WHAT IF
CONVEX
TIME
ENTANGLE THE TEXTURE
UNDO
INDEED
WHAT IF
IMMENSE ROOMS
EMPTY
LOADED
THE DRIFT OF THE EYES, A CHOREOGRAPHY OF ATTRACTORS
submitted to publication (May 2018)
ABSTRACT
This paper concerns the study of a choreographic piece, The Drift of the Eyes, under the lens of the mathematical concept of attractor, on a transdisciplinary landscape. This piece, conceptualized and interpreted by Bruno Senune, was constructed from a geometrical initial idea, that became more and more intricate, giving us, myself as a mathematician, performance studies researcher (as well as also a performer), and Bruno Senune as a dancer, choreographer and performer, the opportunity of exploring the process of integrating several disciplines, as well as several perspectives under the movement narrative being explored. I provide some mathematical background, as well as some thoughts on attractors within dance performance field, and I finally present an approach of connecting mathematics and dance through this piece, as a case study.
INTRODUCTION
To separate disciplines, fields of study and practices was a trend and an ideology to bring the concept of specialization into the table some decades ago. Even if it continues to be a possibility within many specific and specialized areas of study, we have been spectators of an actualization of knowledge, where dualisms are questioned, and identity multiplicities arise as object and subject matters within artistic creation. Including thinkers, philosophers, mathematicians, in artistic creation processes brings up new ways of conceptualizing and mapping actuality, as well as the possibility of informing academic research and generating new transdisciplinary studies.
As a mathematician and a performance studies researcher, as well as a performer myself, I have been dealing with some questions: “Which concepts and theories of mathematics I find in performance art or dance pieces?”, “Can a performance art or dance piece inform a mathematical research process?”, “How can I relate two different fields in a way that is concrete enough to discuss it?”. I have been constructing ways of perceiving artistic creation, as well as generating, through case studies – performance art pieces or dance pieces I participate as a writer and as an academic – relational models within artistic processes. I found some well-known concepts form mathematical analysis, as set, sequence, function, neighborhood, limit, continuity, quasi-continuity, to construct some new concepts, as limbs, cuts, axiomatic image, sub-images, dynamics, that help me in the individual process of relating myself with artistic creation, as well as some other mathematical concepts, as turbulence, for instance, that help me shape my discourse around what can be seen and what can be perceived in a performance art piece. These constructions can be found in (Santos, 2014a), (Santos, 2014b), (Santos, 2017a) and in (Santos, 2017b).
I started to work with Bruno Senune in his project KID AS KING, 2016, based on constructing a narrative generating collapsing expected meanings in transitions, where the excess of information, the power relationships established everywhere, the power of money, are present as objects and as attractors. These attractors are geometrical places, points, locations, that represent several perception and emotional states, that swallows the performer, forced to remain there until he finds out a way to escape, as in a computer game. Concepts as GIF, small isolated movements, inspired by insects and their apparent fragility. In The Drift of the Eyes, Bruno Senune wanted to explore the connection between being an artist nowadays, a geisha and her constrained movements and strict rules, and an ouroboro, eating his own tale. Also, in this piece, several attractors were constructed and defined to convey artist’s ideas within his own way of choreographing, including these moments of drift, where the unknown appears to force him to confront it, to deal with it, to surpass it, and to discover ways of not losing himself, and be somehow functional.
The word attractor is not an unknown one within everyday life, and it roughly refers to a point, a place, a circumstance, a moment, a set, that has a force pulling up the surrounding energy, metamorphosing it into a movement concentrated around the point, place, circumstance, moment, set. It is a well-known mathematical concept: in a discrete-time system, an attractor can take the form of a finite number of points that are crossed in sequence. In a continuous-time sequence, we can characterize it as subsets of the phase space (manifold composed of cotangent spaces) of a dynamic system. In performative arts, several authors and artists use the word attractor to mean the artist’s intention to attract audience or funders attention, but not introducing attractor as a concept and an idea within choreography.
First, I will introduce some mathematical background around attractors, define and characterize it within dance performance. Then, I present the dance performance piece The Drift of the Eyes, where attractors were an essential tool to create choreography, as well as to create theoretical discourse and meaning.
WHAT IS, IN MATHEMATICAL CONTEXT, AN ATTRACTOR?
One of the most known features of mathematics, or the process of studying and learning mathematics is accumulation. In fact, it is difficult to write about several interesting and funny concepts, results and theories, since most of the times we need to introduce first many other concepts, axiomatic, rules and grammar. It is a language: we need to get familiarized, to start learning the rules, axiomatic and grammar, then we can accumulate connections and eventually create new consequent intersubjective matrices. In this direction, to introduce the concept of attractor, we need first to define dynamical systems.
A system is an object studied in some field that can be abstract or concrete, elementary or composite, linear or nonlinear, complex or chaotic. Complex systems are composite ones generated from many subunits interacting and that are often composite themselves. It can survive to some losses along the way, adapting to change. Chaotic systems don’t have, most of the times, many interacting subunits, but they interact in such a way that they generate intricate dynamics. The behavior of these systems is apparently random, but it isn’t. In fact, it is generated by deterministic processes since the complexity is in the dynamical evolution and not in the system itself.
Systems have properties represented by variables: quantities having a range of possible values. The values that these variables take at an instant of time are called system’s state, which is often represented by a point in a geometrical space, or phase space. A trajectory of the system is a path through the phase space, giving the evolution of the system over time, and it describes/is a solution to the system.
A dynamical system is a system whose state, according to some rule or hypothesis, evolves over time. This evolution depends on the rule or hypothesis and on its initial conditions, the system’s state at an initial time. A system is deterministic if it is possible to uniquely determine its past and future trajectories from its initial state. An indeterministic system is one without a unique future trajectory, so that the evolution is random. It is often possible to tell whether or not a system is deterministic by inspecting the time series it generates, plotting states at different times. A chaotic system's time‐series has a fractal‐like structure, meaning that it looks the same at different scales.
An attracting set for a dynamical system is a closed subset A (a closed set is a set including its boundary and limit points) of its phase space such that for several initial states, the system will evolve towards A. As John Milnor writes in On the Concept of Attractor (1985), a closed subset will be called an attractor if there is some positive possibility that a randomly chosen point will be attracted to A and every part of A plays an essential role. This means that A cannot be split into smaller pieces. If the attractor is a point that does not move, it is known as a fixed point. An attractor describing a system that cycles periodically over the same set of states, never coming to rest, is known as a limit cycle. A system does not need to have just a single attractor; the phase space can have several attractors whose “attractiveness” depends upon the initial conditions of the system.
A strange attractor can be characterized by its chaotic dynamics, depending sensitively on initial conditions, diverging exponentially over time. There are some known strange attractor as Lorenz attractors, Hénon attractors, or Ueda attractors. For more insights on strange attractors see (Ruelle, 1995). Systems described by strange attractors do not end up in a steady state nor do they repeat the same pattern of behavior. Chaotic systems have strange attractors; complex systems have evolving phase spaces and a range of possible attractors.
ATTRACTORS WITHIN ARTISTIC CREATION
In dance field, artists have been bringing up to the stage several proposals around the concept of attractor: Attractor (2017), a multidisciplinary piece that is the result of a collaboration between Dancenorth, Lucy Guerin Inc, Gideon Obarzanek & Senyawa. This work intends to consider a set of dancers and musicians as an attractor, creating new patterns and new connection among them and with the audience, blurring boundaries and bringing together apparently distant subjects, turning perceptual environments into mystic experiences. Also, the dance piece Strange Attractors (2000), a dance piece with a departs from a set of music and dancers forming lines and then breaks them apart, as trajectories of chaotic systems.
In theoretical fields, mathematical concepts regarding dynamical systems were brought up within philosophy by Deleuze and Guattari, in Capitalism and Schizophrenia () and What is Philosphy? (), and to performance studies by André Lepecki, in () and () when referring to some actual artistic creation in relation with social-political globalized but yet disconnected environments surrounding it. But this is not done without misunderstandings, judgments and intricately misconceptions on approaching concepts outside some field using that same field’s tools as validating ones, as Alan Sokal and Jean Bricmont discuss in Fashionable Nonsense – Postmodern intellectuals Abuse of Science (1998), where they (apparently) dismantle several errors and abuses regarding the use of several scientific concepts, theories and results, as the case of Deleuze and Guattari when they use dynamical systems and attractor in their theoretical landscapes:
The main characteristic of the texts quoted (in this chapter) is their lack of clarity. Of course, defenders of Deleuze and Guattari could retort that these texts are profound and that we have failed to understand them properly. However, on closer examination, one sees that there is a great concentration of scientific terms, employed out of context and without any apparent logic, at least if one attributes to these terms their usual scientific meanings. To be sure, Deleuze and Guattari are free to use these terms in other senses: science has no monopoly on the use of words like “chaos”, “limit” or “energy”. But, as we shall show, their writings are crammed also with highly technical terms that are not used outside of specialized scientific discourses, and for which they provide no alternative definition.
These texts touch on a great variety of subjects: Godel’s theorem, the theory of transfinite cardinals, Riemannian geometry, quantum mechanics … But the allusions are so brief and superficial that a reader who is not already an expert in these subjects will be unable to learn anything concrete. And a specialist reader will find their statements most often meaningless, or sometimes acceptable but banal and confused.
We are well aware that Deleuze and Guattari’s subject is philosophy, not the popularization o f science. But what philosophical function can be fulfilled by this avalanche of ill-digested scientific (and pseudo-scientific) jargon? (1998, pp. 154-155)
This is the proof that it is problematic to jump from one area of study into another, especially from science to philosophy, without changing, or also jumping the methodologies used to attribute validity. It is dangerous to interpret and judge ways of using concepts without have a deep knowledge about the context in which they are being considered. The authors are mainly concerned with the use of intricate mathematical objects without a previous contextualization within philosophical context, as a criminal who stole an object without explaining why. They demand a moral and ethical explanation as the only possible way of not being condemned. We can think about a reversed way of approaching the debate, denouncing the bad appropriation of mathematical objects done by Deleuze and Guattari, but this leads us to Cartesian thought, which does not make sense anymore.
Also, when referring to science in artistic creation, we tend to settle the importance of justifying our choices, not undergoing the “unknown”, since it is yet to be defined, characterized and classified. In fact, if it is not possible to risk crossings between different approaches and even different fields, artistic and practical areas of theoretical and concrete production of knowledge, without risking the way we perceive methodologies and resulting objects. I claim that not only concepts can be appropriated, as well as the creation and/or research process need to be seen with different lens than the actual or known ones. We cannot pursue the unknown with known tools within strict known rules. To open and blur boundaries, to consider neighborhoods instead of fixed points, to engage in turbulence without fear of the unknown, to accept chaos and attractors as part of the family in a probably totally different perspective from the ones we are used to consider. That is, as Kathrin Bush points out in Artistic Research and Poetics of Knowledge (2009) within the context or art as research,
(…) a particular phenomenon in contemporary art, in particular in institutional-critique, whereby research is considered a part of the artistic process and is carried out by the artist herself. In this case, art is in fact a form of knowledge. It becomes the site of knowledge production and does not restrict itself to integrating previously known concepts. (p. 3)
New research landscapes are only possible under new postulates that can accept continuity and dynamics not only as mathematical concepts only possible under specific theoretical frames, but also as subjectively integrated part of the methodologies and processes considered.
THE DRIFT OF THE EYES, A CASE STUDY
CREATION PROCESS
The Drift of the Eyes started, as a collaboration between Bruno Senune and myself, with a dynamical image and a name. As Bruno said at the time: “I want to construct a periodic path (closed, by the way) with an energy of resistance next to desistance. A resilient path, eyes on a limbo. It is the drift of the eyes and it is a solo dance performance piece”. The music would be essential, as well as the construction of fragility/instability.
This dance performance piece follows Bruno Senune’s first piece, Kid as King, where a schizophrenic figure, a time bomb, an imminent explosion, a figure tied up to the back. The Dirft of the Eyes appears from the ecstasy, speech after speech, the drift of what remained. The figure, now tired, blind, slow, bored, repetitive, on a space that is constructing and establishing itself after an end, punctually psychedelic, punk, rocker, who wants to live.
The fatigue settles in and ritualizes itself. In circle, on a post-ecstatic state, following what already was, music as gravitational center of a series of GIFs constructing speech along repetition. The poetics of repetition leading, at the limit, to a poetics of the actual. The fatigue of the excess of images as narrative of an after-now. Through the blur of an image after its repetition, The Drift of the Eyes generates new ways of seeing. The recreation of fatigue from the repetition of movements-actions around a circular and ritualistic connection of boredom, a puppet of itself metamorphosing along the connections with music, as a co-narrative and an almost-liquidity landscape. An almost amorphous state transforming itself into a new narrative mapping a possible place.
Photo: Tiago Aguiart
The Drift of the Eyes, a crying out, an almost continuous lament, as a place of liberation, of purification of the self. A trial of return to a non-pain of life, being this possibility the only survival way. A lament to live, a poetic construction on chaos, impotence, destruction of desire, extreme fatigue and the path until a possible metamorphosis allowing survival. Researching ways to transform the body, the perception of the same and of what characterizes it, as a metaphor to other possibilities of understanding the body as an imagetic identity motor. The exposure of fragility and of a narrative of survival, the sadness we celebrate, the necessary change to continue walking on life, the constant limbo between equilibrium and fall, the relief within anxiety of impossibility.
ATTRACTORS
The Drift of the Eyes started also as a geometrical piece: Bruno initially wanted to have a musician at the center stage and to construct a ritual around him, as an attractor. In this case, a non-strange attractor. Along rehearsals, Bruno discovered that music had a different part in this artistic process, it would have to be specifically created for this piece. So, he was led to construct a space with several – four - attractors of different dimensions and from different levels.
Four attractors were defined along this piece. It started with an attractor at the center stage, the essential one, “the abyss area”, to where almost all the elements within this piece converge. At the beginning, there was a tear designed on the floor, that would be/represent the physical boundary, where Bruno would walk, trying to balance himself without falling or choosing to be in or out. Then, two more attractors came without problem: one at his left side front, “the getting stuck area associated to fear”, and the other one his right side from front to back, “the getting stuck area along transformation and sadness”. So, on his left side, an attractor was created taking into account that there would be no responsibility towards time and change, in the sense that it would be the first place where he would block and stay some time figuring out how to continue, with which energy and in what direction. The attractor on his right side has an interesting feature, leading us to name it a strange one: the fact that the attractor itself is a landscape, where rhythms, unexpected movements and velocities are explored within the chaos that characterizes that specific space.
Along rehearsals, Bruno discovered the last attractor, center stage back, being the blue zone, the place where he recovers energy to pursue the same path again and again, unifying and constructing strength. This last attractor was, in fact, present since the beginning, but was pointed out along the piece, metamorphosing from a virtual attractor on the wall to an essential space, or as Bruno call it, “the melancholic area”.
All the four attractors do depend on initial conditions and proposals, but not in an exclusive way: the path, despite choreographed, was developed using a random based chaotic methodology, using perception and creating isolated body sites connected strongly to exterior inputs, and also the so-present notion of unbalance. Nevertheless, initial conditions played a very important role in this piece, since “the abyss area” was the premise of being the uncontrolled controller we feel being a strange attractor present frequently in our neo-liberal contemporary societies. “The getting stuck area associated to fear” is related to the always present rule of confronting and questioning notions of braveness and courage today. “The getting stuck area along transformation and sadness” is the resulting connection to the idea of metamorphose, of creating a butterfly within the worst conditions and succeed, despite all. “The melancholic area” is the area that imposed itself as a need to breath in between paths, to engage into melancholy as a possibility of regeneration.
Despite the connection and dependence of initial conditions, the trajectory in The Drift of the Eyes can be defined in between a complex and chaotic dynamical system. On one side, it is made of intricate connections and isolated movements conveying the pursue of different ways of connecting fragility and virtuosity. On the other side, a constant element or risk and a real obstacle to enable the virtuosity to arise in its apparent full potential is introduced, to convey the overcoming/dealing but yet agonizing ways of perceiving meaning and concretize metamorphosing processes of “being there”.
FINAL COMMENTS
Connecting different fields of study within artistic creation brings some discussions into the table, especially in what concerns concepts that are at least relatively know, defined, characterized and classified in some specific field of study as, for instance, dynamical systems in mathematics. As a mathematician, perfomer and performance studies researcher, I began to give myself permission to use these three aspects wtithin my on-going research on the connections between artistic creation and scientific research. Also, I started to ask for permission to document other performer’s works and create new meanings and new reformulations regarding concepts.
Bruno Senune’s work is the perfect challenge in this sense, since it departs, even if in a subtle way, from geometrical configurations and intricate choreographies, where emotional states play an essential role within the piece. It is a transdisciplinary work and assumes a continuous search for authorship through boundary places between known and expected movements and their reconfiguration and recontextualization within different, occasional, or unexpected and chaotic ones, with a permanent subtleness along the way. The Drift of the Eyes is a perfect example to interrogate established ways of knowing and perceiving what is and when does happen a performance dance piece.
One of the most difficult parts of this research was connected with the mapping of this creation process on a documental-based frame as research but intending as a final result something else. In this direction, I should say first that I started to perceive and understand The Drift of the Eyes as a project, with several objects as possibilities of being part of this project. This paper is one of the resulting objects, relating geometrical and mathematical concepts that arouse at the very beginning of the process. I also would like to point out the importance of documentation, not only – or maybe not even – in a classical perspective, but as an essential part of the project, where creation is present in new layers, provoking also new paradigms on actual artistic creation.
REFERENCES
-(2009) Busch; Kathrin; Artistic Research and Poetics of Knowledge, Art & Research – A Journal of Ideas, Contexts and Methods, Vol 2, No 2, pp 1-7 .
-Deleuze, G.; Guattari, F.; Anti-Oedipus - Capitalism and Schizophrenia, University of Minnesota Press, Minneapolis, 1983.
-Deleuze, G.; Guattari, F.; What is philosophy?, Columbia University Press, New York, 1991.
-Lepecki, André; Singularities – Dance in the Age of Performance, Routledge, New York, 2016.
-Lepecki, André; “The Body as Archive: Will to Re-Enact and the afterlives of Dances”, Dance Research Journal 42, no. 2 (2010): 28-48.
-Milnor, John; On the Concept of Attractor, Comm. In Math. Physics 99 (1995): 177-195.
-Ruelle, David; Turbulence, Strange Attractors, and Chaos, World Scientific, 1995.
-Santos, Telma João; On a Multiplicity: Deconstructing Cartesian Dualism Using Mathematical Tools in Performance, Liminalities: A Journal of Performance Studies 10, no. 3 (2014a): 1-10.
-(2014b) Santos, Telma João; “On turbulence: in between mathematics and performance”, Performance Research 19, no. 5 (2014b): 7-12.
-Santos, Telma João; “Mathematics and Performance Art: first steps on an open road”, Leonardo Journal, JustAccepted publication (2017a). doi: 10.1162/LEON_a_01546.
-Santos, Telma João; On Self Codes, a case study within mathematics and performance art, Journal of Science and Technology of the Arts 9, no. 1 (2017b), pp 29-37.
-Sokal, Alan; Bricmont, Jean, Fashionable Nonsense, Postmodern Intellectual Abuse of Science, Picador, New York, 1998.
-Taylor, Robert L. V.; Attractors: Nonstrange to Chaotic, SIAM Undergraduate Research Online 4, http://dx.doi.org/10.1137/10S01079X, 2011.