Escher and Coxeter: a mathematical conversation

On Monday 5 June 2017, Professor Sarah Hart from Birkbeck’s Department of Economics, Mathematics and Statistics gave a prestigious Gresham Lecture at the Museum of London. Andrew Silverman, Learning Development Tutor in the School of Business, Economics and Informatics reports on the lecture.

Gresham College crest in hyperbolic geometry. Credit: Sarah Hart

Gresham College crest in hyperbolic geometry. Credit: Sarah Hart

Gresham College was founded in 1597 and has been providing free lectures within the City of London for over 400 years. Walking down from the dusty roads of the Barbican into the cool and quiet of the Weston Theatre, the audience was transported into a conversation between an artist and a mathematician, Escher and Coxeter. Told with infectious excitement and humour, Professor Hart wove the story of the lives of these two figures and their friendship through the mathematics and the artwork that fed into one another.

Born on 17 June 1898 in Leeuwarden, Holland, the youngest of five brothers and moving with his family to Arnhem when he was five, Maurits Cornelis Escher would eventually go on to study at the School for Architecture and Decorative Arts in Haarlem. His father was a civil engineer and his brothers all became scientists. Escher himself almost became an architect before switching to graphic arts. He later quipped that it was only by a hair’s breadth that he escaped becoming a useful member of society.

Escher began by producing woodcuts and lithographs featuring mainly landscapes. An example of this was the 1931 lithograph Atranti, Coast of Amalfi. But in 1936 his work went in a new direction, becoming more abstract; according to Escher, he had replaced landscapes with mindscapes. The woodcut Metamorphosis I, produced in 1937, exemplifies this change and is clearly a ‘mindscape’ adapted from the 1931 piece.

What could have triggered this change? In 1922, Escher visited the Alhambra palace in Granada, Spain; his second visit was in 1936. The buildings we know today were constructed in the mid-11th century by the Moorish king Mohammed ben Al-Ahmar. One of the key points about Moorish art, and Islamic art more generally, is that it is not permitted to contain images of living things; it is instead rich in symmetry and tessellations of tiles. Escher was able to combine this richness of geometric design with images of ‘living’ things (albeit at times mythical living things), thereby leading to works such as Angel and Devils (1941), produced in ink rather than wood.

Donald Coxeter, or Harold Scott Macdonald Coxeter, was born on 9 February 1907. His name was originally going to be Harold Macdonald Scott Coxeter, until they realised that this would have been HMS Coxeter, more a ship name than a baby name, and so the name was changed. As a schoolboy, Coxeter became so engrossed in geometry, at the expense of other subjects, that one teacher told him he was only allowed to think in four dimensions on Sundays.

In 1936, the year Escher’s art took a new direction, Coxeter took up a post at the University of Toronto. When asked what the point of pure mathematics is, Coxeter responded: “No one asks artists why they do what they do. I’m like any artist; it’s just that the obsession that fills my mind is shapes and patterns.”

In 1954, the International Congress of Mathematicians was held in Amsterdam. To coincide with this, a major exhibition of Escher’s work was held in the Stedelijk Museum in Amsterdam. It was here that Coxeter and Escher first met, when Coxeter bought a couple of Escher’s prints.

Incidentally, another mathematician who visited the exhibition was Roger Penrose, who was a Reader and then Professor of Applied Mathematics at Birkbeck from 1964 to 1973, and later became Professor of Geometry at Gresham College, London. After seeing an impossible staircase in Escher’s Relativity print, he came up with the concept of a ‘Penrose triangle’.

Professor Hart explained the mathematics behind the regular tilings in three geometries: plane (Euclidean), spherical and hyperbolic geometry. She managed to put the concepts across in such a way that even someone with no prior knowledge could walk away with a good basic understanding, and the images presented were an excellent way of getting a more intuitive sense of what was really going on.

Escher learnt a great deal from Coxeter, to the extent that when Escher created a picture based on a new geometrical concept, he would refer to it as ‘Coxetering’. But in turn, Coxeter also learnt from Escher. For example, Escher’s 1959 work, Circle Limit III, led Coxeter to a new understanding of the hyperbolic disc. By looking at the spines of the fish in the image, Coxeter realised that Escher had found equidistant curves and produced them incredibly precisely. In this way their friendship was a true exchange of ideas – a mathematical conversation.

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Motherhood in UK prisons: the devaluing of the maternal

On 8 June, Birkbeck’s Department of Psychosocial Studies in collaboration with the MaMSIE research network (Mapping Maternal Subjectivities, Identities and Ethics), and with support from Clinks, hosted practitioners and academics to consider the challenges that face mothers in UK prisons. PhD student Claire Horn reports on the event.


It felt fitting that this discussion fell on the day of the election. In a time characterized by media amplification of partisan debate and the claims of our political leaders, this was an afternoon to consider the voices of women that often go unheard, and to reflect on an issue that requires collaboration across political divides. Opening the event, Lisa Baraitser (co-founder of MaMSIE, with Cambridge University’s Sigal Spigel), spoke of the importance of addressing the experiences of women who parent and are in prisons as a pressing feminist issue. Setting the tone for the discussion to come, she cited the need to reflect on what incarcerating mothers does to women, to children, and to communities.

Anne Fox, Chief Executive Officer of Clinks, chaired both of the afternoon’s panels and facilitated a focused and thoughtful dialogue. Fox raised an issue in her opening remarks that was reflected again and again throughout the afternoon: as a society, we undervalue and fail to critically consider motherhood. Given this lack of attention to motherhood in general, mothers in prison are in an especially unique situation. In a culture that demonises those who have offended, they are already stigmatized. As mothers, that stigma is amplified.

The first group of panelists delved into this double de-valuing of imprisoned mothers. Naomi Delap is the direct of Birth Companions, an organisation that provides physical and emotional support to pregnant women and mothers in UK prisons. Delap spoke of the lack of basic access to care for many pregnant women, and the necessity of codifying perinatal services and support. While Delap spoke of initiatives to de-carcerate pregnant women, and provide better community services she also illustrated the need for appropriate care for pregnant women who are currently imprisoned. The Birth Charter, compiled by Birth Companions, sets out specific and carefully considered recommendations.

Laura Abbott, the second speaker of the day, is undertaking doctoral work in health research at the University of Hertfordshire on the experience of being a pregnant woman in prison. She is also a volunteer at Birth Companions, and she spoke of the vital importance of midwifery support for imprisoned pregnant mothers. Abbott shared her work on interviewing mothers and staff in prisons, and through her presentation, represented the voices of mothers who spoke of a lack of access to basic care, and also of their sense that sentencing mothers acts as a punishment to their children.

Anastasia Chamberlen, Assistant Professor at the Department of Sociology at the University of Warwick (and past lecturer in Criminology at Birkbeck), discussed the gendered embodied experience of prison and punishment, noting the ways in which the impact of incarceration lingers in and on the body. Among other specifically gendered phenomena, she spoke of how for many women she interviewed, the period of imprisonment extended over what might otherwise have been mothering years.

In the panel that followed, Lucy Baldwin, Senior Lecturer in Community and Criminal Justice at De Montfort University and editor of the Mothering Justice collection, spoke of the emotional impact of incarceration. Baldwin described prison as an assault on a mother’s ability to do the work of mothering. She gave the powerful example of prison visit rules that do not allow for mothers to touch or hold their children during visitation hours. As Baldwin so aptly noted, this lack of contact would be viewed as neglect outside prison walls, yet inside, it is enforced.

Shona Minson, DPhil in Criminology at Oxford University, and author of the “Motherhood as Mitigation” report published by the Howard League for Penal Reform, spoke further of the impact of maternal imprisonment on children. She described the ways in which having a mother in prison is linked to risks to children, and explained the phenomenon of “secondary criminalisation,” through which children experience the punishment and stigma of their mother’s penal sentences.

These five speakers forcefully articulated the potential impact on individual women, children, communities, and society more broadly when mothers are imprisoned. While these panelists are each engaged in the important work of addressing these issues, the concerns they raised also demonstrated the need for further support. Anne Fox closed out the event in a memorable way by asking the room (fully populated by researchers, students, activists, and practitioners) to consider questions for collaboration. Birkbeck is unique among universities in that many students and faculty are invested in producing research that bridges theory and practice. I am hopeful that this event will prompt further work in this vein, and that attendees will heed the call to communicate and collaborate.

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Qualitative Research Methods in Action

This post was contributed by Lucy Tallentire from Birkbeck’s School of Business, Economics and Informatics

birkbeck_qualitative_methods_day_webOn 23 May 2017, Birkbeck’s Department of Organizational Psychology was delighted to host a one-day conference to discuss the latest methodologies in the field – Qualitative Research Methods. Qualitative research is an area of growing interest for organizational psychologists because of its ability to provide unique insight into trends in thought and opinions, and dive right to the heart of a problem. The day’s presentations and panel discussions provided a fantastic platform to engage with the work of academic staff, current and recently completed PhD students from the department, with a particular focus on the challenges and opportunities posed by the participant-researcher relationship that develops over the course of a study.

What are Qualitative Methods?

Where both qualitative and quantitative research methods play a pivotal role in contributing to understandings of work and organizations, qualitative research comprises a wide range of methods developed from a variety of theoretical perspectives and underpinned by a range of philosophical stances. These include critical, postmodern and social constructionist perspectives, reflecting the landscape of influential European philosophies.  More traditional positivist quantitative research methods are used to quantify the problem by generating numerical data for statistical analysis, but qualitative research typically applies inductive methods to explore socially constructed reality, focusing on meanings, ideas and practices.  This approach can provide rich data about sense-making, identity, and lived experience that quantitative studies simply cannot match. Both qualitative and quantitative approaches have their strengths and limitations but essentially they are used to address very different research questions. The Department of Organizational Psychology is committed to a plurality of research methods and their underpinning philosophies as a means of enhancing the methodological options available to work psychologists.

The focus of this event was on the participant-researcher relationship in qualitative research with a particular focus on the role of trust, time and technology.

Building the Trust

So – qualitative data collection takes time, but are there other challenges in qualitative research? Certainly, but the biggest challenge can also be considered the biggest opportunity – the relationship that develops between researcher and participant. The audience heard this first-hand from several presenters, like PhD student Jane Setten. She described trust as the ‘central mechanism’ to her longitudinal study, in which she will meet her participants over several years to map changes in their work-life situations. While the relationship between researcher and interviewee could, in the first instance, create a barrier to authentic and useful data if the participant was wary of the researcher, it can become a window to ‘successfully understand a situation or behaviour from an insider’s perspective’. Qualitative research methods rely on the researcher-participant relationship – it is a key part of co-constructing the data to analyse.

The audience also heard from Paula Fitzgerald, who shared not only her experience of qualitative photo-elicitation and interview methods but a great example of how – regardless of discipline – a research journey is rarely linear. Paula’s qualitative data collection resulted in her beginning to analyse her own experience of conducting research, from the researcher’s perspective – ‘an embodied experience’. Her work shed light on the role of the researcher as nuanced and permeable, and the idea that a researcher can be seen by their participants either as an insider or an outsider.

In Pursuit of Publication

The OP Qualitative Methods in Action day succeeded in providing an informal overview of leading-edge, innovative methods that are currently being used by qualitative researchers in the department. However, qualitative research, like longitudinal case studies carried out over a number of years, and visual approaches, like responses to pictures and participant video diaries, are still considered less traditional than quantitative methods. Student and staff presentations were thus augmented by an insightful keynote on the challenges and opportunities in publishing qualitative research, delivered by Professor Bill Lee, from the University of Sheffield Management School. Professor Lee has not only published widely on topics across management and related disciplines, but is also an associate editor for Qualitative Research in Organizations and Management (QROM) and of the European Management Review (EMR). At EMR he has recently established a new section, “Methodology Matters”, which provides a peer-reviewed outlet for articles that make a methodological contribution.

After leading the audience through the development of opportunities for qualitative research, and the context on which the debate on research methods is based, Professor Lee highlighted the importance of perseverance and resilience in pursuit of publication. New sections such as “Methodology Matters” open up new opportunities for people interested in methods and associated fields, helping to bolster accessibility and diversity within all research fields.

The Department of Organizational Psychology would like to thank all participants and attendees for their thought provoking presentations, posters and questions.

You can follow Birkbeck’s Department of Organizational Psychology on Twitter @bbk_orgpsych or find out more about upcoming events on their website.

The graphic recording of the day’s proceedings was provided by Laura Sorvala, from Auralab. You can follow her on Twitter @_auralab or visit her website.

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Enough is Enough: Secrets of the Warburg Photographic Collection

This article was contributed by Sue Wiseman, Professor of Seventeenth-Century Literature in Birkbeck’s Department of English and Humanities

A visit to the Warburg Institute always provides food for thought; indeed its very system of book classification is designed to precipitate serendipitous findings. So, I should not have been surprised when our Arts Week jaunt to uncover the secrets of the Warburg Photographic Collection, led by its least secretive of guardians, Dr Paul Taylor, provided me with a way to think about our Arts Week lead of art and politics, but I was.

The Warburg Photographic Collection is a library of images that runs from the early Renaissance to the eighteenth-century. As Dr Taylor explained, it covers that period because what Erwin Panofsky called the pre-iconographic and the iconographic exist in a particular relation during that time. So, if we see an image of a woman beside a pool we might think about it pre-iconographicallly – woman with pool. But if we are to think about it iconographically in the period 1450-1790 we are likely to think about the wither Susanna and the Elders (the Bible) or the classical goddess Diana, bathing with her nymphs because that is what Renaissance and eighteenth-century viewers might have had in mind. To stay with classical imagery, we might similarly understand a woman with a helmet on her head in pre-iconographic terms as – just that. However, considered iconographically she might be, or suggest, Athena, and the woman with snakes in her hair that she is killing shares the complex iconography of Hydra, with all the associations of rebellion and misrule. The collection stops in the eighteenth century because, as Paul reminded us, that was when artists and viewers began to think differently about images and for a time these associations faded. Nowadays, of course, they are a smaller but always available part of the thinking of an artist or a viewer and sometimes they give extra resonance not only to the images of artists as different as Cy Twombly and Cindy Sherman, but to the images we find around us in the our image-drenched everyday.

Thinking about pre-iconographic and iconographic looking is an important part of what   modern readers of the internet practice astutely all the time. Renaissance iconography can help us to see more clearly that thinking about images iconographically can add depth and resonance and make us understand more deeply the messages a particular image is offering or, often, selling us. And it is this, of course, that set me thinking about a specific hairy political problem that I have been struggling with. The problem is hairy in several ways –  it is tangled; there is something at stake, and, above all, it is a problem about political hair. Can Erwin Panofsky’s idea of the iconographic help me to understand the iconography of hair as it is playing out. Let us try the method.


Credit: Think London

Hair was brought to the attention of the viewing public when Donald Trump began his political ascent. His hair, like that of Boris Johnson, is unruly and speaks of disruption but also, I think, of the retention of youthful energy into old age. Jeremy Corbyn’s beard, of course, leads us not only to the Bible but to all the images of distinguished masculine age, but the neatness of hair and beard might be a careful distraction. More attention is paid to the appearance of female politicians. It is not that images of men don’t carry meanings but that in many systems of images, not only the Western art that makes up the bulk of the Warburg Photographic Collection, images of women signify instability. Modern politicians know this and part of life for a female politician must be using her image to her advantage – but always with an eye to forestalling problematic associations. Women in the public eye know all too clearly that part of what iconographic thinking does is make visibible some of the living connection between art and politics. Theresa May moves on the stage of world politics beside men whose hair draws comment, and (as Gaby Wood’s fashion-homage to May in Vogue tells us) we know that she is herself interested in fashion. So why has the image projected by her own sleek bob been so little remarked?


Credit: The Margaret Thatcher Foundation

If thinking about May’s hair as a bob is pre-iconographic, it can, nevertheless, take us towards some potentially helpful iconography. Perhaps the most obvious visual antecedent for a female Conservative Prime Minister is Margaret Thatcher and in some ways Theresa May has drawn on that legacy. She does it carefully, though. Just as she has told us ‘Brexit means Brexit’ she has said ‘Margaret Thatcher was a Conservative, I am a conservative’ or on 4 June 2017, apparently marking the latest attack over which she has presided as Home Secretary and Prime Minister, ‘Enough is enough’. These structures of speech are analogous to thinking pre-iconographically in that each rejects an invitation to make an association; they stop short of agreeing to a meaning that comes from outside the control of the speaker. The soundbites strongly associated with Margaret Thatcher such as ‘The lady’s not for turning’ are strong assertions and have over the years been subject to exactly the repurposing and satire that at least the three sentences I have quoted by May resist by refusing image, metaphor, analogy. If May’s speech is distinct from Thatcher’s in attempting insulation and resisting bold, unruly, assertion that might make her vulnerable, I think the same can be said of her relationship to Thatcher’s image. There is little to connect them beyond their use of suits and that is hardly a connection in the wardrobes of political women. May makes little use of Thatcher’s visual image altogether and a comparison on their hair bears out this question. Thatcher’s hair is bold yet fixed in a style now redolent of the aggressive management style of the 1980s when Teresa May was also making her way in politics.

By the time Teresa May caused a stir by putting some spring in her step by wearing a pair of leopard-print kitten heels to the Tory conference in 2002, Margaret Thatcher was long gone. Tony Blair has been elected on a landslide victory. The Tories were nowhere and May was not a particularly important politician. As the eagle-eyed commentator on the politics of fashion, Hadley Freeman, noted in The Guardian there might be some significance in May’s fashion choice. Could May be distancing herself from the difficult Thatcher image by fashioning her own image – in this case by using ‘the old trick of wearing an implausible item in order to create a new image and divert attention from a tiresome past?’ In the end Freeman decided that May liked shoes and liked fun. Yes, but fun and shoes still might have meaning.

princess_dianaFirst coming to politics under Thatcher, May’s formational years were not dominated by the fashions favoured by a woman born in 1925 and whose power hair seemed to grow into a larger and larger helmet as her popularity waned. Much more appealing to a Tory woman born in 1956 might be the fashionable dressing of another, potentially much more sympathetic, political figure – Diana, Princess of Wales. And if we look at the iconography of the People’s Princess in popular memory we can, I think, see much of what May quietly claims from the Thatcher years in her subtle iconography of heel – and hair. Of course, in the 1980s Diana was a troubling and troublesome figure of constrained femininity. However, as time passed, her femininity and fashion sense were remembered. She was beloved, a fashion icon and had the other kind of 1980s hair – various versions of a bob. Any young Tory politician would want to annexe some of Diana’s charisma and good looks. But with May’s subtle appropriation of heel and, especially, hair we might also find something of Diana’s set-apart quality and, of course, it can’t be missed that Diana was political – but she was a princess, not a mucky politician.

Vivienne Westwood by Mattia PasseriDiana was a princess, May was not and was only likely to attain power by a long and careful game. But they both liked fashion, and their tastes were shaped in the eighties. If one Queen of eighties fashion was Diana, the other was of course Vivienne Westwood. Producing madly wearable brilliantly iconoclastic clothes Westwood’s iconography was like Diana’s singular, but much more heavily Elizabethan. She can readily be found as Queen Elizabeth I and II and her Elizabeth gown is at present on display in the Chatsworth chinoiserie. As we now all know, May succeeded in the long game and soon after her accession to rule she made a speech about Brexit – wearing her ‘lucky’ suit. If the trouser part of the suit annexes the panther power of punk, like the leopard she trod in, then its Vivienne aspect fills in the regal dimension from head (hair) to (leopard) toe. If May is a ruler she is a queen – isolate, singular, and deriving her power from a mixture of personality and taste, not from power-dressing, muck-slinging politics.

All this, of course, makes us ask what is the iconography of a female ruler – not helmet haired like Athena or Thatcher, but subtle, powerful, yet without vulnerability? What might Diana and Vivienne do for her?  Well, Vivienne Westwood’s designs draw on the twin iconographies of Britishness (the tartan we find in May’s trouser suit) and the queens Elizabeth. Her designs suggest a rebellion restrained to an iconographic tradition of Britishness that punk both satirised and marketed and that was ultimately absorbed into Cool Britannia and the Olympic iconography of 2012. Westwood’s regal iconography combines deference to royalty and a fantasy of her own regal rule. There is no mistaking that just as her dresses might make a queen, so, too, they mean she is indeed a queen.

If we move up to the hair, then the bob is one of the most common and versatile styles for women’s hair. It can look natural, and that is the way Teresa May wears it (pre-iconographic again) yet it is often saturated with cultural meanings and knowledge. If we look back to the 1980s to think about bobs other than Diana’s, perhaps Debby Harry of the band Blondie offers an example of the bob. Like May, Debby Harry wanted to annexe values to her brand, but none of them were natural; her bob calls up Marilyn Monroe’s. Yet, where Monroe was wrecked by fame, by the time we were all looking at Debby Harry’s bob she had already survived heroin addiction: she was older than most punks; she was a woman and the band, brand and hair were hers – and her hair said it all as the name of that band: Blondie. So if Harry claimed and reworked classic feminine icons of modernity to offer a knowing bid for a modern kind of freedom in which the ‘natural’ was long past, May’s use of the blow dried natural look speaks of a return to nature but also, once again, an avoidance of the iconographic: no helmet for her rather subtle styling, subtle colour and age-appropriacy. Yet for all that May’s hair says to us ‘hair means hair’, the gently groomed and subtly coloured ‘natural’ bob is claiming the power of a quiet femininity that is mixed with brand princess.

Of course, I began by saying that men have an easier time in terms of meaning. Many are allowed to stay in the pre-iconographic. So as I am writing just before an election why can’t I offer some balance by considering Corbyn’s hair? That would be a different topic. Just for the present, though, perhaps we can visualise his head with hair and beard outlining a pineata. As it is repeated and violently struck, sweets and trophies fly through the air at some velocity. It is time to return to the Warburg Picture Collection to ask Dr Paul Taylor for help on the iconography of Corbyn as a pineata …

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