On Turbulence: in between mathematics and performance

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On Turbulence: In between mathematics and performance
Telma João Santos
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Journal of the Performing Arts, 19:5, 7-12, DOI: 10.1080/13528165.2014.958347
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7PERFORMANCE RESEARCH 19·5 : pp.7-12
http://dx.doi.org/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 Turbulence
In between mathematics
and performance
T E L M A J O Ã O S A N T O S
■■ Photo Filipe Oliveira
ISSN 1352-8165 print/1469-9990 online
© 2014 TAYLOR & FRANCIS
PR 19.5 On Turbulence.indd 7 13/11/2014 10:34
Downloadedby[TelmaJoãoSantos]at13:0209December2014

8
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 = 0
Ut + 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
PERFORMANCE RESEARCH 19·5 : ON TURBULENCE

9S A N TO S : O N T U R B U L E N C E
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

10
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.

11
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
S A N TO S : O N T U R B U L E N C E

12
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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