation for the restoration of ecosystems and management of the biosphere. (Reviewed here.)

The Shock of the Anthropocene: The Earth, History and Us, Christophe Bonneuil and Jean-Baptiste Fressoz, trans. David Fernbach. Verso. Two historians dig into technological history, economics and climate science to reveal the role of imperialist ideology in today’s planetary crises. (Reviewed here.)

Serendipity: An Ecologist’s Quest to Understand Nature, James A. Estes. University of California Press. An innovative ecologist unpacks his life’s work tracing the top-down control of ecosystems by sea otters as apex predators. (Reviewed here.)

The Life Project: The Extraordinary Story of Our Ordinary Lives, Helen Pearson. Allen Lane. The Nature editor unravels the 70-year history of the British cohort studies and the crucial insights they offer on socioeconomic inequities. (Reviewed here.)

White Trash: The 400-Year Untold History of Class in America, Nancy Isenberg. Viking.  A historian delivers a searing indictment of the US political forces that persistently marginalise poor whites. (Reviewed here.)

Imbeciles: The Supreme Court, American Eugenics, and the Sterilization of Carrie Buck, Adam Cohen. Penguin. The award-winning writer revisits Buck vs Bell, the notorious 1920s case highlighting the dark history of US eugenics. (Reviewed here.)

Weapons of Math Destruction, Cathy O’Neil. Crown. A data scientist and former Wall Street quant uncovers the biases in the algorithmic overlords that micromanage the US economy. (Reviewed here.)

Bellevue: Three Centuries of Medicine and Mayhem at America’s Most Storied Hospital, David Oshinsky. Doubleday. The historian surveys the key advances and bold open-door policy that have made the New York public hospital a medical beacon. (Reviewed here.)

The Cyber Effect, Mary Aiken. John Murray. A forensic cyberpsychologist examines the mental lures built into sociotechnology and their impact on individuals and society. (Reviewed here.)

The Most Wanted Man in China: My Journey from Scientist to Enemy of the State, Fang Lizhi, trans. Perry Link. Henry Holt. The late astrophysicist and dissident on the scientific passion and quest for freedom of expression that drove his extraordinary life. (Reviewed here.)

Marconi: The Man Who Networked the World, Marc Raboy. Oxford University Press.  The communications scholar investigates the complexities of a giant of technology devoted to both science and fascism. (Reviewed here.)

How Not to Network a Nation: The Uneasy History of the Soviet Internet, Benjamin Peters. MIT Press. A communications specialist plumbs the messy and engrossing history of a Soviet technological failure on the grand scale. (Reviewed here.)

The Great Departure: Mass Migration from Eastern Europe and the Making of the Free World, Tara Zahra. W.W. Norton. An accomplished historian busts myths and adds nuance to the story of the 58 million Europeans who poured into the Americas from 1846 to 1940. (Reviewed here.)

Fine Lines: Vladimir Nabokov’s Scientific Art, edited by Stephen H. Blackwell and Kurt Johnson. Yale University Press. In this collection, a Russian scholar and entomologist trace the novelist’s significant contribution to lepidoptery and how that played out through his fiction. (Reviewed here.)

Bone Rooms: From Scientific Racism to Human Prehistory in Museums, Samuel J. Redman. Harvard University Press. A historian harks back to the nineteenth-century ‘skull wars’ and after, which packed US museums with human remains and fired ethical debates that still burn. (Reviewed here.)

The Euro: How a Common Currency Threatens the Future of Europe, Joseph E. Stiglitz. W.W. Norton. The Nobel laureate and economist analyses the failures of eurozone policymakers and the shape radical reform might take. (Reviewed here.)

Listen to my Nature Podcast interview on the top 20 books with Scientific American’s Steve Mirsky here.

 

For Nature’s full coverage of science in culture, visit www.nature.com/news/booksandarts.

Breaking barriers: the US space programme’s black women mathematicians

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Mathematician Katherine Johnson at NASA's Langley Research Center, where she worked as a "computer" from 1953 to 1986.

Katherine Johnson at NASA’s Langley Research Center, where she worked as a “computer” and mathematician from 1953 to 1986. Her illustrious career included calculating Apollo 11’s trajectory on its flight to the Moon. {credit}NASA{/credit}

Some of the most intriguing stories in the history of US science have emerged over the past few years. It’s about time. These books centre on something long under wraps: the centrally important roles women played starting some 70 years ago in the great technological transition that gripped the twentieth century. Denise Kiernan’s The Girls of Atomic City (Touchstone, 2013) chronicled the contributions of the women who worked at the secret atomic-bomb laboratory in Oak Ridge, Tennessee, during the Second World War. Rise of the Rocket Girls by Nathalia Holt (reviewed here) depicted the mathematicians or “human computers” who crunched numbers at the Jet Propulsion Laboratory in southern California from the 1940s. In this catalogue, Margot Lee Shetterly’s Hidden Figures is more than just another entry.

Shetterly’s book is an exploration of the groundbreaking achievements and shocking discrimination experienced by a group of talented mathematicians in all aspects of their professional and personal lives. These African-American women — Dorothy Vaughan, Katherine Johnson, Mary Jackson and Christine Darden among them — began working from the early 1940s at the Langley Research Center in Hampton, Virginia, then the nation’s premier aeronautical laboratory. That wartime breakthrough was to propel many of them into long and successful careers at the heart of the space race. (A feature film based on the book and starring Taraji P. Henson will be released in January.)

Christine Darden in the control room of Langley's Unitary Plan Wind Tunnel in 1975. Darden became an expert on sonic booms and supersonic flights.

Christine Darden in the control room of Langley’s Unitary Plan Wind Tunnel in 1975. Darden became an expert on sonic booms and supersonic flights.{credit}NASA{/credit}

These stellar scientists broke major political and social barriers. Virginia in the American South was a segregated state. Beginning after the Civil War and lasting until the civil-rights era of the 1960s, “Jim Crow” laws enforced a rigid racial hierarchy. Shops, restaurants, public transportation — all viciously discriminated against African-Americans in matters as basic as where to use the toilet.

The mathematicians whose experience Shetterly unveils came of age in this reality. Members of a thriving African-American middle class, they went to universities such as Howard in Washington DC — historically black institutions where they were taught by eminent faculty trained at universities such as Harvard, who could not secure a position there because of their race. These accomplished young women became teachers, then generally the sole career option for educated black women. (Postgraduate education was not even possible in some states; rather than admit African-American students to its state university for graduate studies, between 1936 and 1950 Virginia paid them “scholarships” to attend graduate school elsewhere.)

Top flight

But after America entered the war in 1941, new professional opportunities opened. Langley, where engineers designed and tested technological advances that permitted US planes to fly higher and faster, needed an awful lot of number-crunchers to calculate, say, the ideal air flow over an aeroplane wing. That crushing demand opened the gates to women. Female computers began working through calculations that kept Boeing’s B-29 Superfortress bomber  aloft and the North American Aviation P-51 Mustang fighter manoeuvering through the skies.

Even here, however, segregation persisted. Vaughan and her colleagues were placed in Langley’s ‘West Computing’ unit. White women computed on the east side. At the back of the Langley cafeteria, a white cardboard sign labeled COLORED COMPUTERS directed the West mathematicians to sit together at lunch rather than mingle. Eventually, “tiny firebrand” Miriam Mann stole the sign, and the table was left unlabelled.

Margot Lee Shetterly.

Margot Lee Shetterly.{credit}Aran Shetterly{/credit}

Shetterly, who grew up in Hampton, illuminates this remarkable group’s professional careers and personal travails. Simply getting housing as a black woman was fraught with difficulty in these decades. It was only by harnessing the strong social networks of the African-American middle class that these mathematicians finally got a toehold in the American dream. Shared work experiences bound the group outside Langley: Vaughan and Mann brought their families together for local activities including a phenomenal performance in Hampton by iconic African-American singer Marian Anderson.

Postwar, the future was unclear, Shetterly shows. Would women be pushed out of the workforce? The Soviet Union’s launch of Sputnik in 1957 changed all that. In 1958, the Langley lab became part of the newly formed NASA and the centre of Project Mercury, the programme for crewed space travel. The West computers scattered to other divisions to begin work on the complex calculations of getting spacecraft into orbit.

HiddenFigures_HC HiResIn 1959 Johnson and her colleague Ted Skopinski first calculated the mathematics of firing a capsule into ballistic flight. The equations described the flight of a spacecraft, from the angle of launch, to point of re-entry, to the effect of Earth’s rotation. Their work underlay the successful 1961 suborbital flight of astronaut Alan Shepard. The following year, when John Glenn was about to make the first US orbital flight, he personally requested Johnson to double-check, by hand, the calculations of his trajectory. Johnson went on to an illustrious career in the US space programme. Her mathematics dictated the trajectory of the Apollo 11 flight to the moon, Neil Armstrong and Buzz Aldrin’s descent to the surface, and their tricky rendezvous with the command module in lunar orbit in order to make it safely home. Later, she worked on the space shuttle programme. In November 2015, she received the Presidential Medal of Freedom, the nation’s highest civilian honor, from President Barack Obama.

Vaughan, who in the 1950s had served as Langley’s first black supervisor, forged a successful career in computer programming. Jackson achieved the rank of engineer, then turned her attention to helping other women and minorities into high-level positions. Darden, one of the next generation to benefit from the barriers broken by this group, became a world expert on sonic booms and supersonic flight.

Hidden Figures is not the definitive history of women in the space programme, nor of women at Langley. It does not need to be. It lies at the intersection of the greatest scientific advances and the greatest civil-rights battles in US history.

Alexandra Witze is a correspondent for Nature based in Boulder, Colorado. Her email is witzescience@gmail.com and she tweets at @alexwitze.

 

For Nature’s full coverage of science in culture, visit www.nature.com/news/booksandarts.

The equations of love

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Gustav Klimt, The Kiss, 1907-8

The Kiss (Lovers) by Gustav Klimt, 1908.{credit}Österreichische Galerie Belvedere/Google Cultural Institute{/credit}.

Few topics are as disparate as mathematics and love — or are they? Modeling Love Dynamics (World Scientific, 2016) by systems theorist Sergio Rinaldi and others playfully, but convincingly, makes the point that even amorous relationships cannot escape the fundamental laws of dynamical systems.

The argument propounded by Rinaldi and colleagues builds on the classical framework of coupled differential equations, which have proven so powerful in describing the essence of relationships in nature such as competition, cooperation and predation. The book’s cover illustration hints at the road ahead: it shows Gustav Klimt’s 1908 painting The Kiss (Lovers). A glance inside reveals that art is an essential part of the analysis of the drama of passion — a drama resulting in large part from the interplay of two strong forces, attraction and repulsion. Simple equations illustrated with elegant diagrams show how, depending on personalities, those forces can result in a transient affair, long-lasting stable equilibrium, or everlasting cycles of attraction and repulsion.

Portrait of Francesco Petrarca (Petrarch).

Portrait of Petrarch, whose Canzoniere can be linked to the  limit cycle.{credit}Via Wikimedia Commons{/credit}

Miniature of Laure de Novis, Petrarch's platonic love, 1463.

Miniature of Laure de Noves, who may have been Petrarch’s ‘Laura’.{credit}Laurentian Library, Florence, 1463{/credit}

The tales and poems chosen masterfully illustrate a range of mathematical features. The limit cycle, known for driving the oscillating dynamics of many economic or biological systems, is linked, for instance, to one of the greatest love stories in Western culture. That is, the cyclical 21-year platonic relationship between fourteenth-century Italian humanist and poet Francesco Petrarca (Petrarch) and the married Laura (possibly the Provençal noblewoman Laure de Noves), charted in Petrarch’s celebrated collection Il Canzoniere.

If three variables are mixed in the differential equations of passion, chaotic dynamics can arise. This is illustrated vividly in Henry-Pierre Roché’s semi-autobiographical 1953 novel Jules et Jim (which inspired François Truffaut’s 1962 film of the same name). Roché documents the love triangle between himself, the brilliant and charming journalist Helen Grund and her shy husband Franz Hessel, his best friend. As with the weather, the course of these dynamics is fundamentally unpredictable in the long run, as the smallest event can put things on a different trajectory. This phenomenon is also known as ‘the butterfly effect’, hinging on the idea that the flap of a butterfly’s wing may eventually lead to a hurricane in a distant place.

Other aspects of relationship dynamics generated by the models are illustrated by a range of classics. The complex tides of emotion between Rhett, Scarlett, Ashley and Melanie in Margaret Mitchell’s 1936 blockbuster Gone with the Wind (and the 1939 film directed by Victor Fleming) are tied to the mathematics of alternative basins of attraction. Gabrielle-Suzanne Barbot de Villeneuve’s eighteenth-century fairytale La Belle et La Bête (Beauty and the Beast) exemplifies a saddle-node bifurcation (often referred to as a tipping point) in Beauty’s slow, barely perceptible progress towards the transition from repulsion to attraction — a pattern also seen in the evolving relationship of Elizabeth Bennett and Darcy in Jane Austen’s 1813 novel Pride and Prejudice. The love triangles in Edmond Rostand’s 1897 play Cyrano de Bergerac meanwhile illustrate how temporary bluffing of one partner can sometimes make the difference needed to move the dynamical system from indifference to attraction for a stable love relationship.

When it comes to making mathematics easy, the book saves the best for last. The 40-page appendix is a complete primer on dynamical systems and their bifurcations. Starting with an example of a love model, a simple, lucid text illustrated by elegant drawings explains everything you always wanted to know but never dared to ask about attractors, repellors, saddles, torusses, strange attractors, tipping points and more.

Scientists and artists alike try to capture the essence of things, whether that is atomic structure, the psychological depths of a fictional character or the crystallization of emotion in music. Perhaps this is why the swirling dance between these two seemingly opposite endeavours works out surprisingly well in Modelling Love Dynamics. Clearly, the arts are superior when it comes to capturing the depths of love. Yet disarmingly easy maths powerfully captures the underlying drivers of stable alliances and transient dalliances. What topic is better suited to seduce a broad audience to play with equations?

​Marten Scheffer​’s research focuses on complex systems and their adaptability. He is an ecologist and mathematical biologist at Wageningen University and Research Centre and is founder-director of the Synergy Program for Analysing Resilience and Critical Transitions (SparcS). His latest book is Critical Transitions in Nature and Society (Princeton University Press, 2009.)  He is also a multi-instrumentalist and composer.
Marten.Scheffer@wur.nl

 

For Nature’s full coverage of science in culture, visit www.nature.com/news/booksandarts.

The making of Alice

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Alice and the White Rabbit.

John Tenniel’s illustration of Alice and the White Rabbit in Alice’s Adventures in Wonderland.{credit}© Macmillan Publishers Ltd{/credit}

On 19 October 1863 an unknown mathematician, Charles L. Dodgson, was introduced to the publisher Alexander Macmillan in Oxford by Thomas Combe, director of the Clarendon Press and printer to Oxford University. Macmillan’s publishing business, established with his brother in 1843, was growing. He had built a reputation among scholars and authors as a leading academic publisher in fields such as mathematics and geology.

Dodgson, a master and tutor at Christ Church, Oxford, was friends with the Christ Church dean Henry George Liddell. On 4 July 1862 Dodgson, his friend the Reverend Robinson Duckworth and Liddell’s eldest daughters Lorina, Alice and Edith, rowed up the Thames to Godstow — the “golden afternoon” when Dodgson responded to Alice’s pleas for a story with the rudiments of what would become Alice’s Adventures in Wonderland. The first version was a handwritten and self-illustrated manuscript, Alice’s Adventures Underground, which Dodgson presented to Alice in 1864. The final book would be published by Macmillan under Dodgson’s pen name, Lewis Carroll.

Carroll insisted on a bold red cover for Alice’s Adventures.{credit}© Macmillan Publishers Ltd{/credit}

Macmillan had published Charles Kingsley’s children’s novel The Water Babies in 1863 to contemporary acclaim. Realising the potential of Carroll’s tale, he agreed to take it on a commission basis: Carroll paid for the printing and marketing, while Macmillan was paid a set commission on sales. Macmillan went on to publish all of Carroll’s books, as well as many of his works written under his own name, on mathematics, geometry and logic (as well as Lawn Tennis Tournaments: The True Method of Assigning Prizes, with a Proof of the Fallacy of the Present Method).

So, intertwined with the story of Carroll’s intrepid heroine Alice is the story of another remarkable journey — the long alliance of a brilliant author and enabling publisher who together created a world tale. Alice’s Adventures in Wonderland has never been out of print with Macmillan since 1865, and has been translated into more than 170 languages.

A deep bond

Carroll and Macmillan had a rare mutual respect, love of literature, and interest in education, new technologies, innovation and scientific enquiry. (A rich source for the depth of this relationship is the volumes of their outgoing correspondence in the Macmillan archive at the British Library.) The writer did not attend Macmillan’s famous “Tobacco Parliaments”, where a scientific magazine, which became Nature, was mooted. He preferred to keep his identity secret, but regularly visited Macmillan and his family and sent them puzzle books to try out before publication.

Chess diagram from Through the Looking-Glass, with the Kings in place (Macmillan, 1871).

Chess diagram from Through the Looking-Glass, with the Kings in place (Macmillan, 1871).{credit}© Macmillan Publishers Ltd.{/credit}

The relationship between the two men was not without some tension, however. Carroll took a great interest in the printing, design and production of his books, discussing all aspects of the process with Macmillan. His eye for beauty, order and perfection and his expertise in the then intricate, difficult technology of photography drew him to such technicalities. This could backfire. He frequently delayed publication because he was unhappy with the quality of production (or instructing the printers to let the paper dry for long enough before binding). In 1878, he insisted that slips should be inserted into copies of Through the Looking-Glass, and What Alice Found There after observing that both the Kings had vanished from the chess-diagram in the front of the book.

Carroll’s meticulous instructions extended to securing parcels: a diagram showing how the string was to be knotted hung in the Macmillan post-room for many years. His careful analysis of accounts for his books, in particular the Alice volumes, caused him to question booksellers’ profits – a concern shared by Alexander and later addressed by his nephew, Frederick Macmillan, leading to the Net Book Agreement of 1899.

A technological tale

At the start of Alice’s Adventures, Alice wonders, “what is the use of a book…without pictures or conversations?” Up to this time, children’s books were sparsely illustrated. The creative Carroll — ever interested in visual impact, particularly on his young readers — realised their importance in storytelling, however, and asked artist John Tenniel, cartoonist at the satirical magazine Punch, to illustrate the book.

These classic drawings have become as well known as the story. Beautifully capturing Alice and the characters in Wonderland and the looking-glass world, they are cleverly incorporated into the text through innovative positioning on the page.

An electrotype of Tenniel's illustration of Alice and the Cheshire Cat in Alice's Adventures, and the printed result.

An electrotype of Tenniel’s illustration of Alice and the Cheshire Cat in Alice’s Adventures, and the printed result.{credit}© Macmillan Publishers Ltd{/credit}

Tenniel first made pencil drawings, then created a tracing from which the main features were transferred in reverse to a woodblock; the drawing was finished on the block, which was then sent to be engraved by the Dalziel brothers, George and Edward. The leading Victorian commercial wood engravers, the Dalziels also worked with artistic luminaries of the day, including pre-Raphaelites John Everett Millais and Edward Burne-Jones. Printing was not done from the blocks; more durable copper electrotypes (electros) were cast to save the wood.

Macmillan suggested their use — ultimately, sound advice from one who could not have foreseen how many copies would eventually be printed. The electros wore out after several thousand printings, after which they were melted down and recast.

The 'Mouse's Tale' in Alice's Adventures was too tricky to set in type, so was treated like an illustration.

The ‘Mouse’s Tale’ in Alice’s Adventures was too tricky to set in type, so was treated like an illustration.{credit}© Macmillan Publishers Ltd.{/credit}

A set of electros was held by the printer Richard Clay, who continued to print from them until the introduction of letterpress in the 1960s. A spare set is still held in the Macmillan archive, including an electro for the ‘Mouse’s Tale’. This section of text had to be treated like an illustration; it was too tricky to set, being narrow and serpentine like a tail. (For more on this, see The Complete Alice.)

After Carroll’s death, the woodblocks were handed over by his estate to Macmillan. In 1932, they were displayed at the Lewis Carroll Centenary exhibition in London, after which they were thought to have been moved to a museum or library.

However, in October 1984 Macmillan’s company secretary was called to the National Westminster Bank to open several metal trunks that had lain in its Covent Garden vault for years. To his amazement, he found the woodblocks, stored there in almost perfect conditions.

It was decided to take one unique printing from the blocks, which had never been printed from directly. This was skilfully done by the Rocket Press — 92 prints in a limited edition of 250 copies, together with a specially commissioned book on the engravings. It is copies of these prints that have been scanned to create the images for the 150th anniversary editions published by Pan Macmillan in 2015. The blocks are now in the British Library.

Alexander Macmillan's descendant Lord Stockton with Lord Boardman, chairman of the National Westminster Bank, look at the original woodblocks made by the Dalziel brothers for Tenniel's illustrations for the Alice books

Lord Stockton (great-great grandson of Alexander’s brother Daniel Macmillan) and Lord Boardman, chairman of the National Westminster Bank, look at the original woodblocks made by the Dalziel brothers from Tenniel’s illustrations.{credit}© Macmillan Publishers Ltd.{/credit}

Publishing disaster — and triumph

In 1865, Carroll was keen for the first edition of Alice’s Adventures to come out as close as possible to the day on which the story was first told, three years before. Despite long delays, it was printed by the Clarendon Press in good time. Dodgson ordered a specially bound white vellum copy to be received by Alice Liddell on 4 July.

Then, on 19 July, the exacting Tenniel wrote to say that he was dissatisfied with the printing of the illustrations. As there were also faults with the printing of the text, all copies were withdrawn. The book, reprinted by Richard Clay, was finally published on 11 November that year in time for the Christmas market (and so bearing the year 1866 on the title page).

It was originally agreed that the unbound sheets of the faulty edition would be sold as waste paper. Instead, US firm David Appleton & Co bought them and 1,952 copies (of the original 2,000 copy print run) were sent to New York. The title page was redone with a New York imprint dated 1866, and the sheets machine-folded and put into cloth bindings. Of the copies not sent to the United States, just a few are known to have survived, and are extremely valuable.

The Macmillan file copy of the rejected printing, including 10 of Tenniel’s original preliminary pencil drawings, was acquired by Lord Swaythling around 1899. Eventually it made its way to collector Justin G. Schiller, who identified the purple markings as those made by Macmillan staff to show corrections for the new printing.

Within three weeks, 500 copies of the corrected November edition had been sold. On 23 December, The London Review deemed it “a delightful book for children” and “for grown-up people, provided they have wisdom or sympathy enough to enjoy a piece of downright hearty drollery”. That it was a trove of mathematical conundrums had yet to be discovered.

At Carroll’s death in 1898, the total number of copies sold by Macmillan exceeded 150,000.

A page from the 'Macmillan Editions Book' showing print runs from the end of 1865 to the end of 1875.

A page from the ‘Macmillan Editions Book’ showing print runs from the end of 1865 to the end of 1875.{credit}© Macmillan Publishers Ltd{/credit}

Almost four years after Alice’s Adventures was published, Nature emerged on 4 November 1869. It is likely that the profits from book publishing, including those from the Alice books, enabled Macmillan to continue to publish the journal for many years. Carroll himself became a contributor to Nature.

The phenomenally fruitful and occasionally fraught partnership of author and publisher lasted for over three decades. It was a bond Carroll publicly celebrated as a factor in the success of the Alice books. In his The Profits of Authorship (Macmillan, 1884), he wrote:

The publisher contributes about as much as the bookseller in time and bodily labour, but in mental toil and trouble a great deal more. I speak…having myself, for some twenty years, inflicted on that most patient and painstaking firm, Messrs. Macmillan and Co., about as much wear and worry as ever publishers have lived through. The day when they undertake a book for me is a dies nefastus for them. From that day till the book is out – an interval of some two or three years on an average – there is no pause in the ‘pelting of the pitiless storm’ of directions and questions on every conceivable detail. To say that every question gets a courteous and thoughtful reply – that they are still outside a lunatic asylum – and that they still regard me with some degree of charity – is to speak volumes in praise of their good temper and of their health, bodily and mental.

The electro of Alice and the White Rabbit.

The electro of Alice and the White Rabbit.{credit}© Macmillan Publishers Ltd.{/credit}

Alysoun Sanders is the archivist for Macmillan Publishers Ltd.

See Macmillan’s ‘Alice: 150 years’ website here. Mathematical Wonderlands: Lewis Carroll, the Alice books and beyond — a new ebook collection of pieces by and about Carroll in Macmillan publications, Nature and Scientific American — is available here. (Morton Cohen’s The Selected Letters of Lewis Carroll, published by Palgrave, also offers fascinating insights into Carroll as mathematician.) The British Library’s exhibition Alice in Wonderland runs from 20 November to 17 April 2016. 

 

All images are from The Macmillan Archive, © Macmillan Publishers Ltd.

For Nature’s full coverage of science in culture, visit www.nature.com/news/booksandarts.

A scintillating shortlist for the Royal Society prize

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{credit}Still Life with French Novels and a Rose, Vincent Van Gogh (oil, 1887){/credit}

As the literati strive to predict the future of the book, one thing is clear in the here and now: the best of popular science writing is still all about clarity, rigour and brio. This year’s six-book shortlist for the Royal Society Winton Prize for Science Books bristles with that mix.

The Society’s annual prize, now worth £25,000, is awarded to the year’s “outstanding popular science books from around the world”. This half-dozen certainly delves into many worlds — the universe inside the skull, the cosmos of numbers, the subatomic, the gene, and the dynamic interplay between biology and quantum mechanics, and people and planet.

Meet the contenders (in alphabetical order of authors’ surnames).

The Man Who Couldn’t Stop by David Adam (Picador)

Seasoned science journalist (and Nature colleague) Adam’s searing study-cum-memoir, reviewed here, is a twin journey through his own knotted, traumatic experience of obsessive-compulsive disorder, and the evolving science surrounding it. A reflective eye on what Adam calls “our siege mentality”.

Alex Through the Looking-Glass: How Life Reflects Numbers and Numbers Reflect Life by Alex Bellos (Bloomsbury)

The erudite and engaging Bellos, a writer and speaker on mathematics, follows up his bestselling 2010 Alex’s Adventures in Numberland with this equally adroit interweaving of maths history, the peculiarities of day-to-day maths, and the mindscapes of mathematicians. (Why is 24 is better than 31 in the context of anti-dandruff shampoo? You’ll need to read the book.)

Smashing Physics: Inside the World’s Biggest Experiment by Jon Butterworth (Headline)

Butterworth, a particle physicist and CERN insider, here (writes my colleague Jo Baker) gives “a personal account of three years that shook his research field – from the switching on of the Large Hadron Collider at CERN in 2009 to the discovery of the Higgs boson in 2012. In a lucid treatment —  part memoir, part primer — he relates the ups, downs and minutiae of everyday life at the particle physics coalface and reflects on the public and political perceptions of science.”

Life’s Greatest Secret: The Story of the Race to Crack the Genetic Code by Matthew Cobb (Profile)
Zoologist Cobb masterfully recontextualises the 1953 discovery of the structure of DNA (reviewed here). One for the shelf bearing seminal early studies by James Watson and Horace Judson, Cobb’s treatment beautifully explicates the contributions of physics, biology and chemistry, and scientists from Oswald Avery to Rosalind Franklin.

Life on the Edge: The Coming of Age of Quantum Biology by Johnjoe Mcfadden and Jim Al-Khalili (Bantam Press)
Al-Khalili (a physicist) and McFadden (a molecular biologist) take on the vexed nexus of quantum weirdness and life itself in this exploration of an emergent field of scientific endeavour (reviewed here). From synthbio to quantum tunnelling inside enzymes, a trip into strange, and strangely compelling, realms of research.

Adventures in the Anthropocene: A Journey to the Heart of the Planet we Made by Gaia Vince (Chatto & Windus)
Writer, broadcaster and former Nature news editor Vince covered six continents over two years to craft this compilation (reviewed here). Bucking the trend to view the environmental challenges of the Anthropocene with terrified or jaundiced eye, she discovered innovators and pioneers working towards new models of adaptation and environmental ‘reverse engineering’. A grand survey of development endeavour through a science writer’s lens.

In looking through this list, it occurred to me anew how popular science writing remains one of the great exemplars of multidisciplinarity. It is the context to the findings — the history, the socioeconomic realities, the psychology of the players and their rivals, the leadup to discovery and the societal implications of its deployment — that reveals the real-world significance of the science.

Scientific storytelling is one of the great artforms of our age. Its roots may stretch back to Mary Somerville’s monumental On the Connexion of the Physical Sciences 181 years ago (reviewed here), but its heyday is now.

The judges of this year’s prize include chair Ian Stewart (mathematician and Royal Society Fellow), Guardian books editor Claire Armitstead, Channel 4 lead anchor Krishnan Guru-Murthy, electronics engineer Jo Shien Ng, science broadcaster and author Adam Rutherford, and novelist Sarah Waters. The winner will be announced at a Royal Society public event on 24 September, hosted by Brian Cox, Royal Society Professor for Public Engagement in Science.

 

For Nature’s full coverage of science in culture, visit www.nature.com/news/booksandarts.