*By attempting everything, Dorothy Wrinch ended up accomplishing nothing.*

For half a century, this was the standard final verdict on the life and work of mathematician and logician Dorothy Wrinch. Undoubtedly brilliant, so the assessment ran, she squandered her gifts by spreading herself too thin, and as a result by life’s end had nothing but a collection of disconnected discoveries and blaring public missteps to show for all her energy, charisma and intelligence.

Had she been a man, her devotion to her pet theories even beyond the point of their debunking would have been shrugged off as an eccentricity that ought not detract from her larger work (as with Newton’s fervour for lost ancient wisdom, Kepler’s mysticism, or Linus Pauling’s obsession with Vitamin C), but as a woman, her refusal to back down was considered so unladylike, so shrewishly unreasonable, that it invalidated everything else she put forth, until by the time of her death, the woman who was a philisophico-mathematical student of Bertrand Russell and an innovator in X-ray crystallographic algorithms was dismissed as a cantankerous crackpot who had paid the just price for her brazenness.

*Photo by Agence Rol.*

The real Dorothy Wrinch (1894–1976), however, cannot be contained within the bounds of such a simple narrative. Neither so out-of-her-depth and congenitally deluded as her detractors suggest, nor so irrationally and helplessly victimised as her sympathisers would have it, Wrinch was an inventive polymath who scored important technical contributions and pushed conceptual boundaries even as her behaviour in defence of her central theory alienated would-be allies and left her increasingly isolated and disillusioned.

Her father was an engineer, and her mother an ex-school headmistress, and, as might be expected, young Dorothy was given access to all the education that the late Victorian era had to offer to a young English girl. She entered school at the age of 4 and made her way steadily through a morality-and-piety heavy curriculum that bore the full imprint of Thomas Arnold’s long influence. Her teachers recognised her brilliance but felt that, all the same, she’d come to a bad end. She did, however, leave secondary school with generally positive recommendations and a barrage of mathematical and linguistic certifications, and so made her way to Girton College.

By 1913, when Wrinch arrived at Girton, the college already had a reputation for producing women of academic brilliance. Hertha Ayrton had graduated from Girton in 1880 on her way to making fundamental discoveries in fluid dynamics and perfecting the arc light, as did our friend Charlotte Angas Scott, whose example Wrinch was (not always favourably) held to in her early career. Future botanist Ethel Sargent had graduated in 1885, and 1893 saw the graduation of mathematician Grace Chisholm Young. These women, by their rigorous work and unimpeachable characters, had cleared the way for a new generation of Girton graduates, young women who felt comfortable not only challenging Victorian assumptions about women intellectuals but some of the most fundamental principles of Western civilisation itself.

Wrinch came to Girton as a student of mathematics at a time when that field was experiencing a complete and invigorating re-examination of its capacities and starting principles. In 1899, David Hilbert created a new axiomatisation of geometry on rigorous modern lines that cast away at last two millennia of Euclidean clutter, while from 1910 to 1913, Bertrand Russell and Alfred North Whitehead published the three volumes of the *Principia Mathematica*, an impossibly dense but bracing attempt to recreate the entirety of mathematics itself from fundamental principles of logic. Logic was maths, and maths was everything, and Wrinch was swept up in the wave of mathematical optimism.

During her first few years of college, Wrinch was being pulled both intellectually and sympathetically into the orbit of Bertrand Russell. His logical genius appealed to her sense of mathematics’ sure and eternal power, his outspoken and solitary stance against the First World War reassured her that her own anti-war stance was not shameful or cowardly, and his brilliant religious scepticism accelerated her own tendencies towards non-belief in the variously oppressive trappings of organised religion. If ever two people were destined to meet and become lifelong friends, it was Dorothy Wrinch and Bertrand Russell, and so they did, in 1916.

By that point, Wrinch had experienced a devastatingly low result on her first Tripos examination as a result of her insistence on taking the exam after one year of study instead of the usual two, but managed to score towards the top of the pack in the second round, while simultaneously studying the leading lights of early twentieth-century philosophy. When Wrinch met Russell, they experienced an immediate meeting of minds. Russell was impressed by her knowledge of, and enthusiasm for, the deep mysteries and puzzles posed by mathematical logic, and included her in the small circle of trusted friends who worked together during the war on mathematical problems even as Russell was jailed for his anti-war statements.

In what was to become a pattern for Wrinch, she devoted the full force of her fertile mind in the name of a dying but beautiful cause. She worked away at problems of logic that bedevilled Russell, diving deeper and deeper into the intoxicating complexity of logic’s infamously impenetrable notation, all in the name of at last constructing a perfectly consistent and self-contained theory of mathematics.

A decade and a half later, in 1931, Kurt Gödel would publish his Incompleteness Theorems, effectively proving that axiomatic systems are inherently limited, and therefore that programs like Hilbert’s and Russell’s are doomed to ultimate failure. Fortunately, by then Wrinch had already moved beyond logic as new fields caught her fancy. Under a pseudonym, she published *Retreat from Parenthood *in 1930, a book laying out her scientific system of parenting which called for a new government department to provide organised support for new parents, allowing women more freedom to work, and increased opportunities for restorative leisure. She tackled the emerging big questions in the history of science about how ideas get accepted and promoted in scientific communities. And, lest we think that she operated purely on the most impossibly vast dimensions of thought, she also made solid contributions to the technical but important theory of trigonometric series.

All very well, some mathematicians thought, but what does it all add up to? G.H. Hardy, one of the most important mathematicians of his, or any, age (and also the promoter of Indian mega-genius Srinivasa Ramanujan), was a devoted feminist and promoter of women in mathematics, and even he struggled to understand where Wrinch was going with her multifarious interests. In his opinion, she had perilously divided her attentions, and as such was achieving far fewer results than she ought, and couldn’t be ranked in the same tier as Charlotte Scott or Grace Young.

All the same, he attempted to help her secure funding, only to be frustrated by the discovery that, while she was telling *him *she was ready to settle down and concentrate fully on mathematics if only he could help her get a grant, she was telling somebody else that she was ready to settle down and concentrate fully on social institutional theory, if only* they* could help her get a grant. Today, the idea of an academic pursuing widely different fields simultaneously, and seeking funding for each, has lost its ability to shock, but in the pre-war years, a mathematician seeking funding was expected to be a lifelong devotee of their field of study. Why give a grant, after all, to somebody who was only going to keep half an eye on her proposed field, when so many pure and committed researchers were waiting in the wings? What today we would laud as a devotion to multidisciplinary research appeared at the time as fickle indecision, and Wrinch would have to wait decades for scientific styles to catch up with her own ways of working and thinking.

In the meantime, in the 1930s Wrinch hit at last upon the field that would claim her more or less undivided interest for the remaining four decades of her life – the intersection of mathematics with biology, and in particular the problem of the structure of proteins. To understand this part of the story, we have to put ourselves back in the mindset of 1935, before the structure of DNA was known, before every biology textbook in the world carried pictures of chains of amino acids folding at the behest of hydrogen bonds and disulphide bridges to form intricate three-dimensional structures known as proteins. At that time, though many were convinced that the key to proteins lay in their structure as long chains of amino acids linked by peptide bonds, others were not so sure. The variety of physical and chemical characteristics of proteins, the sceptics argued, surely couldn’t be the result of a structure as simple and mono-dimensional as a chain.

Wrinch proposed to bring mathematics to the problem – what sorts of geometrical structures fit the given data and had enough variety to explain all the behaviours demonstrated by proteins? Wrinch had entered the field of biology by proposing a genetic structure theorising that chromosomes were essentially fabrics made up of a protein warp and a nucleic acid weft that meet up at the edges to form a tube. It was ultimately nowhere near correct physically (though it did correctly posit that the linear sequence of amino acids is important in a gene’s function), but it was stimulating and elegant, and it demonstrated what might be done when a mathematician and geometer brought her unique perspective to a biological problem.

It was time, then, to enter the debate about proteins and here, as in the case with logic, Wrinch had the misfortune of entering the field when some highly attractive but ultimately incorrect measurements and principles were widely believed. In particular, it was held that proteins consist of amino acid residues whose number is always a product of a power of 2 and a power of 3, and the number of molecular weight classes that proteins belong to is sharply limited. These notions, both of which are entirely wrong but were generally accepted, were seized upon by Wrinch as the mathematical basis for her protein models. Instead of linking up amino acids in long chains, she suggested in 1936 that they formed elegant hexagonal fabrics which could, in turn, form three-dimensional Platonic cages. She called this the ‘cyclol’ structure for proteins, and it created a firestorm of controversy, publicity, and, most importantly, research, as some raced to prove it and others to show that it was utterly impossible.

It was attractive in that it explained in terms of symmetry and structure why the amino acids should show up as powers of 2 and 3, and also carried a physical interpretation for how protein denaturing worked (the cage simply reverting to its two-dimensional fabric state, which carries a different chemistry). It was a beautiful and seductive model, sporting symmetry and mathematical substance, but it was vehemently opposed almost from the start, and no voice against it was more influential than that of chemistry’s reigning monarch, Linus Pauling.

Pauling saw several immediate problems with the cyclol model, and when he had the chance to question Wrinch about them directly, was so disturbed by her unwillingness to engage with the nuts and bolts chemistry of her theory that he made it a point to publicly weigh in on the matter with a crushing critique. He pointed out that the bond energies of the cage model were too high to be nature’s preferred structure, that the density of a cyclol cage didn’t match the density of known globular proteins, that the covalent bonds she proposed as holding together her cage couldn’t be undone easily enough to account for the regularity of protein denaturisation as effectively as a hydrogen bond hypothesis, and that her 288 (2^5 * 3^2) amino acid base unit didn’t faithfully reflect the diversity of known proteins.

Pauling’s 1938 paper listing what he believed were the most glaring faults of the cyclol model all but sealed its fate as far as the larger scientific community was concerned. But Wrinch was not convinced. She wrote to journals pointing out how some of Pauling’s calculations were flawed (they were, but not enough to change the evidence by enough to warrant a reconsideration of the cyclol model), requested that chemists she knew personally devote time to performing experiments that might prove her correct, and when those chemists ultimately grew tired of her unwillingness to accept their results when those results went counter to the cyclol model, asked for funding to hire her own assistant to do the experiments herself.

The subsequent years were filled with bitterness and controversy. She sparred unsuccessfully with crystallography legend Dorothy Hodgkin, insisting that insulin displayed a cyclol structure, compelling Hodgkin to definitively remark that there was no evidence for such a claim. Colleagues expressed frustration whenever she waived away their chemical objections as unimportant details not relevant to the larger theoretical picture. There was a scandal when she redirected her second husband’s research funding to pay for her own experiments, and her last book, *Chemical Aspects of Polypeptide Chains* (1965), continued to argue for a cyclol model nearly three decades after Pauling’s paper, prompting one reviewer to comment, ‘If a reviewer had been asked to consider a book on the “phlogiston” theory years after Lavoisier had shown that it was incorrect, he might have felt the same way as I have with this one.’

These were hard years, yes, but not barren ones. Though, to the scientific community, Wrinch was That Sad Cyclol Obsessed Lady, and though her increasingly desperate unwillingness to admit defeat certainly added to that impression, these were also the years that she was quietly working a potential revolution in yet another field of science: X-ray crystallography. She brought her knowledge of mathematics and topology to the labour-intensive process of interpreting Patterson maps to intuit a crystal’s molecular structure from the pattern obtained when X-rays are shone through it. Her algorithm was technically correct but was, unfortunately, not of immediate use for the protein crystallography of its time, lacking as it did the computing power to perform the underlying computational gruntwork. Decades later, the result would be rediscovered and widely applied in the computerised interpretation of molecular structures.

At this point it is hard not to feel a bit like Hardy did almost a century ago: what are we to make of all this? What does Dorothy Wrinch’s life amount to? Must we tie her reputation entirely to the success or lack thereof of the cyclol model, as she herself did, or can we see her as something more than that one beautiful hypothesis and its disastrous aftermath? Certainly, we are in a position now to understand her better than her contemporaries ever could. She contributed to philosophy, the history of science, geometry, molecular biology and chemistry, and was the first in a wave of talented multidisciplinary thinkers to bring the advanced topological tools of mathematics to the realm of biology. Her ideas, even when wrong, spurred people to develop and apply new techniques to their study, and her lectures were roundly reported as inspiring master-classes that taught a generation to possess the insight of a chemist and the eye of a geometer simultaneously. She spoke out against destructive and unnecessary war even as her home nation intoxicated itself on jingoistic euphoria, pressed for reform in governmental policies towards mothers, and held religion accountable for its manifold intellectual and spiritual abuses, and did all of that WHILE acting as an interdisciplinary scientific pioneer.

Dorothy Wrinch’s only daughter died in a fire in November 1975. Dorothy lived for three more months thereafter, unspeaking, and died herself in February 1976 after five years of retirement from a career that had brought pain, and beauty, though who can say in what ratio.

FURTHER READING: The major book about Wrinch’s life and science is 2012’s *I Died for Beauty: Dorothy Wrinch and the Cultures of Science *by Marjorie Senechal. It brings Russell’s intellectual companions and those of the Vienna Circle to robust life, and does well in explaining the points of contact between these schools and Wrinch. It is written in a loose and artistic style that definitely isn’t the norm in scientific biography, culminating in a proposed libretto for an opera about Pauling and Wrinch that takes up a full chapter. Her characterisation of Pauling is, perhaps not surprisingly, not entirely balanced, for which Thomas Hager’s *Force of Nature: The Life of Linus Pauling *(1995) provides a corrective – he has a dozen pages devoted to the cyclol controversy where the portrayal of Wrinch is, perhaps not surprisingly, not entirely balanced. Between the two books, one comes out at a happy and probably accurate medium wherein Pauling’s concerns were largely legitimate and Wrinch’s theory, when you consider the reigning assumptions it was nested in, was not nearly as improbable as it sounds now.

If you want read more stories about women in mathematics like this, you can pre-order my *A History of Women in Mathematics* from Pen and Sword Books, due out October 2023!

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