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On Turbulence: In between mathematics and performanceTelma João SantosPublished online: 24 Nov 2014.

To cite this article: Telma João Santos (2014) On Turbulence: In between mathematics and performance, Performance Research: AJournal of the Performing Arts, 19:5, 7-12, DOI: 10.1080/13528165.2014.958347

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7PERFORMANCE RESEARCH 19 ·5 : pp .7-12ht tp : / /dx .do i . o rg /10 .1080/13528165 .2014 .958347

Turbulence is a multidirectional and multilayered concept, defined in several fields of study and which may be roughly understood as a chaotic-in-appearance – albeit deterministic – reaction to changes, especially on high-scale settings. In this article I will focus on two fields: mathematics and performance art, which I will relate through the ideas of remediating concepts and using multimedia tools. I argue that we may construct any performance art piece as a turbulent flow, which may be seen as a new object that arises from research on how we may perceive the concept using different approaches.

One of the first ideas that comes to mind when relating mathematics with performance is to apply geometry or some direct physical – in terms of body or space – translations of how turbulence is defined and researched in mathematics to a specific performance piece. In fact, this article aims to change that ordering used in the idea of direct translation and to introduce the idea of a continuous order, which changes dynamically as the connections between definitions and concepts, as well as new possible derivations and points of view around them, also continuously change. Thus, one of the main goals is to add new knowledge to and to raise new questions in performance studies, especially regarding new intersections among disciplines, and also to add the use of mathematical knowledge as a tool within artistic processes. In order to do so, it is important to start again from the beginning, remediating classical concepts, considering them in new environments where some crossing points may generate turbulent states implying new concepts, as well as new connections among them.

There are four main directions that will be crossed in this article: remediation of concepts, the use of technology, software and the internet, performance studies concepts, and mathematics related to the study of Navier–Stokes equations, one of the directions of research on turbulent flows. In media and software culture studies the concept of remediation was introduced and developed by Jay David Bolter and Richard Grusin as a ‘way to complicate the notion of repurposing’, where repurposing is in this context referred to as ‘the use of the same content across different media forms’ (Bolter and Grusin 2000). In this article I refer to remediation as a concept (re)defined and (re)characterized within one field of study that is considered and studied as embedded within another field of study. In this article, I also understand multimedia as a tool for documentation purposes as well as a tool to theorize on the way we perceive concepts through several media forms with previously edited perturbations, as well as real-time ones.

The first section is dedicated to the introduction of some preliminary concepts from mathematics and from performance art related to turbulence. The next section is dedicated to the creation of a turbulent flow within remediation of some concepts (that will be introduced in the next section) from both mathematics and performance art. The final section will be devoted to the application of the turbulent flow within a concrete case study: the performance On a Multiplicity, created during the final year of my PhD dissertation in mathematics , which I present as an example of a multilayered, multimedia, multi-defined and multi-characterized experimental turbulent

On TurbulenceIn between mathematics and performance

T E L M A J O Ã O S A N T O S

■■ Photo Filipe Oliveira

I SSN 1352-8165 p r in t /1469-9990 on l ine© 2014 TAYLOR & FRANCIS

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flow. The turbulent flow, from the spectator’s point of view, may be seen as the movement of the final narrative of the performance, the creation of turbulent flows being one of the main goals of this performance.

O N M A T H E M A T I C S

Even though the word ‘turbulence’ was already used in the Old Testament (Ezekiel 5.2–12 for instance) to describe an unusual, fluid behaviour, only around 1500 did Leonardo da Vinci first present some sketches and a preliminary echnical definition of turbulence as a distinct physical phenomenon (see Richter 2008). New developments were presented only in the late nineteenth century, with works by Joseph Boussinesq in 1897, and the experimental (1883) and theoretical (1895) works by Osborne Reynolds, among others. In particular, Reynolds’ experimental results led to the discovery of what is currently known as the Reynolds number, the only physical parameter involved in transition to turbulence, considering a simple incompressible flow over a smooth surface. The author also concluded that turbulence was too complicated to be fully understood, and he proposed a random description of turbulence. At the same time, Henri Poincaré proved that some simple non-linear dynamical systems, which could exhibit a chaotic behaviour, were, in fact, completely deterministic. The analysis-based or deterministic point of view draws on the study of Navier–Stokes equations and their solutions, generating new ways of understanding turbulence. Several developments along the way led to the current and precise definition of turbulence establishing the ‘sensitivity to initial data’ as an essential requirement: ‘Turbulence is any chaotic solution to the 3-D Navier–Stokes equations that is sensitive to initial data and which occurs as a result of successive instabilities of laminar flows as a bifurcation parameter is increased through a succession of values’ (Chapman and Tobak 1985).

After this brief historical review, let me define some terms. A ‘flow’ is the continuous movement of a fluid – liquid or gas – from

one place to another. There are two types of flows: ‘laminar flows’ and ‘turbulent flows’. In a ‘laminar flow’ the molecules move smoothly, all in the same direction at a constant speed; in turn, in a ‘turbulent flow’ the molecules move in many different directions at different speeds. There are many examples of turbulent flows in nature and in daily life. One of the simplest examples of the transition from a laminar to a turbulent flow is the behaviour of water when it is heated: after a while the water starts to move constantly and it forms a laminar flow, but if we wait longer, bubbles start to rise from the bottom and the movement of the water becomes very complicated and not predictable. Water, like air, is a non-viscous fluid, but if we perform the same experiment with honey or syrup, for instance, we see that they tend not to become turbulent. Turning a fluid movement into a turbulent flow depends on the viscosity of the fluid: the more viscous a fluid is, the less it becomes turbulent.

This said, the following physical properties of turbulent flow are especially relevant to our topic: it is disorganized, chaotic and random-in-appearance; it is sensitive to initial conditions, which may also be understood as non-repeatable; it exhibits a broad range of length and time scales; there is enhanced dissipation in the mixing of fluids; it mobilizes three-dimensional space, is time-dependent, rotational and intermittent in both space and time.

Navier–Stokes equations were introduced by Claude-Louis Navier and George Gabriel Stokes by the mid-nineteenth century. In one of its simplest forms an incompressible flow of a fluid is defined as one that has constant transport properties, and the Navier Stokes equations can be written as follows:

∇U = 0Ut + U.∇U = –P + υ.ΔU + FB ,

U is the velocity vector; the equation ∇U = 0 relates to the incompressible flow; the left-hand expression Ut + U.∇U refers to the convective acceleration (the acceleration of an element of fluid); P is the reduced pressure; υ.ΔU refers to the viscosity of the fluid; and FB is a body-force

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term. The operator ∇ denotes the gradient operator, the vector that contains the partial derivatives, which are the velocities along each coordinate,1 and Δ denotes the Laplace operator that is the divergence of the gradient. That is, it is the sum of second partial derivatives to each of the coordinates. Under some specific assumptions, it is possible to prove that, if there is (at least) one solution to a Navier–Stokes equation, then it is a turbulent flow. Thus Navier–Stokes equations may be considered, under specific assumptions, as describing turbulent flows. These equations can be mathematically very difficult to work with and there are still some open problems regarding the existence of solutions in a three-dimensional case.2 Their relevance here, though, is to the main goal of this article, which is to propose a performance methodology that provokes the appearance of Navier–Stokes equations with turbulent solutions, and then to generate the appearance of action/reaction techniques in performance art that generate an understanding of turbulence.

O N P E R F O R M A N C E A R T

Turbulence has always been implicitly embedded in creative processes, but only recently have we witnessed the emergence of theoretical texts exploring concepts such as turbulence and methodologies that give value to certain subjective aspects of performance that while they appear random can be shown to exhibit a deterministic nature. Only recently has there been a determined effort, as later articles in this special issue demonstrate empirically to outline the role turbulence plays in the subjective experience of the performance.

According to Marvin Carlson, performance art may be seen as a field of work and study where

its practitioners do not base their work on characters previously created by other artists but on their own bodies, on their autobiographies, on their specific experiences in a given culture or in the world, that become performative in that practitioners are aware of them and exhibit them before an audience. (2004: 4-5)

As RoseLee Goldberg writes it may

take the form of a solo or group show, with lighting, music, or visual elements created by the performer him/herself or in collaboration with other artists, and be presented in places such as an art gallery, a museum, an “alternative space”, a theater, a bar, a café, or a corner. (2011: 9)

In this article, I also consider that, in the performance art context, a performance piece may be seen as any artistic object in which the consciousness of sharing/showing something is present, and the source of this awareness is the notion of performative action as coming also from daily social behaviour. That is, we may also approach a performance piece from the daily-life point of view regarding the performance itself and the way it is perceived by spectators, as well as by performers. As Erving Goffman writes,

the legitimate performances of everyday life are not ‘acted’ or ‘put on’ in the sense that the performer knows in advance just what he is going to do ... But [this] does not mean that [the person] will not express himself ... in a way that is dramatized and performed. (1999: 73-74) ,

So, I argue that a performance may be any action that is performed and that has been thought and prepared (choreographically or not), and which may be shared through several mediums: in real time and/or live streaming, and in several locations, from traditional places, such as a theatre or a gallery, to non-traditional ones, such as a bar, a corner of the street, among others.

1 For instance, considering the three-dimensional case, the operator ∇ applied to a function U is a three-dimensional vector containing the functions that determine velocity of U along each of the axes.

2 Richard Feynman described turbulence as part of the unsolved problems coming from physics, and still today researchers are trying to prove the existence of a solution for three-dimensional Navier–Stokes equations in general.

■■ Photo Tiago Frazão

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In 2000, Eugenio and Judy Barba introduced the concept of turbulence to describe ‘what appears to be a violation of order; in fact, it is order in motion’ (Barba and Barba 2000), 61. In the same paper, they characterized turbulence as a succession generated by ‘the vortexes that upset the current of narrative action’ (61). Their article offered a way of mediating between concepts normally presented oppositionally in a performing arts context – storm and meticulousness, catastrophe and density, coherence and confusion. Like Henri Poincaré before them, albeit in a different context, Barba and Barba assert that the apparent randomness that surfaces in performance is an effect of our intellectual dualism. Once these opposed properties are recognized as related aspects of turbulence, their deterministic nature emerges. The implications of this analysis are considerable. Turbulence now appears as one of the essential characteristics of performance. Performances arise from personal and artistic research-based questions and exist in a feedback relationship with the evolution and reappraisal of the questions and their solutions.

As Richard Schechner writes:

Performance studies is ‘inter’ – in between. It is intergeneric, interdisciplinary, intercultural – and therefore inherently unstable. Performance studies resists or rejects definition. As a discipline it cannot be mapped effectively because it transgresses boundaries, it goes where it is not expected to be. It is inherently ‘in between’ and therefore cannot be pinned down or located exactly. This indecision (if that’s what it is) or multidirectionality drives some people crazy. For others, it’s the pungent and defining flavor of the meat. (1998: 360).

Turbulence in performance is not simply the seemingly chaotic blurring of conventional boundaries between different states and feelings; it is the capacity to hold both in play, to maintain the play of ambiguity. New digital technologies, and new ways of sharing and disseminating material have lent turbulence a virtual dimension.3 As Barbara Kirshenblatt-Gimblett writes,

Performance as an organizing idea has been responsive not only to new modes of live action, but also new technologies ... [We need to] take issue with the assumption of human agents, live bodies, and presence as organizing concepts for Performance Studies ... If boundaries are to be blurred, why not also the line between live and mediated performance? (2004)

M A T H E M A T I C S A N D P E R F O R M A N C E

W I T H I N T U R B U L E N C E

In considering any possible intersection between the mathematics of turbulence and turbulence as a tool of performance research, the classical, linear approach to knowledge formation needs to be replaced by a model of knowledge that is rhizomatic and liquid. Over thirty years ago Gilles Deleuze and Félix Guattari (Deleuze and Guattari 1980) proposed to reconceptualize the social production of knowledge rhizomatically, arguing that new ideas were best imagined as sets of interconnections without centre or logical development. Rather than operate as an ordered set where we can define operations, such as the sum and the product, so as to connect different values and to generate new values, the new mode generated multiplicities. Similarly relevant to the challenge of handling a multiplicity of variables, any one of which can alter all the rest (in a seemingly random, although, in reality, deterministic way), is Zygmunt Bauman’s notion of liquidity used to describe the way we have become socially amorphous, no longer embedded in durable structures of thought but moving instead in a liquid way among them (Bauman 2000). Liquidity is a post-rhizomatic concept, which allows us to conceive new forms of interconnections on continuum settings.

The corollary of this is that any value variable or new concept (to speak in classical terms) exists in a set related to what Daniel N. Stern calls an intersubjective matrix:

our mental lide is cocreated. This continuous cocreative dialogue with other minds is what I am calling the intersubjective matrix (Stern, 2004: 77)

3 See website www.turbulence.org for examples.

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and which exhibits a rhizomatic-tending-to-liquid behaviour. Adopting this model of social production, a rhizomatic-tending-to-liquid approach to liaising between mathematics and performance involves constructing a continuum setting (or, at least, a setting that is dense in a continuum setting), where turbulence may arise. In this methodological frame, turbulence is inherent in the new intersubjective matrices generated by the pursuit of specific open questions and concrete knowledge. This new order that emerges in this way is not settled but dynamic, and changes continuously as new connections are continuously established. To study and characterize turbulence, then, we need to construct an intersubjective matrix of definitions, properties and approaches that, in their interconnections, allow turbulent states to be present. As already indicated, the aim is not to translate the Navier–Stokes equations into performance directions, but to devise physical experiments that generate the conditions of turbulence. The challenge is to establish the parameters or frameworks – the axiomatic preconditions – likely to produce turbulent flows, whether these are understood physically, psychologically or technologically, for in any performance art project, we have different levels of turbulent flows: those associated with the movement of the body, the movement of words, space, narrative, drawings, texts, multimedia, possible interactive tools, etc.

In a recent article (Santos forthcoming), a possible methodology for effecting this mediated and continuous translation between creative processes and mathematical notions is presented. The development of a performance, it is suggested, can be described in terms of the ‘Axiomatic Image’, ‘Sub-Images’ and ‘Dynamics’. An ‘Axiomatic Image’ has to do with the main concept of a specific performance art piece. It is not exactly the starting point for experimentation but something more abstract, a set of possible directions that consciously guide the creative process. ‘Sub-Images’ are concrete, three-dimensional but also dynamical and abstract. They occur when the ‘Axiomatic Images’ – the mathematical notions, together

with movement improvisation techniques – give rise to concrete ideas, or concrete images. Lastly, we consider the ‘Dynamics’, the narrative form of the work. Critical to this schema is the definition of ‘axiom’. An ‘axiom’ is a ‘proposition that is not proved, but considered either self-evident or subject to necessary decision. Therefore, its truth is taken for granted and serves as a starting point for deducing and inferring other (theory-dependent) truths’ (see Santos forthcoming).

Hence, if we want to look at a performance as a set of Navier–Stokes equations, describing, in this case, the motion of the performance along the time of its duration, we have to consider the ‘Axiomatic Image’, the ‘Sub-Images’ or the ‘Dynamics’ as initial conditions of those type equations. We know that initial conditions are important, since Navier–Stokes equations are very sensitive to them: if we change infinitesimally an initial condition, we can obtain turbulent, complicated, not-predicted flows. Therefore, despite the fact that an initial condition is axiomatic – and, thus, we take it for granted – it may also generate very turbulent solutions. Especially in the ‘Dynamics’, where axiomatic moments are mainly body-related, almost every moment, almost every movement, may be seen as an axiomatic initial condition of some Navier–Stokes equation within the whole performance. We may, then, perceive a performance piece as a set of Navier–Stokes equations, which describe the several flows within the performance, and that have turbulent solutions. These turbulent solutions start, obviously, as laminar flows but, subsequently, become turbulent ones, and then they become laminar again until some new laminar flow appears and the process re-starts.

In illustration of this methodology I conclude with a mention of a work of my own.

On a Multiplicity is a project in which I was involved in late 2010, when I decided to film myself regularly and for about a year in improvised movement in defined and restricted spaces in each of three houses where I lived. The rules were that a specific space of the house was set aside for the performance, that

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I should previously have done at least five hours of work related to my PhD on the Calculus of Variations and that the video recording should be made within ten minutes of finishing my work. Having made video recordings of motion improvisations – which I called Improvisations Series – using various movement techniques, such as Laban, techniques of real-time improvisation and composition, they also benefited from access to the lab Being Present/Making Present, supervised by Nicole Peisl (Forsythe Company) and Alva Noe (University of California-Berkeley), in August 2010, Frankfurt . Then I edited the videos, shared them using social networks and then reformulated them for public presentation. I also decided to record in video and sound the verbalization of the research I was developing in the Calculus of Variations, as well as some thoughts on this particular future performance. In its final form, On a Multiplicity was presented as a performance/installation, where real-time improvisation was combined with video and sound projection of the thesis of the research study I had carried out throughout 2011 and 2012.

On a Multiplicity mapped a multiplicity of self-representations from two fields usually seen as too different to be joined together. Its main object, in fact, was to test this claim, to question preconceived ideas about artistic creation and scientific research. The rationality of mathematical endeavour was recontextualized in the emotional milieu of improvisation: timeless abstraction was put in dialogue with embodied time. On the other hand, improvisation in real time was progressively distanced from the present of its performance as it was documented and the videos edited to meet different audience expectations. Finally, returning the process of documentation to the real time of self-presencing, there was the interference between recorded sound and live sound. The result, I submit, was a genuinely heuristic tool for investigating the phenomenon of turbulence as it might appear between the usually separated fields of mathematics and performance.

This work is financially supported by Portuguese National Funds through FCT – Fundação para a Ciência e Tecnologia – in the ambit of the project Pest-OE/MAT/UI0117/2014.

R E F E R E N C E S

Barba, Eugenio, and Judy Barba (2000) ‘The deep order called turbulence: The three faces of dramaturgy’, The Drama Review 44(4): 56–66.

Bauman, Zygmunt (2000) Liquid Modernity, Cambridge: Polity Press.

Bolter, David Jay, and Richard Grusin (2000) Remediation: Understanding new media, Cambridge, MA: The MIT Press.

Boussinesq, Joseph V. (1897) Théorie de l’écoulement tourbillonnant et tumultueux des liquides dans les lits rectilignes a grande section, Paris: Gauthier-Villars et fils.

Carlson, Marvin (2004) Performance: A critical introduction, New York: Routledge.

Chapman, Gary T., and Murray Tobak (1985) ‘Observations, theoretical ideas and modeling of turbulent flows – past, present and future’, in D. L. Dwoyer, M. Y. Hussaini and R. G. Voigt (eds) Theoretical Approaches to Turbulence, New York: Springer-Verlag.

Deleuze, Gilles, and Félix Guattari (1980) Mille plateaux – capitalisme et schizophrénie – Volume 2, Paris: Minuit.

Goffman, Erving (1999) The Presentation of Self in Everyday Life, Peter Smith.

Goldberg, RoseLee (2011) Performance Art: From Futurism to the present, London: Thames & Hudson.

Kirshenblatt-Gimblett, Barbara (2004) Performance Studies, New York.

Reynolds, Osborne (1883) ‘An experimental investigation of the circumstances which determine whether the motion of water in parallel channels shall be direct or sinuous and the law of resistance in parallel channels’, Philosophical Transactions of the Royal Society of London, 174: 935–82.

Reynolds, Osborne (1895) ‘On the dynamical theory of incompressible viscous fluids and the determination of the criterion’, Philosophical Transactions of the Royal Society of London 186: 123 – 64.

Richter, Irma A. (2008) Leonardo da Vinci notebooks, 2nd edn, Oxford: Oxford University Press.

Santos, Telma João (forthcoming) ‘On a multiplicity: Deconstructing Cartesian dualism using mathematical tools in Performance’, Liminalities.

Schechner, Richard (1998) ‘What is performance studies anyway?’ in Peggy Phelan and Jill Lane (eds) The ends of performance, New York: New York University Press.

Stern, Daniel N. (2004) The present moment in psychotherapy and everyday life, New York: W. W. Norton & Company, Inc.

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