There are two very interesting essays by this person, which land excitingly close to what I’m doing:

Why Knot? Towards a Theory of Art and Mathematics (2001) in Beyond Aesthetics: Art and the Technologies of Enchantment edited by Christopher Pinney and Nicholas Thomas

Reflections on Art and Agency: Knot-sculpture between Mathematics and Art (2006) in Contemporary Art and Anthropology edited by Arnd Schneider and Christopher Wright

In the first, Kuechler begins by noting Alfred Gell’s shift toward considering how art can be ‘thought-like’. Kant and Schiller come up again immediately, seeing art as a part of contemplative or active life respectively, and she describes these as spawning modernist art and anthropology’s contextualisation of art respectively. She talks about Gell’s *Traps as Artworks, Artworks as Traps*.

“In ths chapter I will critically address some of the fall-out springing from the modernist preoccupation with mathematics and the visualisation of spatial cognition which was born out of the nineteenth-century rediscovery and popular acceptance of axiomatic representation. An overview of the relation between art and mathematics will be followed by an exploration of topology, also known as knot theory, and its implication for an anthropological theory of art. Knot-theory models the behaviour of systems that have the capacity for self-organisation, for generativity and autonomy – of systems, that is – whose behaviour cannot be explained by simple and identifiable laws and of which image-systems outside of the Renaissance are as much an example as the weather. Knots are, as Alfred showed, prone to become artworks for reasons which are pointed out by mathematics, yet which support Alfred’s theory of ‘abduction’: it is binding which exemplifies the coming together of affective and cognitive processes, enabling ‘syntonic’ learning and the ability to think with objects. Knot-theory, moreover, promises to offer a model which can help us to get away from the dated mechanical model of reproduction, often associated with the term ‘template’, which has hindered anthropology from recognising that style may indeed be independent of culture (Gell 1998: 160).” p. 58-9

Geometry and Relationality

Alfred Gell apparently suggests that “modernism arose out of the exploration of the implications of the reality of n-dimensional worlds for representation” p. 59. She refers to the Yupno of Papua New Guinea who “use different reference systems at the same time in ways that are only comprehensible when apprehending spatial conception as decentered from the body in everyday life” (from Wassman 1994). She refers a lot to Duchamp after Gell. Kuechler discusses string figures and the wane of visual focus in anthropology, contrasting this with new visual techniques in computer modelling in mathematics and art which gave rise to eg. hypersolids (developed by the artist Brisson in 1978, apparently). Some good stuff here about mathart and Bridges types.

“Like most mathematically inspired work which came to represent the modernist spirit, the imagery of hypergraphics is generative and multiple.” p. 61

The possibility of ‘seeing’ an object in two, three, or four space, she describes as decentering spatial conception / the human body.

Knot-theory and the Topology of Cultural Form

Gell used the ‘Wen-diagram’ from topology to analyse Umeda dance and Marquesan art. Nonlinearity comes up.

“Mathematics found itself thinking of the knot as this kind of object that is likely to touch people’s lives in connecting with personal and ‘affective’ aspects of thought and thus as forming the springboard for associations that are both abstract and concrete in their mimetic capacity.” p. 63

O rly?

“For long, the understanding of how objects carry abstract ideas into the head has been obscured by what has been labelled the ‘projectionist fallacy’ which proposed that objects are the product of the externalising of concepts that are pre-existent to those objects (cf. Davis 1986).

[…]

Long before the full impact of technology led to a re-examination of the theory of the unconscious and its implication for an understanding of memory (cf. Rosenfield 1992), the mathematician Papert (1980: vi) offered a new approach to the cognitive validity of representations. He reminded us that we do not start with concepts, but think with objects, both in childhood and in everyday experience – by using them as image in thought, and maintiaining in thinking the properties that these objects would demonstrate in the physical world.” p. 63

Papert talks about his experience with gears and differentials – great! Carol Strohecker in the 1980s ran a school thing called the ‘knot lab’ in which children had dialogues and debates about ways to think about knots.

Towards an Ethnography of the Knot

Kuechler describes some properties of mathematical knots, refers to Alexander the Great cutting through the knot, underlines the importance of negative space around the knot (knot complement).

“As the knot is contained within the negative space created by patterned surfaces, it lends itself to be applied to the conceptualisation of sculptural form. And, as it is prone to retaining its geometric properties under deofrmations, the sculpted know enables intellectual economies to unfold around a polity of images.” p. 65-6

I like that. RT. She talks about Brent Collins and making wood into the stretchable rubber knot form.

“Such examples give credence to the suggestion that the knot epitomises what Levinson (1991) called ‘knowledge technology’ responsible for externalising non-spatial, logical problems in a distinctly spatial manner. Associative or inferential thought provoked by the knot may thus condition spatial cognition because of the textured and deformative properties of the knot.” p. 71

A knot is not referential but synthetic. The chapter refers to Malanggan, New Ireland, Andy Goldsworthy, Valeri on the Hawaiian king and a lot more.

Great! Not the easiest read for reasons of sentence structure, but useful nonetheless.

The second essay is shorter and uses some of the same material. She focuses on Brent Collins and John Robinson.The latter’s work came to be used as a teaching aid and so she refers to it as a ‘tractor’ of thought.

Art and Mathematics: New Perspectives on an Old Theme

Here she traces a history of mathematics and art from the Renaissance.

“The anthropological investigation of art and mathematics has been hindered, rather than helped, by the seemingly ubiquitous presence of mathematics in cultural activity […] Anthropology has developed approaches to mathematics as [art of wider concerns with cognition, mapping and navigation. Yet these studies tend either to exclude artefacts as data suitable for analysis or approach atrtefacts as evidence for the reconstruction of technique.” p. 89

The Knot: Generative and Non-Linear Forms in Art and Mathematics

Kuechler looks at Victor Vasareli and the links etween knots and DNA, then John Robinson and his influence on mathematician Rinald Brown, and Carlo Sequin (a Bridges regular)’s sculpture generator.