Tinkering with Curves
Tinkering with Curvesexplored curves as mathematical and material, as abstract and anthropological, through hands-on basketwork. Participants were invited to “‘Make a curve in space”; “Explore how materials make decisions on their own”; “Plait in different planes.” “Work alone, or in pairs….” This was interspersed with sharing ideas and hands-on projects, discussion of papers, and simply engaging with the materials and the library in the workshop.
This two-day ‘practorium’ took place at the Byre Theatre in the University of St Andrews. The Byre acted as our studio, a rich learning environment ofnatural, recycled and resourced materials, books – a library for individual research, drawing paper and pencils, charcoal, crayons, pens, paints. And tools. We worked independently and together – people said: “We worked in conversation with others.”
Broad themes that arose through the process included… ‘What are curves?’ ‘What is mathematics?’ and ‘What does anthropology have to do with all this?’ Specific discussions included; Continuities between mathematics and craft; Is mathematics everywhere? Patterns and rhythms; Singularities; Drawing and shadows.
Our first activity, ‘Make a curve in space’, led to discussion about positive and negative aspects of curves – friendly; convoluted; learning curves…
Johanne Verbockhaven drew, and drew, and drew. She drew shadows, movement, colour. She said she let her hands move ‘into the shadows’ when she was beginning a curve. Michel Serres talks of how ‘nature’ is full of curves, there are no certainties. A pure circle, he says, does not exist. Nor is it possible to have a static curve. Generative processes have curves in them.
Looking at Johanne’s gestural drawings, we asked, Is there a difference between a finished curve and drawing it? For example, is drawing a curve representational, while curving a basketry material such as a willow rod something different? The willow curve is an outcome of a force, an expression of energy between material and maker. Yet the drawing is dynamic … Or is drawing, too, the force of a movement or gesture? Can there be a curve without movement?
Perhaps there is a meeting point – a continuity as Ricardo might say – between drawing and making, between two activities of different orders? Both, after all are gestural…
And where do diagrams come in?……
Memory curves. In basketry, it would appear that curves have memory, in that the willow holds its form once bent into shape, even if it slightly unfolds. A curved willow form ‘expresses the memory of what has been done to it,’ said Victoria Mitchell. Yet, as Mary Crabb (see her thought pieces on looping and knotting) commented, basketmakers use the varying strength of the curved inside and outside of a willow rod to give strength to a structure. Different curves give different strengths, so there is a meeting of forces in basket making where the outcome can be expressed in curves.
Ricardo’s paper(see full text) addressed the theme of where the continuities between mathematics and craft practices might lie. He began discussion of the paper by suggesting that we treat mathematics as if it is somehow a more perfect, unperceivable expression of the world than we experience in our daily lives. As if ‘how things are’ is not even visible. As if there were an everyday world and a mathematical one, so that most bodily skills, from dance, to basketball, to basketry, all seem to be separate from maths.He proposed that such a perspective on mathematics rather parallels the notion of ‘mind-body split’, where mind links to the mathematics, and the body links to actions such as craftwork. – Yet, this perspective, he said, is a cultural invention. For example, we could ask the question… ‘When is a curve mathematical?’ – Or to consider whether a curve is, of its nature, mathematical?
For Ricardo, explaining how one curve relates to another is what makes it mathematical, while for Tim (Ingold) it would help to specify a ‘mathematical take on a curve’. For example, to consider whether the ‘mathematical take’ is to do with analogous thinking. See. for example, D’Arcy Thompson’s approach, where he identified common principles underneath structures such as skeletons and bridges, or different kinds of spiral, for example.
Ricardo’s view wasthat mathematicians are doing a craft. Diagrams, equations, are physical things, and there is mathematical making too. Maths is a creative instrument…
‘So, when I say that, for example, we all know that mathematicians are all the time working with diagrams, with equations, with symbolic expression. Well, all these diagrams are physical things, are things that are drawn also, are looked at from different angles. And the same drawings that you make for the shadow of the shape… Also, things that are in three dimensions, they do the diagrams in too. So, I am trying to say that (if) the work of craft consists of making curves, then this is mathematical making as well. So, I think that….For example, we think of how to use different materials, how to create different products. So, there is something mathematical about how to weave them together in a different way. So, the creation of something new, probably replicating something that exists and making a series productions is not necessarily mathematical But, mathematics, we can think of it as a creative instrument, or many instruments, that allow us to develop new ways of weaving, new ways of using materials.’
For Victoria, the question then arose as to whether mathematics was a form of basketry. As opposed to basketry being a kind of mathematics? For her, what was interesting was the way that mathematics uses words like ‘braid’ or ‘net,’ or ‘weave’, or ‘loop’.
This brought us back to Ricardo’s ‘continuities’, where one could not necessarily say, ‘Well at this moment basketry becomes mathematics’, but rather that mathematics works on a continuum, where specific kinds of entities or practices meet.
Michelle questioned the notion of continuities further, by asking about arithmetic, and whether there was a difference between arithmetic and mathematics, or if is there a continuum between them too? This was an interesting question, producing an interesting response, linked to drumming and rhythm. For example, as Ricardo illustrated, – drummers need to be able to count in order to learn new rhythms. Counting up to 4 is easy, but counting the beat while you perform it is a whole new world. You might count on- and off-beats, repeating these, and so on… This has to do with arithmetic because arithmetic is counting to a large extent. But it is strikingly different (and difficult) trying to generate the rhythm and count it at the same time. So, counting may always be embedded in the continuum, as is music, butas a learning musician, one is always learning to discriminate aspects which make the output musical.
Pattern and reasoning
Our discussion about reasoning was short, but it brought up the question of pattern. For Lucie, mathematics was always a method… “It’s logical reasoning. Do you reason through touching things or can you just do it in the mind? “You get a good sensation through things that join which are correct…”
Or is there a problem with ‘correctness’, ie reasoning ‘correctly’, in that it can ‘finish’ things which stops exploration? And if maths is closed, then it’s limited…
But then is it logical reasoning which allows us to see patterns?…
This raised the question that a computer can reason, but it cannot grasp the infinite. Which perhaps makes the case for bringing feeling into mathematics?…or not separating reason from how something feels. Above everything, rhythm has to carry on or make connections.
Curves and singularities
There is a critical point, we learned, at which overall behavior changes from one side to the other, and in mathematical terms, this is known as ‘a singularity’. Elsewhere, slightly to one side, is a point of continuity. A lot of the mathematics of curves is concerned with identifying singularities. For example, a point of inflection is a kind of singularity. If you bend a piece of willow into an s-shape, if you curve a piece of willow from left to right and from up to down at the same time (see drawing), and hold the centre point still, the point of inflection is the point where the willow goes up in a curve and from the same point also goes down.
A singularity could also be a crease in a piece of paper…Or a Gothic window which is vertically straight, and then moves into a curve
Where does the anthropology come in?
Tim Ingold took us through this question which Stephanie had been puzzling on, since people have often said, since the start of the project, that they can see where maths and basketry link, but less obvious is what is anthropological about their connection. For Tim, anthropology asks philosophical questions about the nature of the world by engaging in conversation with people. So, if mathematics and philosophy are closely related, then could we have a ‘mathematics with the people in it’?
But then mathematics has parallels with arts, music, anthropology… it’s a vocation. So maybe it is generative, in that it generates forms and ideas. And maths is an aspect of the generation of forces, relations, growth. Maths is in the world we inhabit so there can’t be maths without movement, force, growth, tension and material. Stasis doesn’t work. Can there be a curve without music? Or a maths without tension? It is hard is to have maths without some kind of force. Maths entails a sense of enquiry, a relationship between the intellectual and intuitive, in this regard it is very close to music, something which is lost if you try to pin it down. We want to keep it open-ended.
So, asking ‘Where’s the anthropology?’ is like posing a false problem. Like asking whether skills are more creative than improvisation. A false problem has a solution already in it. Real problems are generative and produce new answers.
Mary Crabb’s thought pieces
Those attending Tinkering with Curvesincluded Stephanie Bunn, Ricardo Nemirovsky, Geraldine Jones, Mary Crabb, Hilary Burns, Victoria Mitchell, Tim Johnson, Michelle Feder-Nadoff, students from the University of St Andrews anthropology department, students from MMU and members of the University of Aberdeen KFI Project. Thanks to the University of St Andrews KE&I fund and the University of Aberdeen KFI Project for supporting this event.