Figure 1.5.1 Hexagonal structure of benzene. Image courtesy of the Institute for Figuring.
Margaret Wertheim
Figuring is a word with deep interdisciplinary resonances in mathematics, literature and science; and as an activity it encompasses diverse histories and contexts: from textile makers weaving patterned figures with jacquard looms to courtly dancers spinning dynamical figures across a ballroom floor. The human figure, long a staple of artistic representation, is now a locus of constant measurement as we count steps and calories to figure our physiology. As cognitive beings, of course, we are continually figuring things out. In this essay I want to focus on the act of making figures that instantiate scientific and mathematical principles, a practice-based methodology that lies at the heart of the Institute for Figuring (IFF), a Los Angeles based organization I co-founded and direct with my twin sister Christine Wertheim. The IFF – its acronym being the logical symbol for ‘if and only if’ – is an enterprise dedicated to the aesthetic and poetic dimensions of science and mathematics (www.theiff.org). With my background in physics, and Christine’s in literature and philosophy, the institute was born from our entwined desire to explore processes of figuring as a way of thinking beyond symbolic form.
Although Western thought has long privileged symbolic modes of representing information – the mathematical equations of physics, the DNA code of molecular biology, the binary codes of computation – we suggest that the insights illuminated by material figuring extend traditional disciplinary approaches and often reveal surprising facets of knowledge. In our work, we explore how sign systems can manifest in concrete figurative forms. For instance, fractals can be constructed out of business cards, platonic solids can be woven from bamboo sticks, hyperbolic surfaces can be crocheted, tessellations can be cross-stitched, the projective plane can be knitted. Even the logic underlying digital computing has a geometric analogue that can be represented by a three-dimensional network. Not infrequently, abstract relations have correspondences in material objects that lend themselves to concrete play, and to a consequent playing with ideas. As a practice of making concrete figurative forms, figuring calls attention to the wisdom of embodied objects, whose qualities are not merely reducible to, or predictable from, descriptive codes.
Partly stimulated by Friedrich Froebel’s revolutionary nineteenth-century ‘kindergarten’ system of pedagogy (Brosterman 1997), with its focus on tactile geometric construction, another inspiration for our practice has been the field of chemistry. As the study of atomic assemblages, chemistry is inherently combinatoric; there are only a hundred or so atoms but these can be arranged in an infinite variety of molecules. Not everything, however, is possible, for various laws, patterns and regularities assert themselves. To articulate such relationships, chemists have developed a variety of symbol systems, including a lexical notation for specifying any particular compound as well as a graphical notation for representing the arrangement of its atoms in space. Take the case of benzene, a vital organic molecule made up of six carbon atoms and six hydrogen atoms. Benzene’s chemical formula is C6H6, and its graphical representation is shown in Figure 1.5.1.
Figure 1.5.1 Hexagonal structure of benzene. Image courtesy of the Institute for Figuring.
In figuring out how molecules work and how they can be assembled, chemists also make three-dimensional models to represent how chemicals occupy physical space. In the past, these were hand-crafted out of balls and sticks, for example Watson and Crick’s famous model of DNA, yet contemporary chemical modelling is mostly now done on computers, including state-of-the-art virtual reality set-ups, such as the CAVE where researchers can move around and through a simulation to explore the physical figure of their molecule (Cruz-Neira, Sandin and DeFanti 1993). The design of new drugs, for example, is largely premised on understanding the shapes molecules make and the specific shapes of the body’s receptors into which they must fit. A good deal of pharmacy is applied geometry and here function literally follows form.
At the IFF we aim to create circumstances where participants can experiment with similarly embodied activities, generating objects that delight the eyes and stimulate our haptic sensibilities while also illustrating formal sets of relationships. Such acts of figuring allow for creative exploration within a context of rules and constraints having their own internal logics. Interested in the dance between codes and forms, like chemists, we also seek insight by figuring out problems through the structures of our models. In one key project, our Crochet Coral Reef (Wertheim and Wertheim 2015), we examine hyperbolic geometry through crochet, an unlikely conjunction between mathematics and feminine handicraft inspired by a discovery by Daina Taimina, a mathematician at Cornell (Taimina 2009). In hyperbolic space (an alternative to the Euclidean space we learn about in school), geometric forms behave in novel ways: parallel lines can diverge while the angles of a triangle may sum to 0°. In Taimina’s models such theorems may be visually illustrated by sewing diagrams onto the woollen surface, thereby concretizing abstruse mathematical concepts (Wertheim 2005).
Geometrically precise, Taimina’s models are generated from a simple algorithm – ‘crochet n stitches then increase one’, where ‘n’ may be any fixed number. Additional stitches increase the surface area, generating the geometric opposite of a sphere. But what happens if we deviate from this rule? Let us say we increase one in every three stitches, then one in every ten? Here we no longer get a hyperbolically exact surface, for it is no longer geometrically regular. Just as there is only one sphere (an object with constant positive curvature), so there is only one pure hyperbolic surface (an object with constant negative curvature). By morphing the code and manually introducing deviations, we crochet Reefers move away from mathematical ‘perfection’ into a domain of organic possibility. In the Crochet Coral Reef project, we are indeed exploring negative curvature analogues of wonky spheroid forms such as the oblate shapes of sea urchins and the asymmetries of eggs. Nature itself has been playing with such forms in the frilly surfaces of corals, kelps and sea-slugs for hundreds of millions of years. Yet mathematicians, with their formalized rules, spent hundreds of years trying to prove that such structures were impossible.
Figure 1.5.2 Model of the hyperbolic plane constructed from hexagons and heptagons. Image courtesy of the Institute for Figuring and Cabinet magazine.
Across the academic spectrum, we see a growing interest in and sensibility towards embodiment and the qualities of material being. The philosopher of science, Evelyn Fox Keller, for instance, has noted the limitations inherent in the ‘master molecule’ theory of DNA and drawn attention to the dynamic role of the cell cytoplasm in embryonic development (Fox Keller 1995). Feminist science studies scholar Donna Haraway celebrates entangled webs of ‘sympoesis’ (a word meaning ‘to make with’), in which embodied critters together create and nurture environmental health (Kenney 2015). In mathematician Brian Rotman’s radical proposal for a non-Platonist account of mathematics, articulated in his book Ad Infinitum (Rotman 1993), he suggests that doing maths is itself an active embodied process. Much like the act of crochet, Rotman declares that even mathematics results from cognitive acts carried out by physical agents subject to physical limits, and thus, he says, there is no such thing as a perfect sphere or perfect straight line. For him, all mathematical objects – including numbers – are finite entities that arise when concrete actors construct them. Rather than a static and transcendent domain waiting to be discovered, Rotman considers mathematics as an evolving landscape of forms continuously brought into existence by communities of practising mathematicians.
By morphing a crochet code, and exploring the potentialities of a woolly DNA, we at the IFF have created communities of localized knowledge and expertise who branch out from geometric perfection to generate vast simulations of coral reefs. To date, more than 10,000 women in a dozen countries on five continents have collaboratively stitched a crochet ‘tree of life’. ‘Iterate, deviate, elaborate’ has been the motto for this handiwork, which now constitutes a globally extensive experiment in applied geometry and emergent algorithmic complexity. As life on earth begins with simple cells and evolves into ever-more complex forms, so our crochet forms have evolved. Figuring with our fingers – a literal digit-al technology – has opened a diversity of patterns that provide a yarn-based analogue for thinking through Darwinian ideas.
These crafted objects are underpinned by a DNA-like code (the pattern of stitches that can be written down in symbols, much like the symbols articulating molecules). Yet the code is not wholly determinant. When figuring with materials, the properties of substances impress themselves on structures, causing chains of consequence that often cannot be predicted in advance. A form figured in stiff acrylic thread might stand pert like a stony coral, but constructed in silk it might flop like a piece of kelp. Real figures – as opposed to idealized mathematical ones – result not just from the codes and equations scientists use to describe them, but also from the qualities of their components. As in chemistry, matter matters, and the structural properties of molecules also result not merely from their chemical formulae but, critically, from physical interactions between their parts. This is why proteins are so hard to model on computers; here scientists must engage with the real-world physics of complex atomic interactions. As a methodology, our insight is that figures themselves come into being through acts of figure-ing that involve rules and deviations, material substances and dispersed communities of practice.
Brosterman, N. (1997). Inventing Kindergarten. New York, NY: Harry N. Abrams Inc.
Cruz-Neira, C., Sandin, D. J. and DeFanti, T. A. (1993). Surround-Screen Projection-based Virtual Reality: The Design and Implementation of the CAVE. Siggraph’93: Proceedings of the 20th Annual Conference on Computer Graphics and Interactive Techniques, 1993, pp. 135–142.
Fox Keller, E. (1995). Refiguring Life: Metaphors of Twentieth-Century Biology. The Wellek Library Lecture Series at the University of California, Irvine. New York, NY: Columbia University Press.
Kenney, M. (2015). Anthropocene, capitalocene, chthulhucene: Donna Haraway in conversation with Martha Kenney. In H. Davis and E. Turpin (Eds.) Art in the Anthropocene: Encounters among Aesthetics, Politics, Environments and Epistemologies (pp. 255–270). London: Open Humanities Press.
Rotman, B. (1993). Ad Infinitum . . . The Ghost in Turing’s Machine: Taking God Out of Mathematics and Putting the Body Back In. Palo Alto, CA: Stanford University Press.
Taimina, D. (2009). Crocheting Adventures with Hyperbolic Planes. Wellesley, MA: A. K. Peters.
Wertheim, M. (2005). A Field Guide to Hyperbolic Space. Los Angeles, CA: Institute for Figuring Press.
Wertheim, M. and Wertheim, C. (2015). Crochet Coral Reef. Los Angeles, CA: Institute for Figuring Press.