Re-constructing space
as a network of itinerant assemblages.
In “thauseand Plateaus”
Deleuze and Guattari allude to an epistemology which promotes the development
of a “nomadic” or a “secondary” science.
A dynamical fluxus of
continuous relocations substitutes the procedures of iteration and
reproduction. It is the becoming as such which is the constitutive remark of
the model, and not a series of static and immutable shapes. The world is not a
reproduction or emulation of fixed and regular figures constitutive for the
assessment of a hierarchical order in which everything is metred and
masterable. The smooth space, in the words of Deleuze and Guattari, replaces,
or more properly extends the striated.
An example of the
striated space, as Deleuze and Guattari state, might be the fabric. A fabric is
constituted by a regular intertwining of parallel stripes and constructed by a
progression of back and forth movements that necessarily delimitate and enclose
the space.
This strategical
reduction has organized a closed space in which natural complexity is tamed: A
predetermined rasterization decomposes a complex system into a series of simple
subsystems, that can be examined individually by defining a small number of
parameters, laws, equations. The outline of the whole system equals the sum of
the outlines of any singular subsystem (principle of superposition)(cp. Tsuda
1994). Description of nature follows the rules of mimesis: natural phenomena
are described by taking account of the deviation from an exact ideal shape. It
is furthermore overruled by a sequence of linear relationships between cause
and effect. A single law, in the words of
Michel Serres “runs through causality like a star of sequences and consequences
which start out from a source or reference point” (Serres 2000,42). The
observer takes the place of the laplacian god, able to see or to predict by
deterministic laws the order of things.
And the smooth space. Deleuze and Guattari outline the example of the felt. No separated threads can be distinguished; it is obtained by an entaglement of fibers on a microscopic level. “An aggregate of intrication of this kind is in no way homogeneous: It is nevertheless smooth, and contrasts point by point with the space of fabric (it is in principle infinite, open and unlimited in every direction; it has neither top nor bottom nor center; it does not assign fixed and mobile elements but rather distributes a continous variation)” (Deleuze, Guattari 1988, 476). It is akin to what Serres calls “ a topology of interlacings, a hydrology of what flows trough the network”, constructs a smooth space, “space without edges” or borders, with no preferential sites: a complex web of divisions, bifurcations, knots and confluences (Serres 2000, 51). On a global holey circuit many local circulations arise and are subsequently disentagled or reshaped. Molecular assemblages interact with global structures in a continuous exchange of order and disorder, disorder and order. The euclidean space and time system which embeds the whole form, unfolds – by closer examination of molecular details - into a multiplicity of different space and time structures. Boundaries are submitted to a continuous change, and the laws of analogy and repetition become baseless.[1] .
Significant is the role played by chance. Irregularities are not ruled out by an “ideal of reproduction, deduction and induction” (Deleuze Guattari, 511) but are a constitutive element of the dynamical complexity of the phenomenon itself: new information is ceaslessly produced by microscopical uncertainities systemically raised up to a macroscopic dimension. No element can be extracted from the system and considered separately. Complexity cannot be grasped by any ideal point of view.
Mathematical models
following these laws, establish themselves in the “more” that exceeds the space
of reproduction. (Deleuze Guattari, 511) As they continuously construct
new pattern of relationships between singularities of space, they can´t
constitute ideal shapes mimed by natural phenomena. No quantitative measure of
mimetic relation between model and nature makes sense. Rather, an artificial
structure which allows a qualitative characterization of transcendental
properties underlying natural phenomena of higher complexity is constructed.
Here i intend to present qualitatively a mathematical model by which a kind of “smooth” space, as depicted previously, is constructed. It is part of a class of models defined by Smale in the late 60´s and “rediscovered” in the mid 90´s by Rössler and Tsuda, who saw them as a tool of description of high dimensional dynamic systems (that is systems that have more than three degrees of freedom (number of independent variables characterizing the system).
The idea of presenting this model in an art context is to produce a visualization of a kind of space where the rules of linear causality are disentagled.
The organization of the paper
is the following: i begin to sketch some basic features of the model in
discussion, out of which the conceptual frameview related to the mathematical
and physical formulations can be outlined. I will then try to show a
relationship with the idea of inorganic life, which Deleuze and Guattari
develop out of Bergson´s thought, and which is constitutive of the BWO.
These features are shown
in the following figures.
Fig 1: Equations of
the model. The x variable generates chaos by stretching and folding. In the
bottom table values for the variables x, y, z, w obtained by inserting
arbitrary initial conditions in the equations are shown.
Fig 2:
The y-z-w space might assume very different shapes depending on its rotation
relative to the observer.
(Every
point is given by coordinates as shown in the table of fig 1).
The
scenario of “chaotic itinerancy” is presented: the figure as whole constitutes
a big stable attractor (hyperbolic set), which contains many little temporally
stable attractors (“clouds” of points).
Fig d) constitutes a projection of the space into the y-z plane.
Fig 3: Illustration of the property of sensitivity of
initial conditions, which is constitutive of chaotic processes: two points
which at the first step differ only by an imperceptible value, increase their
distance by ongoing time steps. Here two time series obtained from the first
equation in fig 1 having respectively initial conditions x(1)= 0.5 and
x(2)=0.5000001 are shown.
We deal with a system
whose shape does not remain fixed in time, but which is in constant evolution.
Its dimension is not an integer, but a fractal characterized by the property of
self similarity. This system is locally determined by arbitrary initial
conditions, which, if changed produce a different set of points. That is, each
local assemblage assumes an other configuration or position in space (although
the global structure would remain the same). We see thus that change is a
constitutive element in the generation of the physical structure. This makes it
impossible to define an exact outline of the figure, which would change by the
addition of the slightest perturbation. The lines connecting the points do not
follow a pre-defined structure, nor a referential axis: the angles between the
axes might take any value, neglecting any possibility of preferential
direction.
In one figure we have
seen the property of chaotic itinerancy; a the authors themselves define
it in the following way: ” In the midst of highly disorganized states, ordered
motion governed by a few degrees of freedom often emerges, which, however, does
not last forever due to the second point (the dynamic change of relationships).
Again, high dimensional motion comes back, until another structure
emerges.”(Kaneko, Tsuda 1994) This scenario poses an other point of view than
the strict distinction between an “ordered” low dimensional macroscopic state,
and a “disordered” high dimensional or microscopic state, (which one can find
for example in the theory of dissipative systems developed by Prigogine): order
and disorder affect, in an ongoing interaction, both macroscopic and
microscopic levels. Crucial changes in the dynamics of the system might not
only be due to instabilities intrinsic in the structure of the system itself,
but also to slightest changes in the observation precision.
Is it then still
possible to understand modelling as a descriptive tool at all? Only under the
condition of the introduction of a further kind of instability, not related to
any intrinsic feature, but dependent on the assumed point of view of the
observer (the authors call it descriptive instability). As a consequence, a
point of view taken from the inside of the system necessarily would differ from
a point of view taken from the of the outside. That makes it once more
impossible to define an outline of the boundaries of the figure.
This frameview
illustrates that there exists no fixed immutable ideal shape free from natural
evolution which assumes the position of a natural archetype, able to reproduce
natural systems by a mimetic process of stamping.
The class of models from
which the one that has been shown is an evocative example, constitute, rather
than a representation, a description of
the interaction of variables which determine the degrees of freedom of a
complex system. This interaction, as the notion of chaotic itinerancy implies,
is subjected to a continuous evolution in time: the forces in play cannot be
described by a fixed pattern of constant vectors, but are in continuous
evolution between stable and unstable states. A becoming unstable of the
pattern might be generative for the following becoming stable.
These models are part of
an artificial world which might provide a metaphor for natural processes, and
assess means to describe transcendental properties of a wide range of complex
behaviours (f.e. turbulence or brain). They are constructive in the sense that
new information is produced by every iteration. Every step is a creative
process, an act of growing which cannot be simulated by any prior pattern, as
information which was not available before is constructed. A virtual condition
determined by the law of dynamics becomes actual. It is not a process of
limitation, which characterizes the transformation of the possible into the
real ( not every possibility acquires a reality, that is, from a larger
multiplicity, a minor one is extracted). The virtual creates by itself, in the
process of becoming actual, the lines along which the process of actualisation
occurs. The actual is therefore not necessary similar to the virtual as it was
in the match possibility/reality ( reality is per definition the realization of
the possible, that is its representation). Deleuze states clearly: “Im
Prozeß der Aktualisierung zählt in erster Linie der Unterschied – der
Unterschied zwischen dem Virtuellen, von dem ausgegangen wird, und den
aktuellen Momenten, zu denen man gelangt, sowie der Unterschied zwischen den
sich ergänzenden Linien, an denen entlang die Aktualisierung verläuft. Kurz,
Virtualität hat die Eigenart, in der Weise zu existieren, daß sie sich
differenzierend aktualisiert und daß sie, um aktuell zu werden, sich zu
differenzieren und ihre Differenzierungslinien erst zu schaffen hat.“ (Deleuze 1989, 123).
We deal with a construction
of a structure which cannot be encapsulated into a fixed rasterization, as it
is in continuous evolution. Molecular assemblages, by actualising themselves
moment by moment into new shapes, forms or configurations, are in constant move
on the plane of immanence . No “organism” –a predefined and static structure of
assemblages – can be formed.
In every time step an
inorganic transformation produces a destratification in which the interaction
of assemblages produces a circuit of exchange and circulation. Time becomes a
generator of space.
Literature:
Deleuze G., Guattari F., A
thousand plateaus, translated by B Massumi, Minnesota press 1988.
Deleuze G., Bergson zur Einführung, hgst. und übers. von
M. Weinmann, Hamburg 1989.
Serres M., The birth of
physics, translated by D. Webb, Manchester 2000.s
Tsuda, I., A new type of
self organization associated with chaotic dynamics in neural networks, Int.
Journ. Neur. Syst. 7(4) 1996:451-459.
Rössler O. E., Knudsen
C., Hudson J. L., Tsuda I., Nowhere Differentiable Attractors, Int Journ.
Intell. Syst. 10 1995: 15-23.
Smale S., Differentiable dynamical systems, Bull. Amer. Math. Soc., 73 1967: 747-817.