Hearing Things and Dancing Numbers: Embodying Transformation,Topology at Tate Modern

Can we hear geometrical and topological shapes? Is it possible to dance numbers? How can we grasp topological shapes and mathematical ideas through our senses of sound and movement? These were some of the questions explored in a recent sound sculpture and dance performance event at Tate Modern that Brian Rotman and I conceived as part of the Tate Topology speakers’ series. With the so-called turn to embodiment and the issue of affect, the role of embodied sensory expression and impression is important for understanding ‘thinking’ outside the rubrics of cognitive processing, discourse or representation. For philosophy, traditionally vision has been privileged as the most ‘noble’ of the senses – on the basis of the distance it gives from corporeal contamination, unlike touch for instance. But with topology the mind’s eye loses its grip, it could be said, yielding to the possibility of auditory, gestural, kinetic and other forms of conceptualization. While a Möbius strip can easily be made by twisting a length of paper, a Klein bottle cannot be perfectly produced in the three dimensions of Euclidean space. The mathematical imagination leaps further than the places anyone can visualize. This is evidenced by the fact that a significant number of geometers and topologists, including the great Euler, have been blind (see Jackson, 2002). ‘Embodying Transformation’ was concerned with how topological relationships and mathematical concepts exceed the bounds of the visual, but by embracing rather than eschewing the senses.

The distinctive feature of the sound sculpture was to explore ‘acoustic space’ and the spatial dimension of sound. ¹ An auditory image or gestalt can be described in many ways – a melody or a rhythm, the timbre or tone of a musical instrument or voice, or what a particular sound, a creaking door for instance, denotes. But with Knots and Donuts discrete sound sources were programmed to travel all around and through the listeners who were immersed in a three-dimensional auditory field filling the entire East Room of the Tate Modern (see Figure 1). Before the start, the audience were briefed to expect what the travelling sounds would describe (as a sparkler does in the dark). These were the geometrical shapes of a cylinder, sphere and figure of eight; and the topological figures of a Borromean Knot and torus (or donut). The sounds themselves consisted of various rolling balls, a roulette wheel, a bee buzzing and what was often interpreted as a rumble, swoosh and crash of an ocean wave. The audience were invited to sit, lie down or wander around in the near-darkness as they pleased. OPEN IN VIEWER

The specialist hardware and software allowed the location and movement through space of a discrete auditory source to be controlled quite precisely. ² This had the effect of converting sound – most often a diffused ephemeral effect – into a form of graphic expression – an ‘audiographic’ medium, that is, writing with or in sound (as distinct from the phonographic writing of sound). But this audiographic line, unlike one on paper, was drawn in three-dimensional space. At the pre-performance programming stage, tracing out its path as visualized on a screen with a retro computer game joystick felt like constructing a wire frame sculpture. The movement of sound sources across space (as with a marching band, or a carnival float) is not very often considered within the repertoire of musical or sound art composition techniques. But with Knots and Donuts sound became a sculptural material, in the way that wood, metal or marble is normally considered to be.

One inspiration for the sonic sculpture was the avant-garde music tradition of Varèse, Stockhausen, Xenakis, Nono and others who treated the sound spatialization as a compositional element. Another was Bernhard Leitner’s sound sculptures (see Leitner, 1998). But most important was the sonic engineering of the hugely powerful outdoor Jamaican dancehall sound systems (see Henriques, 2011). Here, the engineers often make a point of exploiting the circular travel of sounds around the triangular configuration of the three stacks of speakers. These sound effects, often incorporating gunshots, sirens and the like, serve as the signature of a particular sound system. In addition, Jamaican and other music producers often talk of ‘building’ a riddim (rhythm) track. So this idea of spatialization of sound and the consequent sonification of space could be said to have travelled from Trenchtown to the Tate. ³

For the Ordinal 5 dance, the audience returned to their seats to view the performance on the stage area they vacated. The mathematical concept danced was a commuting diagram for Ordinal 5, that is, the number five as counted in a sequence (see Figure 2). Rotman’s approach was distinctive in that it was the choreographic expression of a concept, rather than a geometric shape as such. ⁴ The danced expression of the mathematical concept of Ordinal 5 requires: a minimum number of six movement directions (or dancers), a specific number of positions in the space (on the dance floor) and sequencing in time of coincidence between dancer and position. The dancers all start on the same spot, and as long as they each get to their next assigned position and finish at the same spots, the mathematical concept can be said to have been expressed – to exactly the same extent as would be done with an equation or drawing with a conventional pen on paper diagram (see Figure 3). OPEN IN VIEWER

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Figure 3.

Dancers in Brian Rotman’s Ordinal 5. Photo courtesy of Douglas Moody.

The material from which both the sonic sculpture and the dance are built is embodied movement. With Ordinal 5 this was the choreographed movement of the six dancers, including their vocal and facial expression and gestures, readily recognized as transforming their bodies through space and time in the dance on the stage area. With Knots and Donuts the movement was that of the sound source across the 3-D auditory field as well as at the more micro scale of the periodic motion of compression waves themselves. The point made here is this: while actualization is dependent on some mark, gesture, sound or other noticeable difference, it is independent of which particular medium of expression embodies the transformation. ⁵

One reason for interest in this kind of event is methodological, as an experimental investigation. Audiences for the five performances were most forthcoming in the Q and A and informally afterwards. While sound alone is not particularly good for pinpointing location (and is typically used in cooperation with vision), the sound sculpture’s novel demand for attentive listening was experienced as enjoyable. The auditory impression of the sound shapes proved to be quite a robust phenomenon. Several remarked on their being at the centre of the sound field and feeling its depth around them – rather than having sound frame a stage or screen at some distance from their point of listening. Another comment was that the naturally appropriate sound samples of a roulette ball, for instance, made locating its circular path through space quite easy. In addition, for several listeners the ‘ocean wave’ sound evoked deep associations of childhood seaside memories specifically, as distinct from the more general kind of association that a listener might have for a particular piece of music.

The most important point to be made from the sound installation and dance, however, concerns the issue of embodiment for topology and mathematics itself. ‘Embodying Transformation’ was inspired by the idea that mathematical thinking should be conceived as an accomplishment of an enminded body, doing what bodies do, such as making gestures or touching things, rather than any purely abstract processes (whatever that might be), generated by a mind isolated or even opposed to material extension of the actual world. Such a line of thought – entirely contra to the orthodoxy of how most mathematicians might understand what they are doing – is what Brian Rotman has been establishing through several monographs (see Rotman, 1987, 1993, 2000, 2008). Rotman develops an argument that hinges on the distinction between, on the one hand, notational media that depend on metaphor, similitude and language system, and, on the other, capture media with their metonymy, synecdoche and analogue variation (see Rotman, 2008: 42). Embodied minds and enminded bodies then underpin what Rotman describes as:

A psyche that is at once porous, heterotopic, distributed and pluralised, permeated by emergent collectives, crisscrossed by avatars and simulacra of itself. In short, a para-human agency which experiences itself as an ‘I’ becoming ‘beside itself.’ (Rotman, 2008: 134)

The challenge of this idea of mathematics as an embodied activity is addressed not only to conventional ideas of the mathematical mind, but also to those about bodies themselves. Once the conventional divided subject of mind/body has been dethroned, or decentred (Adlam et al., 1977), then all manner of exciting possibilities are opened up. Topological bodies concern relationships rather than identities, qualities rather than quantities, subjectivities and objectivities at the same time, enfolded insides and outsides, together with pasts, futures and presents – all in transitions and transformations. As one slogan put it: ‘Occupy the Future’. ⁶ The implications of this lead to the consideration of embodied, situated and social ways of knowing, the nature of knowledge itself as techné and phronēsis (as discussed in Henriques, 2011), as distinct from more formal text-based epistemologies by which academic research still tends to define itself. ⁷

Topological generalization – sacrificing the measure and angle of Euclidean geometry for the invariance of relationships that survive transformation – has been taken as evidence for the fundamental, even innate, nature of topological relationships rather than their abstract character. Jean Piaget argued that the child has a topological concept of space before he or she develops the conventional idea of Euclidean space. ⁸ According to the mathematician Alexei Sossinsky: ‘the blind person who regains his sight does not distinguish a square from a circle: he sees only their topological equivalence’ (Sossinsky, 2004: 13). Steven M. Rosen goes further, calling for a phenomenological topology, drawing on Husserl, Heidegger, Serres and others. Quoting Sheets-Johnstone (‘Topology … is rooted in the body’; Sheets-Johnstone, 1990: 42) and Connor (‘No matter how abstract it may become topology remains fundamentally bodily’; Connor, 2004), Rosen uses this topological body as a critique of the ‘categorical separation’ of classical cognition, for which ‘the axiomatic base serving as its unquestioned point of departure is the self-evident intuition of object-in-space-before-subject’ (Rosen, 2004: 12)..

The idea of invariance in topological transformation is indeed particularly useful for undermining such traditional ideas of consistency as being based on objects and their properties. Rotman uses this to develop the idea of a ‘quantum self’ that

exists as a co-occurrence of virtual states, an ‘I’ which becomes actual or ‘realised’ and fixed as an ‘objective’ whole precisely when it is observed, subjected to psychic measurement or social control, or otherwise called upon to act, respond, be affected, and project agency. Such an ‘I’ would be a mass of tendencies, an assemblage in a perpetual state of becoming, rather than a monolithic being. (Rotman, 2008: 135)

It is also interesting to note that this and the idea of topological embodiment appears to be consistent with a Deleuzian conception of the body – which would most often be entirely antithetical to phenomenology. Brian Massumi in his account of the ‘body topologic’ puts it like this:

The problem is that if the body were all and only in the present, it would be all and only what it is. Nothing is all and only what it is. A body present is in a dissolve: out of what it is just ceasing to be, into what it will already have become by the time it registers that something has happened. The present smudges the past and the future. It is more like a Doppler effect than a point: a movement that registers its arrival as an echo of its having just passed. The past and future resonate in the present … The past and future are in continuity with each other, in a moving-through-the-present: in transition. (Massumi, 2002: 200)

Topology thus comes to be about bodies, but perhaps not quite as we usually think we know them. This is not the conventional idea of a body as something or somehow without mind, an inert object, the Cartesian res extensa, lump of flesh, or sack of organs. Rather, it is the enminded body. So, on the other side of the false dichotomy, the topological body has little to do with the mind as res cogitans, or the abstract disembodied faculty of mathematics. The conclusion I draw from ‘Embodying Transformation’ is that the greater our sensitivity to the senses, the greater the sense of embodiment for topology.