On Monday February 26, we’ll be presenting the results of last term’s Three Pillars of the Digital project as a Monday Evening Lecture (5:30pm in E002, KX). The last talk we gave in this programme, two years ago, outlined the goals and outlook of the Fine Art Maths Centre as a whole. Here’s Andrew with Part 2 (you can read part 1 here).

We approach mathematics as a creative practice, one that has been transformed in the last 100, 150 years. Increasingly, mathematicians would subscribe to the line that their subject is the ‘science of pattern and structure’.

As such, its possible connection to art and design should come to the fore.

We reiterate – school mathematics is largely comprised of eighteenth century techniques of calculation. It is not modern. FAMC counts itself amongst a number of pedagogical initiatives complaining about post-16 mathematics education and the dominance of calculus.

We live with the inherited consequences of an education system that forces poor choices on youngsters at 15 and ones still dominated by the desires of universities that teach science, medicine, engineering: *they* don’t want to teach calculus, so algebra and calculus suffocates the school curriculum.

There is lots of other maths. All of it is legitimate: much of what we teach is cloistered in the maths undergraduate curriculum for no good reason: it is appealing and amenable to students for whom numbers, algebra and calculation is a turn off.

This situation is exacerbated by poor academic organisation across the English university sector – marked by increasingly unproductive and entrenched divisions of intellectual labour.

Art – relative newcomer to the university structure – suffers here more than the established academic disciplines, because in truth art is not a discipline. It is not formed and defined by well-defined objects of study, techniques of presentation or methods of research.

Art lives on the absence of such boundaries and delimitations. All life, intellectual and otherwise, is open to art practice. Mathematics is no exception. This means that, like philosophy, art is *non-disciplinary*. Everything is open to art practice – every other discipline can become its material.

The prevalence of the digital and the complexity of contemporary life – where mathematics serves as the handmaiden to transformed technology and modern political economy of communication, information and complex asset structures – means that artists are increasingly drawn to mathematics and associated domains of knowledge such as computing.

This is a promise, a challenge and a demand.

Artworks are intellective and expressive: they relate to the world, raise and address questions, and can, in addition, appropriate philosophy and mathematics, *inter alia*, as material for their productions. Art can turn all to its own ends, but also is required to if it is to maintain the promise inherent in art – that it can express truth in previously unrealised or more powerful manner.

Viewed in this way, we can return to an older debate within German Idealism and its legacy from the nineteenth century onwards. The Jena Romantics, for example, characterised Art as the medium within which all knowledge, including Philosophy and mathematics, can be gathered and extended.

For the nineteenth century Romantics, the modern novel best captured this possibility: its form allowed for the results of modern disciplines to become material in a larger-scale construction. It led to the notion of the Gesamtkunstwerk – the composition that brings all manner of arts together with new and expansive expessive power.

What new forms are offered by computational power and the mathematics underpinning and organising it? By expressive power I mean to include all subjective capacities – sensibility, intellection etc. – the concept, in turn, underscores the debates around the superiority of particular arts.

**This fundamental character of artworks is the truth underlying the magpie instincts and healthy autodidacticism of artists. **

But this demand is threatened by a dilettantism or casual consumerism – these materials cannot be held together by the subjective attitude or enthusiasm of the individual artist.

What expressive power can be marshalled? Can the results of those technical disciplines be organised adequately?

In a slightly different context, the German philosopher, Theodor Adorno wrote:

Modern works … must show themselves to be the

equalof high industrialism,not simply make it a topic. … The substantive element of artistic modernism draws its power from the fact that the most advanced procedures of material production and organization are not limited to the sphere in which they originate. [AdornoAesthetic Theoryp. 33-34].

These comments about high industrialism – think the film or computer game industry for a ‘visual’ comparison – apply also to the creative practices of mathematics. Can such a technical procedures be made amenable to the sphere of art? Arts and humanities has been shedding technical components of the curriculum over the last 30 years, missing the manner in which these technicalities embody alternative ways of thinking.

Maths as technical discipline does not give itself up to idlers in the garden of knowledge – its elements cannot be absorbed by osmosis or consumed in a single gulp.

As mathematicians say, you cannot ‘grok’ mathematics – it is something with which one wrestles continually and gradually one develops skills and insights.

But that resistance to appropriation is perhaps what is missing from the arts and humanities, which still seem beholden to a nineties idea of Theory, capital T, that offers little more than a jargon or a standpoint on the world, postmodernism still – in that it is consumerism at the level of ideas.

Much contemporary theory *evangelises* for maths and science, in the sense you might find in accelerationism or speculative realism. But this evangelism is not what is needed.

The evangelist brings the good news – but is not the source of the good news. The evangelist treats mathematics and science as a *authoritative revelation* rather than something that can become integral material for artistic production. Evangelism offers the semblance of participation – it worships maths and science inappropriately.

“Theory” replicates what Manny Farber in 1962 called the temptations of ‘gilt culture’, the ‘white elephant art’ that wants to make the big claims, wants *too hard* to be meaningful. Farber instead extols a

termite- tapeworm-fungus-moss art [that] goes always forward eating its own boundaries, and, likely as not, leaves nothing in its path other than the signs of eager, industrious, unkempt activity … eating away the immediate boundaries of art, and turning these boundaries into conditions of the next achievement.

[Farber, ‘White Elephant Art andTermite Art‘(1962)]

**Here then is the challenge and demand for undergraduate pedagogy. **

In the abstract we can return to a vision from 100 years ago – Walter Gropius’s take on the Bauhaus:

The

Bauhausfelt it had adouble moral responsibility: to make its pupils fully conscious of the age they were living in; and to train them to turn their native intelligence, and the knowledge they received, to practical account in the design of forms which would be the direct expression of that consciousness.

Walter Gropius (1923)

Fully conscious of the age they were living in. Its workings, procedures and techniques. What mathematical topics are amenable without prerequisites to an authentic relation to art practice?

Our remit: Works get made that otherwise wouldn’t. Our primary aim is to create mathematics provision that sits naturally within the students’ disciplines. Our approach exposes students to new ideas, but also provides technical grounding when appropriate. Hopefully, artworks are then produced which would otherwise not get past initial ideas.

Success would be judged on that criterion – to open possibilities for skill development otherwise closed down for lack of technical support.

We have pioneered an innovative ‘no prerequisites’, ‘practice oriented’ approach to meeting the authentic mathematical needs of undergraduate and postgraduate art students.

Our workshops have concentrated on different aspects of geometry and programming. In tutorials recurring topics have included: group theory and symmetry; combinatorics; data analysis and visualization; 3D modeling and graphics engines; programming; perspective (especially in the context of sculptural and installation work).

Our students have also asked about: differential geometry; topology; linear algebra; financial mathematics; traditional calculating devices such as abacuses; game theory; Latin squares; tilings (including spherical tilings); polyhedra; optical illusions; hyperbolic geometry; symbolic logic; graph theory and the mathematics of rivers.

A 30 or 45min tutorial cannot exhaust any one of those questions. In many cases, we can only give guidance on where to start so that the magpie instincts can be directed initially in the most productive ways. Sigma fund us to cover any topic that might be covered on the first-year undergraduate mathematics curriculum, but we endeavour not to be bound by that disciplinary setting.

As we develop our project we plan to develop workshops in abstract algebra, group theory, graph theory and linear algebra.

Another caveat: Nor do we assume that our task is a one-way transmission of mathematical knowledge to artists, or the policing of legitimate application. Perspective and its mathematical underpinnings forms a central part of our provision – in this field, renaissance Italian artists pushed geometry forward by investigating new methods for organising the depiction of three dimensional space in the two-dimensional picture plane. Centuries later, mathematicians developed those insights into Projective geometry.

Instead of being a minor province in the empire of Euclid, perspective geometry was logically a much vaster empire. – Cayley

Tellingly, today, Perspective is disappearing from art education and what remains in diplomas and foundations appears ill-conceived and divorced from any intellectual content. We have had to go to comic book illustrators to pick up some of the fundamentals of three-point and curvilinear perspective or alternatives such as isometric and axonometric projections.

The ways of thinking encapsulated in alternative perspective systems are important for *thinking* about space – they are not simply outmoded representational techniques (or a kind of foreshortening).

Here, though it may seem we have championing a return to skills-based education, we are actually championing learning a broader range of thinking techniques. A broadening of vision. This is what we emphasise in our first FAMC publication on Euclidean Geometry – drawing-led, practice-led mathematics which brings ways of thinking to the fore, along with practical applications.

New (and old in the case of geometry) ways of thinking are needed – we are living through a phase where whole media may collapse as sites of autonomous meaning production. Not by any natural process, but because those involved did not meet the challenges and were content to do the comfortable thing, the familiar but moribund. Though the commercial afterlives may be sufficient for some.

**Conclusion **

It is self-evident that nothing concerning art is self-evident anymore, not its inner life, not its relation to the world, not even its right to exist. … It is uncertain whether art is still possible; whether with its complete emancipation, it did not sever its own preconditions.”

Aesthetic Theoryp. 1

The separation of mathematics from art subjects is ingrained, but is a historical accident – a result of the histories of educational institutions and policies rather than anything that follows from the disciplines themselves.

Both art and mathematics are involved with creating concepts and new ways of seeing. What we hope to achieve by highlighting visual and spatial mathematics is that potential connections are closer than commonly imagined.

There is no shortcut to this mediation and what we offer at FAMC is only a first step towards achieving the potentials that may lie therein.

Mathematics is not about being good at mental arithmetic as it was until 200 years ago – it is about learning to see and present things in new ways through the creation of abstract concepts.

Grothendieck described mathematical creation as follows:

a vision that decants little by little over months and years, bringing to light the ‘obvious’ thing that no one had seen, taking form in an ‘obvious’ assertion of which no one had dreamed

When I read that, it seems as if art’s terrain had been stripped from it. The challenge for art is to demonstrate that this is not the case. And it may be that this new age of computer-led visual or spatial mathematics again, as in the renaissance, opens the possibilities for new artist-led discoveries.

The modern world is very different even from that of twenty years ago. There are plenty of quick takes on the contemporary world being peddled, but a practice that is adequate to today may need to take new steps into the abstract the better to underpin practice with intellectual firepower.

On Monday February 26, we’ll be presenting the results of last term’s Three Pillars of the Digital project as a Monday Evening Lecture (5:30pm in E002, KX). We’ll say a bit about what we did but, more importantly, raise some points for discussion about the place of maths and the digital in creative arts education.

The last talk we gave in this programme, two years ago, outlined the goals and outlook of the Fine Art Maths Centre as a whole. Rich started off with Part 1, accompanied by a silent edit of this video of John Milnor lecturing:

*Click here to view the embedded video.*

Part 2 is here.

The hand we’ve been watching belonged to John Milnor. Its movements were made in January 1965, in a lecture theatre in Denver, Colorado, before a large audience of professional mathematicians from a wide variety of fields. It writes, draws, scribbles and gestures; its shadowy presence reminds us that mathematics is something we make with our hands.

The talk lasted about an hour, and this film shows more or less everything he wrote. We see symbols that look somewhat like algebra, perhaps the kind of thing most of us expect maths to look like. But we see other things too: diagrams involving arrows, representational sketches and sentences written in something close to ordinary English. These different codes are not isolated but cross-refer: together they map out a sort of conceptual territory.

What was being charted in this talk was a new and mysterious country to all but a handful of those present: the field of differential topology. It had been invented by Milnor and his contemporaries over the previous decade. In 1956 he had proved a surprising result about high-dimensional spheres using radical new techniques, and it was these that he sought to describe to his audience.

The field Milnor had invented was a new branch of topology. Topology studies how spaces can be connected together; not just real, physical space but any space that can be imagined. Topologists can study high-dimensional spaces, spaces that are curved in seemingly impossible ways, spaces that are radically discontinuous, or that have degenerate points where strange things happen. Topologists ask: just what could space be like, and how could we describe it?

It’s a qualitative and global subject. The spherical topology of the Earth’s surface, for example, can only be appreciated from a distance. Up close, it might just as well be flat. Topology stands back and looks at the whole space and asks questions like: Does it have holes in it? Or: Starting from one point, can I find a path that leads to any other point? If I walk around the space long enough, will I find I’ve come back to where I started, or perhaps that I’ve been reflected from left to right, as if in a mirror? What kinds of pictures can I draw inside this space?

These are qualitative questions in the sense that they don’t seem to ask for numerical answers. They have nothing obviously to do with measurement or calculation. They ask about the properties of a space, not lengths, angles, areas, volumes. This is what separates topology from geometry.

In fact, at the beginning of the subject in the late nineteenth century this focus separated topology from almost all fields of maths. Existing techniques weren’t very obviously applicable to it, and it got a reputation for being a very exotic and rather specialised subject. Yet for this very reason, topology can be learned by anyone: mathematical expertise really isn’t needed, or even very helpful.

What Milnor was introducing to assembled members of the Mathematical Association of America was an application of calculus to topology. This was an intrinsically strange idea. Differential calculus studies only “local” phenomena: the things that happen very close to a single point. What’s more, it had always been part of geometry: it measures and calculates. It seems designed to ignore the very qualitative, global properties of space that topologists were interested in.

Calculus, though, was a much older and more mature set of mathematics. It had its roots in the physics of Isaac Newton and was developed into a fully-fledged theory by French mathematicians of the early 1800s. Throughout the nineteenth century it had evolved in parallel with science and engineering, becoming a sophisticated language that could express theories of electromagnetic fields and the statistical behaviour of gasses. By 1915 even the most abstract parts of geometry had found applications in Einstein’s general relativity. These tools were well-understood and widely taught.

But nobody thought they might be useful for topology. The spirits of the two endeavours seemed diametrically opposed. Yet there were precursors. In 1931 Georges de Rham had proved that the behaviour – measured by calculus – of a process in a small region around a point depends on the topology of the whole space: features like holes, even if far away, exert a mysterious influence. The qualitative structure of the whole space produces detailed, local phenomena.

Milnor and his contemporaries turned this around, asking what calculus could do for topology. It turned out it could do a lot.

For all the richness of the text Milnor produces, you may notice that there are only a few numbers, no more than you might see in a talk on any topic, and there are no complicated calculations. This is not merely because the talk is intended to be an overview. Modern mathematics is at least as much about concepts as it is about numbers. Problems that can be solved by mere number-crunching or algebraic juggling are easy; even before the invention of the computer, individuals with great reserves of patience and meticulous attention to detail tended to solve such problems fairly quickly. They may have interesting consequences but they aren’t usually intellectually thrilling in themselves.

The culturally important face of mathematics is almost always conceptual, in the literal sense that mathematicians invent concepts and connect them together in ways that make new ways of thinking possible.

Constructing such heavy machinery is hard work; part of the appeal, of course, is that it lightens subsequent labour. A famous contemporary of Milnor’s, Alexander Grothendieck, also favoured this approach:

The unknown thing to be known appeared to me as some stretch of earth or hard marl, resisting penetration. . . the sea advances insensibly in silence, nothing seems to happen, nothing moves, the water is so far off you hardly hear it. . . yet it finally surrounds the resistant substance.

An ocean of abstract mathematics was, on this analogy, capable of eroding away the most obstinate of difficulties. Although Grothendieck’s machinery was at least as hard for his contemporaries to learn as Milnor’s, it too showed its worth by producing an understanding of geometry that seems impossible without it. Perhaps it is no coincidence that one of its central concepts, the structure known as a “sheaf”, is a device that explicitly connects the local and global aspects of space.

The early twentieth century was dominated by the rise of formal logic: the philosophical legacies of Frege, Peirce and Russell. In mathematics David Hilbert’s formalist project cast a long shadow, and in France the Bourbaki group set to the task of rewriting the entire subject in a language of dry austerity. They were suspicious of appeals to intuition and their books – even books on geometry –shunned illustrations of any kind, as if putting aside childish things. Their mathematics was above all symbolic: a modernist mathematics, obsessed with its own purity. It retreated into a state of pristine isolation from everything else, turning firmly inwards and becoming accessible only to a tiny elite.

It was here that the central task of the mathematician was defined as the making of *proofs*.

The post-war years saw a flowering of visual and spatial mathematics. It has not led to the abandonment of symbolic methods. Rather, a polysemous practice has emerged in which various codes coexist and interact. Milnor produces this practice in his lectures spontaneously and, I suspect, largely unconsciously; it’s just how the subject looks to him. Much later he said:

I think most textbooks I have written have arisen because I have tried to understand a subject. […] I have a very visual memory and the only way I can be convinced that I understand something is to write it down clearly enough so that I can really understand it.

Although he says he “writes it down”, he connects this with a “visual” rather than “verbal” form of cognition; perhaps this is not surprising when we see what he means by “writing”.

Many maths textbooks read like this, especially those dealing with specialised topics. They are like conceptual portraits: images of the territory of the subject that the author has managed to form by her own idiosyncratic process. Each reader needs to form their own map, and each will be different: this is why maths books are not to be read like novels, or even like works of philosophy.

In the distant past, a mathematician spent much time engaged in laborious calculation: there are tales of seventeenth and eighteenth century discoveries that were only made possible by almost superhuman feats of computation. Such tasks can now be carried out by the phone in your pocket, and their status as “mathematics” has fallen very low indeed. Though we still make schoolchildren learn to carry out long division by hand nobody really does it that way, and many of us struggle to imagine why it’s still on the curriculum. Certainly such drudge-work is unlikely to strike a mathematician as particularly mathematical.

The change of focus from calculation to proof was a response to two things: a rise in the importance of formal method and a decline in the importance of computation. The latter was accelerated enormously by the appearance of ever more practical and powerful calculating machines.

But as the twentieth century went on, some proofs began to emerge that were incredibly long and complicated. Famously, the proof of the Four Colour Theorem was produced using a computer that made billions of calculations; no human being could ever read and understand every step. Even proofs produced entirely by humans became so complicated they could turn out to contain very obscure but serious flaws. Andrew Wiles’s original proof of Fermat’s Last Theorem is one example – he managed to fix the problem once it was discovered. In 1999 Vladimir Voevodsky discovered a severe mistake in one of his results that had been published years previously and used by many other researchers; this, as he puts it “got him scared” and led him to consider how the extremely subtle, complicated proofs of contemporary maths can be better organised and managed.

And today we may be on the brink of another revolution. Computer software already exists that can check a mathematical proof for correctness, although the process is currently unwieldy and time-consuming. In my lifetime, I have no doubt that it will improve vastly, to the point where highly complex proofs can be checked rapidly. Perhaps, too, the same timeframe will bring us software that can make its own proofs; work is already well underway in that area, too.

This will not just be a matter of greater convenience or efficiency; it will completely revolutionise our idea of what it means to do mathematics.

If the pocket calculator rang the final death knell of the mathematician as athlete of arithmetic, perhaps the computerised proof system will bury the old joke that a mathematician is “a machine for turning coffee into theorems”. Perhaps we will turn this around and come to think of proofs as something best left to machines. If so, maths will become more qualitative, descriptive, intuitive, synthetic, conceptual.

It’s already all of these things: what I mean is, these aspects may come to predominate over proof just as it, in its turn, predominated over calculation.

The following saying is attributed to the geometer Federigo Enriques: “It is a nobleman’s work to find theorems, and it is a slave’s work to prove them. Mathematicians are noblemen.”

Today, of course, remarks about noblemen and slaves may strike us as rather ill-considered. He probably said it, if he did at all, some time around 1900, when the tide was turning against him and proof-making was becoming the primary activity of his field: so his contemporaries and immediate successors would have found it jarring, too. Perhaps in our present century it will be recast, and people will come to say: It is a computer’s work to prove theorems, but an artist’s work to find them.

*(Part 2 will be published on Monday.)*

We’re halfway through the *Three Pillars of the Digital* programme, and yesterday we ran a workshop on hardware architecture emphasising the physicality and embodiedness of the digital world. Students began by dissecting some desktop PCs that were slated for recycling (kindly on loan from IT for the afternoon).

The point was in part to simply see what’s inside: modern consumer electronics devices strongly discourage you from such activities, at least if you want your warranty to stay intact. Furthermore, desktop PCs still have big enough components that you can dismantle them and identify the parts.

By the end everyone in the class could explain the differences between RAM and disk storage, the roles of the northgate and southgate on a motherboard and why even a desktop a computer needs a tiny battery inside (it’s to keep the CMOS ROM alive). One PC had a sophisticated graphics card (probably it came from a media lab) and quite a few groups managed to take their hard disks completely apart, discovering why they’re called “disks”.

We also spent some time looking at how software is built to run on a CPU’s instruction set, and the idea of high-level languages that provide a human interface to the bare metal. A few of the students in the class had recently completes our crash course in Processing; we hope a few more will be able to take it up next term.

There’s power in knowing the jargon and how it fits together even if you don’t intend to build your own computer. The “digital” and — even worse — the “virtual” can be obfuscatory terms, creating the sense of a mystical otherworld (“cyberspace”, as it used to be called) created and ruled over by a technocrat class whose secrets are incomprehensible to the uninitiated. It’s easy to foget that all this is mostly just electricity flowing through wire and silicon. And while there are engineering wonders in the humble desktop PC, there is nothing conceptually difficult about how it works.

This was the very first workshop we planned when we initially put the *Three Pillars* programme together, and it symbolises our whole approach this term: “opening the box” as a gesture of curiousity, playfulness but also a little bit of defiance. Many people feel disempowered by the opacity of contemporary technology and this approach is designed to lift the veil, if only a tiny bit, and encourage further peeking.

*Three Pillars of the Digital* is an experimental programme of one-off workshops for BA and MA students in the School of Fine Art, generously sponsored by the Teaching and Learning Fund. So far we’ve run sessions on data, networks and architecture open to all students and advanced sessions (with a programming prerequisite) on image manipulation, OO programming and twitter bots. Our final beginner session will cover algorithms and we have advanced sessions coming up on Arduino, video and maths for computer graphics.

For our upcoming series of workshops, *The Three Pillars of the Digital*, we needed a quick introduction to writing Twitter bots. The result was Markov Duck, a sort of parody of Twitter bots that nevertheless embodies some useful tricks and techniques.

Markov Duck combines three simple Twitter bot themes in a completely arbitrary way:

- Text generation using Markov chains
- Word replacement
- Algorithmic image generation

The first is done using a Markov chain built from Hermann Melville’s epic *Moby Dick*, Robert Barnwell Roosevelt’s *The Game-Birds of the Coasts and Lakes of the Northern States of America* and Thornton W. Burgess’s *The Adventures of Poor Mrs Quack*. The first of these is much longer and so dominates, but occasionally a random quail or Mrs Quack makes a welcome appearance.

This kind of text generation (though not necessarily this exact technique) is used by bots such as @lowanimalspirit, @MythologyBot, @str_voyage, @oliviataters, @thinkpiecebot and Allison Parrish’s The Ephemerides, about which she observes:

both space probes and generative poetry programs venture into realms inhospitable to human survival and send back telemetry telling us what is found there. For space probes, that realm is outer space. For generative poetry programs, that realm is

nonsense.

The second theme is very simple: it replaces whales with ducks. This is a simple matter of string manipulation that can easily be done in almost any language. Many simple Twitter bots do this and it’s a good thing to try as a first project.

The third is realised by some messy code with a lot of random parameters; it still isn’t very good. It proves a point about how to go about the mechanical task of generating an image procedurally and getting it onto Twitter, though, which is the key technique used by such popular bots as @mothgenerator (algorithmic botany), @pixelsorter (glitch), @veilbymist (abstraction) and @tiny_star_field (a sort of minimalism).

Twitter bots using these techniques have become more or less clichéd; we suspect they offer far more possibilities than have been explored so far.

Our “Write a Twitter Bot” workshop runs on 17 November and will cover the basic code and libraries needed to do what Markov Bot does, aside from the image generation (which graduates of our Processing course already know how to do).

35 “Participation should be guided by the principle that all students should study the mathematics they need for the future.”

The Smith Review, originally commissioned by George Osborne at the March Budget of 2016, was published on Thursday 20 July, the final day before parliamentary recess. Its headline remit was to review how to increase mathematics participation in England after GSCE, including considering the feasibility of making mathematics compulsory to 18.

There are a interrelated set of concerns here (paragraph numbers refer to the review):

- England is an international education outlier in having such a low percentage of students study mathematics to 18. Nearly three quarters of those achieving a good pass at GCSE drop maths at 16.
- Politicians and policy wonks are concerned that England’s low maths skills contribute to recent poor productivity and that there are new demands not being met, particularly around the quantitative skills needed to make the most of ‘big data’: “Higher levels of education and skills raise productivity
*directly*by enabling individuals to accomplish difficult tasks and address complex problems. They also raise productivity*indirectly*by facilitating technological diffusion and innovation. Mathematics and quantitative skills are a vital component of this. Furthermore, analysis of student performance in international mathematics and science tests from 50 countries over 40 years shows that higher performance is significantly and positively related to growth in real output per capita.” (§80) - Universities are concerned that courses which have some mathematical content (135,000 students enter courses with “medium” maths demands) are accepting students who have done no mathematics for two years.
- Worse: “International studies show that around 10 per cent of all university students in England have numeracy or literacy levels below GCSE level, indicating a major numeracy challenge, and suggest that this is often not resolved at the point of graduation.” (§103)
- In general, there is concern about the absence of statistics from the school curriculum with post-16 being the most likely place where it can be embedded. In relation to university study and workplace needs, the Smith Review concentrates on the ‘primary need’ for the ability to analyse and interpret data.

“By age 18, every young person should have secure numeracy – the ability to use and apply basic mathematical knowledge to make decisions and engage in society. Most should have gained, and retained securely, the fundamental mathematics needed to thrive in the modern workplace – for example, the ability to analyse, interpret and present quantitative and statistical information and reason with data.” ( §56)

** **Smith decided that making mathematics compulsory was not currently achievable:

- 223 “There is not a case at this stage, however, for making it compulsory:
- the appropriate range of pathways is not available universally,
- teacher supply challenges are significant
- and it is unclear when sufficient specialist capacity will be in place for universal mathematics to become a realistic proposition.”

In sum, currently we don’t have the qualifications or teachers to achieve the target. From Smith’s self-penned Foreword: “my clear conclusion is that we do not yet have the appropriate range of pathways available or the capacity to deliver the required volume and range of teaching.” Instead there should be an ambition to have mathematics become universal by 2030, ‘by changing expectations and culture by ensuring a more appropriate range of pathways’.

Besides that headline, the review was mostly well received – particularly since the government has pledged £16million to address the review’s concerns about how funding reforms to Alevel (limiting each student to three funded Alevels) were affecting Core Maths and Further Mathematics take-up.

The review itself did little to explain why it felt Core Maths was ‘essential’, beyond being the only initiative aimed at increasing understanding of statistics and the application of mathematics. The limitations of Core Maths for increasing participation are clear. The six qualifications currently available under this rubric are pitched to natural and social sciences: “Core maths offers the opportunity to apply mathematics and statistics to examples from economics, sociology, psychology, chemistry, geography, computing, and business and management.” (§116)

From the perspective of Fine Arts Maths Centre, too little attention was paid to the need to expand the range of pathways so that the mathematics offered is more relevant to those choosing options in the arts, humanities and design. Although the review repeatedly stated that it meant mathematics “in the broadest sense”, the only topics referenced here were statistics, data analysis and numeracy. If we are serious about identifying the mathematics that students will need for the future, we will need to be doing more than trying to argue that all students need better statistics skills.

Smith constructed a defence of current policy measures around quantitative skills but more fundamental questions need to be asked. Firstly, the problem is clearly that A levels are too narrow: students in England are specialising too soon and dropping mathematics is part of that problem. Secondly, if you are going to keep the current post-16 framework then you have to offer mathematics that is closer to the legitimate subject and career interests of students. Since students are exercising choices at 16, we need to develop pathways that cater better to those choices. Running a PR campaign around the benefits of mathematics, is not going to work if the maths is dreary and seemingly irrelevant.

At FAMC, we teach over sixty students a year (from an undergraduate and postgraduate cohort of 400), many of whom gave up mathematics the first chance they got. They come back to mathematics when they realise that mathematics is broader and more interesting than what they were taught pre-16 and when they have projects – ideas for artworks – where the technical impediments can be resolved with mathematical approaches. We run short course workshops in geometry (Euclidean, projective and non-Euclidean), topology, infinity and programming and logic along with new courses planned for 2017/18 featuring basic mechanics, data and algorithms. These topics could form the basis of an alternative post-16 qualification.

The mathematics students need for the future is not just statistics. There is a rich tradition of mathematics that can be much more appealing to art and design students. Our submission to the Smith review emphasised the importance of logic and studying axiomatic systems such as Euclidean geometry. We believe strongly that “A systematic study of geometry provides a foundation for thinking about and working with space that the disconnected fragments that remain on the GCSE syllabus do not provide.”

Core Maths could be expanded to offer a broader range of options and shouldn’t just be left in the hands of the quantitative lobby. We need to listen to what mathematics the students might think they need and look at what mathematics fits better with their future studies and careers, not simply tell them to learn their stats. From our submission: “Topics we believe are relevant for many creative practitioners and industries in the twenty-first century include geometry, programming, group theory and symmetry, abstract algebra more broadly, graph theory, probability and game theory, linear algebra, topology and set theory.”

Last year we provided a submission to the Smith Review into post-16 mathematics, based on our experience running Fine Art Maths Centre for over 3 years. The five-page document is available here – it starts with some background on FAMC and then addresses the specific review questions.

Extracts:

If more students are to engage with mathematics beyond GCSE, there will have to be an offering that is relevant and appealing to them.

[…]

Students have not been exposed to a sufficiently broad range of mathematics and have a too narrow conception of the subject and, crucially, what it means to be ‘good at’ mathematics (which can mean a mastery of a broad range of calculation-based techniques with very shallow understanding). Almost all of school mathematics material is at least three centuries old, which often becomes an issue when dealing with issues and ideas thrown up by the contemporary world and, of course, when attempting to work critically and creatively with modern technology.

[…]

In particular, Art students lack familiarity with geometry and logic. Many would have benefited from being exposed to some ideas associated with modern mathematics; many find axiomatic systems and foundational questions appealing. A systematic study of geometry provides a foundation for thinking about and working with space that the disconnected fragments that remain on the GCSE syllabus do not provide.

[…]

We believe that more exposure to logic, broadly conceived, and its place in programming, deduction and practical reasoning would benefit most HE students, especially (but not only) across the arts and humanities.

[…]

We have found that, despite their absence from traditional curricula, the topics most relevant to art students often lie at the core of modern mathematics. Incorporating these topics in their study can bring multiple benefits: alongside domain-specific knowledge and technical skills, students learn transferrable skills in reasoning, problem-solving and abstract thought. They also learn what GCSE maths might not have taught them: that maths can be interesting and intellectually exciting, and is open to anyone.

[…]

For arts and humanities students, we recommend a focus on geometry, logic, reasoning and mathematical ideas (e.g. infinity and the nature of the real numbers). These may not normally be considered under notions of numeracy and ‘quantitative skills’ but the general relevance of logic and forms of reasoning and axiomatics represent a very suitable compromise for post-16 study.

[…]

We also note that such students often find the historical and philosophical background to some mathematical ideas helps them contextualise them and relate them to their other studies. This kind of material is usually conspicuously absent from existing pedagogical practice. Indeed, would-be artists and designers are currently being deprived of the intellectual underpinnings of important components of the history and culture of art and design. Euclidean geometry is central to Islamic design and Gothic architecture; projective geometry is intimately connected to linear perspective. No Home students appear familiar with these mathematical dimensions. Straightedge and compass constructions are touched on in GCSE mathematics but in deracinated and emaciated form. We also find that students who have been taught some ‘perspective’ only have a smattering of foreshortening techniques and no familiarity with the geometric principles underpinning Renaissance innovations in depiction. This is not always the case with international students.

Lauren Maxey is a graduate of the BA Fine Art programme who studied FAMC’s introductory programming course. In this post she introduces the work she exhibited for the degree show, ‘Murmuration’.

This work explores the alternate familiarities of natural existences both abstract and scientific, exploiting how science has codified our existence to make it comprehensible, making a scale and in doing so, forgetting natures unfathomable scale and mystic, my work addresses this neglect.

‘Murmuration’ is a combination of scientific research and abstraction, accentuating the allure of natural phenomenon’s, exploring and representing how starlings flock, through the subjects of infinite space, microcosm and macrocosm.

Physicist Giorgio Parisi explains that starlings flock in a scale-free correlation, increasing their perception range of their neighbours. Every bird moves in relation to the seven birds around it, so on and so forth, like a magnetic reaction.

Coding on ‘Processing’ allowed for an interesting oxymoron narrative, representing the unpredictable habit of starlings through an instructive language. This code is an implementation of Craig Reynold’s and Daniel Shiffman’s Boid program, simulating the flocking behavior of birds. Each boid steers itself based on rules of avoidance, alignment, and coherence.

The poetical dimension of growing crystals onto organza material was mimicking the theme of correlation, every crystal growing in reaction to its neighbour, spreading and increasing in size.

The ambiguity of the salt sculpture allows its existence to be malleable for the viewer, existing as a naturalistic symbol the starlings exist on. A delicate meditative aesthetic, blurring between a microcosmic representation of sky, sea and landscape.

Biomorpha (Evolving Structures) is a digital interactive installation created by Neus Torres Tamarit in collaboration with computer scientist Ben Murray as her final project for the MA Art and Science. It explores evolutionary interactions between an organism and its environment.

A digital lifeform resides within a landscape of evolutionary pressure to which it is well adapted, until audience members enter its environment. Captured by a Kinect camera sensor, the location of each audience member is treated as a new evolutionary threat to which the lifeform responds through a series of selected mutations, reaching for the audience in order to be well adapted to their presence.

In the private view and event held on Saturday, Neus invited Aimee Dulake and Jessie Richardson, two contemporary dancers from the London Contemporary Dance School, to interact with Biomorpha in a complex and evolving dialogue:

*Click here to view the embedded video.*

As part of the project, Neus produced acrylic and bioplastic sculptures combined with holographic effects of the Biomorpha forms. Neus started learning how to program in 2015 and FAMC’s introductory programming course helped her build the skills needed to develop early versions of the software, and to then collaborate successfully to complete its more technical aspects.

In October, thanks to funding from the Mercers’ Award, Neus begins a residency at Dr Max Reuter’s laboratory in evolutionary biology at UCL, developing artworks and workshops that explore the topics of evolution and genetics. About this, she says:

“Our objective is to be true to science, but using artistic mechanisms, which directly links with my artistic practice of creating artworks that bridge the gap between genetics and popular culture and that evoke curiosity and wonder about the mechanisms of life; I want to synthesise the concepts into mixed media installations that create an emotional response so that people react to the subject of genetics as a human experience.”

*Pictures by José Ramón Caamaño. In the pictures: Aimee Dulake and Jessie Richardson during the performance; Ben Murray and an audience member*

*This is a guest post by graduating MA Art & Science student Nicolas Strappini — see more of his work at nicolasstrappini.com.*

I have recently completed the MA Art and Science course at Central Saint Martins. For my degree show, I wanted to present work that mapped and visually traced a variety of processes including oscillations, Lichtenberg figures and pendulum movements using a variety of mechanisms like harmonographs and Wimshurst machines. My practice involves finding ways of visualising mathematical concepts and the nature of physical laws, from electromagnetism and sound to elementary particles. I have been researching and selecting different types of natural phenomena that can be described using equations.

**MAXWELL’S EQUATIONS AND LICHTENBERG FIGURES**

I applied to show my work at Imperial College as part of the Center for Doctoral Training event. I displayed some pieces that are direct visualisations of static electricity (Lichtenberg figures, see below). During my time at the college, I spoke to MRes student Jeevan Soor about my works. He spoke to me about Maxwell’s equations and how they help to describe Lichtenberg figures. I wondered if the toner dusting process had been used in forensic science and he mentioned that footprints are recorded using an electrostatic lifter. Forensic scientists use a device that generates static charge, and the charge draws the dust from the print on to the black plastic.

I have been exploring the possibilities of using electricity as an artistic tool. Through using a Wimshurst machine, I have been charging up plastic surfaces with static then dusting powders on the surface, thus visualising the invisible Lichtenberg figures left in the plastic. I then exposed the patterns onto photopolymer plates, resulting in works that are visually similar to the piece above. The works are direct visual representations of electricity.

**TUNING FORK DRAWINGS**

I demonstrated and recorded sound oscillations. This is a recording of sound oscillations on a sooted glass plate. One of the two prongs was equipped with a metal tip. I also used the tuning fork on a zinc etching plate. (below)

**KEPLER’S LAWS DRAWN IN SULPHUR**

The artwork below depicts different phases of the Belousov-Zhabotinsky reaction (2016). The zebrafish is a model organism for pattern formation in vertebrates. First found in chemicals in dishes, (Belousov-Zhabotinsky) then in the stripes and spirals and whorls of animals, Turing patterns are everywhere. Perhaps these patterns extend to ecosystems and galaxies. My plotting electrode and its graphical depiction of Kepler’s laws (image above, 2017) is also a visual representation of Turing inhibitors because the electrode is constantly turning on and off – hence the zebrafish texture.

I’m interested in making links between processes, using the micro to explain the macro – for example, Lissajous figures drawn in sand could be illustrative of Lissajous orbits – the orbital trajectories of planets. My work unravels like Ariadne’s thread, proceeding by using multiple means and attempting exhaustive applications of logic.

Some of the processes are mathematically chaotic in nature, and leave behind a fractal pattern. The idea of chaotic patterning is fascinating and may seem contradictory – one pendulum may represent chaotic motion, the other harmonic – the Lichtenberg figures are chaotic discharges, but may also display self-similarity.

I’m interested in the idea of the mechanical prosthesis between the artist and the art – the work being able to describe something of the natural world. The performative aspect of the work also takes the form of scientific demonstration to be able to describe something about the inventor or discoverer of the equipment or process I am demonstrating.

The delineation of time is also important – simply through visual analysis, the individual strokes of some of my pieces can be given time stamps. The marks produced by plotting electrodes change in reference to its speed – the same can be said for the tuning fork works.

How do these small (Wimshurst machine) and giant (the Large Hadron Collider) technological devices help us to understand the physical universe on different scales?

The relationships that connect this world together are mysterious, indeed, why do these relationships exist? Why and when does mathematical structure appear? Is it that the structure of physical laws is transmitted from a solitary point – the symmetry that becomes diminished and scatters as the universe unwinds itself to the viewer?

This term, for the first time, we used some video in our introductory Processing class. This came from requests from students; I admit I hadn’t really played much with this part of Processing.

I was actually pretty impressed by how fast it is. It even works, up to a point, for carrying out real-time transformations on a live video feed. Today’s experiment was simple enough: it takes the feed from the webcam on my laptop but scrambles the pixels around according to a mathematical transformation.

We’ll definitely do more of this sort of thing next term,in part of course because there’ll be another intro class. But also, as part of the Three Pillars of the Digital, we’ll be running a workshop on the most useful maths for programming visual work. That includes the modulus operator and the trigonometric functions sine and cosine, all of which are in the mix here. They help us break the tyranny of straight lines that the organisation of a computer’s visual display tends to impose.

This code is parameterized, allowing the warping algorithm *itself* to warp in response to the position of the mouse. This makes for an interactive experience that’s positively addictive. It’s even more fun when, in addition to waving things at the webcam and moving the mouse, you get to intervene at the code level too.

This particular experiment is really just a bit of idle fun, but I’m half-thinking about ways to hook it up to something more responsive. I’m also wondering whether my new phone will have enough power to do this kind of thing in VR. But another lesson is that coding can be be playful and exploratory. I probably only continued with this because the effects made me smile.

Anyway, below are some more screen captures from my “warped” webcam, and at the bottom is the code so you can try it yourself. Let me know if you make something cool with it!

_______________________________________________________________________

```
```import processing.video.*;

Capture cam;

int camWidth;

int camHeight;

boolean newFrame=false;

float[] xCosArr;

float scalar = 1.0;

float offset = 0.0;

void setup(){

size(displayWidth, displayHeight);

cam = new Capture(this);

cam.start();

camWidth = cam.width;

camHeight = cam.height;

xCosArr = new float[width];

initXCosArr();

}

void captureEvent(Capture cam){

cam.read();

camWidth = cam.width;

camHeight = cam.height;

newFrame = true;

}

void draw(){

if (newFrame){

warp();

}

}

void warp(){

setParams();

loadPixels();

for(int y = 0; y < height; y = y + 1){

float sinVal = abs(sin(float(y)/scalar)*cos(scalar));

for(int x = 0; x < width; x = x + 1){

color c = cam.get(int(x*sinVal) % camWidth, int(y*xCosArr[x]) % camHeight);

pixels[y*width + x] = c;

}

}

updatePixels();

}

void mouseClicked(){

saveFrame("##########.png");

}

void setParams(){

scalar = map(float(mouseX), 0, float(width), 0.1, 350.0);

offset = map(float(mouseY), 0, float(height), -10, 10);

initXCosArr();

}

void initXCosArr(){

if(scalar == 0){

scalar = 1;

}

for(int x = 0; x < width; x = x + 1){

xCosArr[x] = abs(cos(float(x)/scalar)*sin(scalar));

}

}