5 Assemblages and Virtual Diagrams

The abstract machine is like the diagram of an assemblage. It draws lines of continuous variation, while the concrete assemblage treats variables and organizes their highly diverse relations as a function of those lines. The assemblage negotiates variables at this or that level of variation, according to this or that degree of deterritorialization, and determines which variables will enter into constant relations or obey obligatory rules and which will serve instead as a fluid matter for variation. We should not conclude from this that the assemblage brings only a certain resistance or inertia to bear against the abstract machine; for even ‘constants’ are essential to the determination of the virtualities through which the variation passes, they are themselves optionally chosen.

Deleuze and Guattari, A Thousand Plateaus1

In our discussion in previous chapters we left out a crucial component of an assemblage: its diagram. Every assemblage is, as we saw, a concrete historical individual, from individual atoms and molecules to individual cities and countries. As such, assemblages are characterised by enduring states defined by properties that are always actual, existing in the here and now. But in addition to properties, assemblages also possess dispositions, tendencies and capacities that are virtual (real but not actual) when not being currently manifested or exercised. Moreover, when the concept of assemblage is endowed with parameters, the zones of intensity defined by the latter, and the critical values of the parameters mediating between zones, have the same ontological status as dispositions. When at a given point in time the setting of a parameter happens to be a critical value and the assemblage undergoes a transition (to become, for example, a stratum), the zone of intensity it finds itself in, and the crossing of the threshold, are actual states and events. But most of the time the zones and thresholds that structure the space of possible parameter values are not actual but virtual. Thus, the ‘virtual is not opposed to the real but to the actual . . . Indeed, the virtual must be defined as strictly a part of the real object – as though the object had one part of itself in the virtual into which it is plunged as though into an objective dimension.’2

But if we already had all the elements to define a virtual diagram, then why the delay? Because in addition to existing as part of concrete assemblages, diagrams are connected to a space of pure virtuality, a cosmic plane of consistency that exists as a limit of deterritorialisation, and it was necessary to perform a detailed analysis of this other concept before we could proceed. That is the task we will attempt in this chapter. We can conceive of this immanent plane in our minds by mentally forcing all movements of deterritorialisation to their absolute threshold. The concept of the plane is the result of carrying out this operation of taking to the limit. But we can also perform the opposite operation: start in thought with an ideally continuous cosmic plane and then derive all assemblages (and their material and expressive components) as the products of a process of actualisation, a process that breaks up the continuous plane into discrete or discontinuous entities. Deleuze and Guattari refer to these discontinuous, segmented entities as ‘lines’, and usually refer to the components of an assemblage as segments or lines: some rigid (with a high degree of territorialisation), some supple (low degree of territorialisation), while still others act as lines of flight, marking the directions along which an assemblage can become deterritorialised. As they write:

in all things, there are lines of articulation or segmentarity, strata and territories; but also lines of flight, movements of deterritorialization and destratification. Comparative rates of flow on these lines produce phenomena of relative slowness and viscosity, or, on the contrary, of acceleration and rupture. All this, lines and measurable speeds, constitutes an assemblage.3

Thus, there are two directions along which we can follow the concept of a diagram. Let’s begin by following the first direction, starting in the actual world, the world of properties, currently exercised capacities, and currently manifested tendencies, and slowly move towards the absolute threshold and the immanent plane that is revealed once that limit is reached. Since we are dealing with bundles of lines – every actual assemblage or component of an assemblage is the product of a segmentation of an ideally continuous virtuality – what we must do is to create maps of those lines, that is, we must follow a cartographic strategy. So our point of departure should be maps, as well as the different kinds of spaces that are mapped. As biological organisms and as social agents we live our lives within spaces delimited by natural and artificial extensive boundaries, that is, within zones that extend in space up to a limit marked by a frontier. Whether we are talking about the frontiers of a country, a city, a neighbourhood, or an ecosystem; or about the defining boundaries of our own bodies – our skin, our organs’ outer surfaces, the membranes of our cells – inhabiting these bounded extensive spaces is part of what defines our social and biological identities. We also inhabit other spaces delimited by intensive boundaries, critical points at which quantitative changes becomes qualitative. Examples of these spaces include the zones of high pressure explored by deep-sea divers; the zones of low gravity inhabited by astronauts; the zones of low temperature experienced by Arctic explorers; the zones of high speed traversed by test pilots. These are all, of course, rare professions, but we all populate these intensive spaces even if at more moderate intensities.

Extensive and intensive spaces can both be mapped, but the maps will necessarily be different. An extensive map captures features of the Earth that are extended in space, such as coastlines, mountain ranges, or the areas of land and volumes of airspace defining the sphere of sovereignty of a given country. By contrast, an intensive map captures differences in the intensity of properties as well as the dynamic phenomena driven by such differences. A well-known example, appearing on our television screens every night, is a meteorological map showing zones of high and low pressure, cold and warm fronts, air masses moving slowly or rapidly. Although the distinction between extensive and intensive properties is old, dating back to medieval scholastic philosophy, in its modern form it has been developed mostly by physicists. So we can begin our discussion with the textbook definition:

Thermodynamic properties can be divided into two general classes, namely intensive and extensive properties. If a quantity of matter in a given state is divided into two equal parts, each part will have the same value of intensive properties as the original, and half the value of the extensive properties . . .4

A typical extensive property, such as length, area, or volume, is divisible in a simple way: dividing an area into two equal parts results in two areas with half the extension. But if we take a volume of water at, say, 90 degrees centigrade, and divide it into two half volumes, we do not get as a result two parts having 45 degrees of temperature each, but two parts with the same original temperature. Put differently, while two extensive quantities add up in a simple way, two pieces of land adding up to a proportionally larger piece of land, intensive quantities do not add up but rather average: two volumes of water or air at different degrees of temperature, when placed into contact, trigger a diffusion process that tends to equalise the two temperatures at some intermediate value. Gilles Deleuze is the only modern philosopher who grasped the importance of this distinction, not only adopting the textbook definition but extending it to highlight its philosophical significance. In particular, Deleuze established a genetic relation between the extensive and the intensive: the diversity of entities that we can perceive directly are entities bounded in extension, but they are generated by invisible processes governed by gradients of intensity. A good example is the diversity of entities that populate the atmosphere: hurricanes, thunderstorms, cloud formations, wind currents. These entities inhabit our consciousness as meteorological phenomena but we cannot normally perceive the gradients of temperature, pressure, or speed that are responsible for their genesis. Similarly, while many diverse animals appear to us as entities bounded by their skin, we are not normally aware of the gradients of concentration of gene products that, as part of an embryological process, created those animals in the first place. In short, all the diversity that is given to us in experience depends for its existence on something that is not phenomenologically given. Or as Deleuze puts it:

Difference is not diversity. Diversity is given, but difference is that by which the given is given . . . Difference is not phenomenon but the noumenon closest to the phenomenon . . . Every phenomenon refers to an inequality by which it is conditioned. Every diversity and every change refers to a difference which is its sufficient reason. Everything which happens and everything which appears is correlated with orders of differences: differences of level, temperature, pressure, tension, potential, differences of intensity.5

Although the distinction between intensive and extensive properties belongs to thermodynamics, it can be argued that it is only in the context of a materialist philosophy that the distinction acquires its proper metaphysical significance. A similar point applies to extensive and intensive maps. Let’s discuss in some detail these two types of map, focusing first on their scientific aspects, then extracting the relevant philosophical problems they pose. Since the time of Ptolemy, mapmakers have struggled with the problem of capturing in a flat representation the spherical features of our planet. One could, of course, simply use a globe, a spherical map, in which the spatial relations can be represented directly. But if the goal is to create a flat map that can be folded and carried around, the spherical form of our planet must be transformed somehow, because spheres are not the kind of shapes that can be unrolled and made to lie flat. Cylinders and cones, on the other hand, are just those kind of shapes, so if one could transform a sphere into a cylindrical or conic shape then the problem would be solved. The special transformation that achieves this objective is called ‘projection’. Ptolemy projected the planetary sphere on to a cone, much as one would project a slide on to a screen, while Mercator, fourteen hundred years later, used a cylinder as his screen. Although once unfolded and flattened both the conic and the cylindrical representations gave the desired result, a new problem emerged: one can preserve the original spatial relations yielding specific shapes, like the shape of a coastline or a mountain range, or one can preserve the original areas covered by land or water masses, but not both. Opting for the former (in what is called a ‘conformal’ map) we lose the true relations between areas or between lengths, while choosing the latter (an ‘equal-area’ map) gives us shapes that appear distorted on the map. For the purposes of navigation along a coastline, where visual recognition of landmark shapes is what matters, a conformal map is the right choice, but for statistical purposes, to depict the density of population per square mile, for example, we need an equal-area map. Other uses call for a compromise, a projection that does not preserve anything unchanged, but in which the errors are small enough or balance each other out.6

This description of extensive maps contains two concepts that are important to clarify the notion of a diagram: the existence of transformations – two in this case, a projection operation corresponding to shining light on to a piece of film, and a section operation, the equivalent of intercepting those light rays on a screen – and the fact that once applied, these transformations leave some of the features of the original form unchanged. In the traditional Mercator projection, for instance, shapes remain invariant, as do some lengths (the distances along the line of the equator) but areas and distances away from the equator do not remain the same. These two concepts, transformations and invariants, have become an integral part of the bodies of knowledge of different scientific fields in the twentieth century, but to deploy them philosophically, the term ‘invariant’ should always be used in a relative way. That is, we should never speak of invariants by themselves but always relative to a specific transformation.7 The reason is that an invariant does not refer to a constant property, but to a property’s capacity to be unaffected by a transformation. These two concepts are necessary to characterise an assemblage’s diagram, because the latter captures the structure of a possibility space, and when thinking about the latter it is important not to bring any intuitions about physical space. One way of enforcing this requirement is to use only spatial features that remain invariant under the largest number of transformations.

Let’s discuss this requirement in more detail. In the late eighteenth century most philosophers and scientists agreed that Euclidean geometry was not only the most fundamental of all geometries but the one that captured the features of real physical space. Euclidean geometry is one example of a metric geometry, that is, a geometry in which properties like the length of a line or the angle formed by two lines are fundamental concepts. But in the following century, some mathematicians realised that concepts like length, angle, and shape could be logically derived from concepts in projective geometry, concepts that were non-metric.8 What up to that point had been a humble geometry belonging to minor scientific fields, fields like cartography or architecture, turned out to be more fundamental. We can use the two concepts just introduced to clarify this. All metric geometries, Euclidean and non-Euclidean, form spaces the properties of which remain invariant by a group containing rotations, translations, and reflections. In other words, all lengths, angles, and shapes remain invariant under this group of rigid transformations. In projective geometries, on the other hand, those properties do not remain invariant but others do, such as linearity, collinearity, and the property of being a conic section. Moreover, the group of transformations that leave the latter invariant is a larger set, including rotations, translations, and reflections, but also projections and sections. It was this realisation, that the group characterising metric spaces is a subgroup of the one characterising projective spaces, that established the logical priority of the latter.9 When differential geometry and topology were invented, the followers of Felix Klein – the mathematician who first used transformation groups to elucidate the relations between the different geometries – realised that the new geometries had invariants under even larger groups, including transformations like bending, folding, and stretching, and hence that they were more fundamental than projective geometry. Extending Klein’s original insights, it became clear that all known geometries could be organised by the size of the group of transformations that left its features invariant, or to use the technical term, by their degree of symmetry.10

Although mathematicians view this classification as a logical construction, useful to establishing conceptual priorities, it is possible to extract a metaphysical lesson from it by making the relations between the different spaces genetic. In this metaphysical version, metric spaces, the ones closer to our spatial intuitions about real space, are literally born from non-metric spaces as the latter progressively lose symmetry and gain invariants. Or to express this thought using the terms with which we began this chapter, the geometries with the greatest degree of continuity and flexibility, the ones forming the smoothest of spaces, give rise to those that are striated or rigidly segmented.11 To return to the main point, what we needed from our examination of extensive maps were insights into the relations between the segments or lines that are the components of an assemblage and the unsegmented spaces of possibilities defining an assemblage’s diagram. The concept of spatial invariants under transformations supplies the needed bridge. An assemblage’s lines vary in their degree of rigidity, some more territorialised than others, much like the geometries in Klein’s classification, and we can follow the less territorialised ones in thought, pushing their deterritorialisation to the limit, until we reach the most deterritorialised component, the diagram, much as we can climb Klein’s classification until we reach topology. In addition to their logical and genetic priority, there is another reason to favour the least metric spaces when considering candidates for an assemblage’s diagram: they allow us to think of the diagram as immanent as opposed to transcendent.

In the early nineteenth century, when mathematicians wanted to study a curved two-dimensional surface, they first embedded it into a three-dimensional space that was structured by a set of Cartesian coordinates. In other words, they placed the object of study inside a space with one additional dimension, then used the fixed axes to assign x, y, and z coordinates to every point of the surface. But then the mathematician Friedrich Gauss realised that the differential calculus could be used to study a surface using only local information, that is, without a global embedding space. And by doing that, Gauss ‘advanced the totally new concept that a surface is a space in itself’.12 In particular, if relations between the changes of two or more quantities could be expressed as a rate of change, then the calculus allowed finding the instantaneous value for that rate. Applying this to geometry involved thinking about a curved surface as an object characterised by the rate at which its curvature changes between two points, and then using the calculus to compute ‘instantaneous’ values for this rate of change. Treated this way, a surface ceased to be a set of coordinate values and became a field of rapidities and slownesses, the rapidity or slowness with which curvature changed at each point. The philosophical significance of this can be brought out when we consider that, traditionally, transcendent forms of determination, whether the God of creationists or the formal essences constituted by Aristotelian genera and species, operate on a higher dimension than the space in which a material process unfolds. If we use the variable ‘n’ for number of dimensions, transcendent formal or divine causes tend to operate in a space with n +1 dimensions. But as Deleuze and Guattari argue, the diagram of an assemblage ‘however many dimensions it may have . . . never has a supplementary dimension to that which transpires upon it. This alone makes it natural and immanent.’13

Let’s move on to examine intensive maps to find out what insights into the concept of a diagram they can provide. What needs to be mapped in this case are not the borders of entities possessing a spatial organisation, like the boundaries of an ocean, a lake, or another body of water, but thresholds of intensity causing changes from quantity to quality in the spatial organisation of those bodies. Let’s imagine a frozen body of water, a solid piece of ice, linked to an outside supply of energy that we can control. As we increase the amount of energy flowing into the system, its temperature reaches a critical point at which, suddenly, the ice begins to melt. At that intensive threshold a solid spontaneously changes into a liquid as its spatial organisation, its manner of occupying space, mutates. If we continue to increase the amount of energy we reach another critical threshold, the boiling point of water, and the liquid turns into a gas, with accompanying changes in extensive properties: the amount of space the water molecules occupy, their volume, greatly expands. Finally, as the temperature reaches yet another threshold, first the molecules of water dissociate into their component atoms, then even the atoms of hydrogen and oxygen lose their own identity, the entire population becoming an electrified cloud of charged particles: a plasma. A plasma exemplifies an intensive continuum, a body of matter and energy that is not segmented into differentiated atoms but which can give rise to the latter (as well as to larger segments made out of atoms) as it progressively cools down and undergoes phase transitions.

A map of these intensive thresholds is called a phase diagram. The number of dimensions of the map is determined by the number of intensive parameters used to affect the body of water. Using a single parameter, temperature, yields a map that is one-dimensional, that is, the temperature values form a linear series in which the thresholds appear as points: the point at zero degrees centigrade marking the melting point of water, and the one at 100 degrees centigrade marking its boiling point. (The names of the points vary depending on the direction in which the thresholds are crossed: in the opposite direction they are the freezing and condensation points, respectively.) These two singular points are constant – so constant that we use them to mark our thermometers – but only as long as we keep other possible parameters unchanged. In particular, zero and 100 degrees mark thresholds at sea level, but the numerical value changes if we measure them on top of a tall mountain because the pressure that the air exerts diminishes with altitude. This implies that adding a second parameter, pressure, changes the map into a two-dimensional space in which the thresholds cease to be points and become lines. And similarly when we add a third parameter, like specific volume, the thresholds become surfaces in a three-dimensional map.

Typically, as we add more dimensions an intensive map reveals further complexity in the behaviour of matter. The two-dimensional phase diagram of water, for example, is not structured by two parallel lines running through the zero and 100 degrees points of temperature. If it were, adding pressure as an extra intensive parameter would not add any new information. In reality, the lines are not parallel but form a shape with the form of the letter ‘Y’. At the pressure present at sea level, the map is structured by the upper part of the Y, so a perpendicular line of temperature values intersects its two arms at the two points just mentioned. But at lower pressures the map is shaped by the lower part of the Y, so a line of temperature values intersects it only once. This means that at very low pressures there are only two distinct phases, solid and gas, one transforming directly into the other in a phase transition called ‘sublimation’. Finally, despite the fact that the thresholds are now lines, singular points are also present: the point in the Y where the two upper arms meet the lower vertical is called a ‘triple point’, a zone of coexistence at which all three phases simultaneously occur and can be readily transformed into one another. Similarly, the right arm of the upper Y does not cross the entire map but terminates at a critical point creating a zone of indiscernibility within which the liquid and gas phases of water become indistinguishable.14

Let’s give a different example of an intensive map, one using speed as a intensive parameter. The behaviour of fluids in motion exhibits sudden changes in form at critical thresholds of speed, that is, it undergoes transformations between different regimes of flow: at low speeds the flow is uniform or steady (laminar); then past a threshold it becomes wavy or periodic (convective); and past yet another threshold it becomes turbulent, displaying a complex structure of eddies within eddies. Because we are using a single parameter, an intensive map of these transformations would be one-dimensional, a line of speed values divided into three different regimes by critical points. Using a special apparatus consisting of two transparent concentric cylinders between which a fluid is sandwiched, this one-dimensional map can be enriched. The higher degree of control allowed by the so-called Coutte-Taylor apparatus reveals seven distinct regimes of flow: laminar, Taylor vortex flow, wavy vortex flow, modulated wavy vortices, wavy turbulence, turbulent Taylor vortices, and featureless turbulence. If we modify the apparatus so that we can spin both the inner and outer cylinders at the same time, we can create a two-dimensional map. As before, adding an extra dimension reveals much hidden complexity. Two new intensive zones are created on both sides of the line with its seven regimes of flow: to the right there are variations produced when the two cylinders spin in the same direction, including ripples, twisted vortices, corkscrew wavelets; to the left there are variations produced by spinning them in opposite directions, such as simple spirals, interpenetrating spirals, and spiral turbulence.15

Several insights can be derived from these intensive maps. The first is that, as Deleuze argues, intensive properties are not so much indivisible as that which cannot be divided without changing nature.16 Critical thresholds do segment or divide an intensive map, but each subdivision corresponds to a limit of quantitative change, after which it gives rise to qualitative change. Next is the idea of creating maps by assigning to each dimension of a space the possible values that the variables or parameters of a phenomenon can have. The variables represent the properties of the phenomenon, while the parameters stand for the properties of its immediate environment. The state in which a phenomenon can be is a combination of both of these values, so the intensive map captures the space of all its possible states. Finally, there is a lesson here related to the remarks made above regarding transcendent and immanent spaces. Critical thresholds are always one dimension lower than the map itself, that is, intensive thresholds always have n –1 dimensions: points in a line, lines in a surface, surfaces in a volume. Thus, unlike forms of determination that come from above, from a space with n +1 dimensions, and which give an artificial unity to that which they determine, the n –1 entities at which quantity is determined to change to quality do not unify the space but, on the contrary, allow it to remain multiple. As Deleuze and Guattari write: ‘The multiple must be made, not by always adding a higher dimension, but rather . . . with the number of dimensions one already has available – always n –1 (the only way the one belongs to the multiple: always subtracted). Subtract the unique from the multiplicity to be constituted; write at n –1 dimensions.’17

The insights extracted from this analysis of extensive and intensive maps can be combined if we take the idea of mapping possible states into points, but we map them instead into a continuous topological manifold. In fact, we need two such possibility spaces, one in which the dimensions of the manifold are assigned the values of variables (state space) and the other in which the values of parameters are assigned (control space). Intensive thresholds appear in the latter, not the former, but some of the topological invariants structuring the former also follow the n –1 rule. Unlike phase diagrams, which are graphic representations of laboratory data, we need to establish relations of dependency in the way variables (and parameters) change so that the points in the space represent possible states by these relations, that is, so that the combination of values for each variable defining a given point is coherent relative to the dependency relations. This allows us to go beyond simply assigning a point to each possible state, and generates series of points (forming curves or trajectories) that stand for possible histories. Spaces of this kind, in which the dependencies between variables were captured using differential equations, were investigated by the mathematician Henri Poincaré towards the end of the nineteenth century. As he explored the behaviour of trajectories, Poincaré noticed that curves tended to converge at special points in the space, as if they were being attracted to them: it did not matter where the trajectory had its origin, or how it wound its way around the space, its long-term tendency was to end up at a particular point.18 These special, remarkable, or singular points were eventually named attractors. When a state space has several attractors, these singularities are surrounded by an area within which they affect trajectories, an area called a ‘basin of attraction’: if a trajectory begins within a particular basin of attraction then it inevitably ends up at the attractor.

This implies that attractors and their basins define zones of stability, since they pin down trajectories to a small set of states (coherent combinations of the values for the variables) and do not let them escape. Thus, unlike phase diagrams that simply record the empirical observation that an entity (such as a body of water) has certain recurrent states (liquid, solid) or recurrent regimes (convective, turbulent), state space diagrams provide an explanation for the recurrent nature of these states or regimes: they recur because they are stable tendencies characterising the body of water, and there is an empirically testable isomorphism between these dispositions and the convergent tendencies of the trajectories in state space. What about the intensive thresholds? These appear in the other space, the control space in which dimensions are assigned to parameter values. N –1 entities in this other space indicate the critical values at which one distribution of singularities is transformed into another, topologically inequivalent one. A particular distribution of attractors is a topological invariant of state space: it does not matter how we fold, stretch, rotate, or project this space, the distribution remains unchanged. But crossing thresholds in the associated control space destroys invariants, which is why the thresholds are referred to as symmetry-breaking bifurcations. Control space is thus subdivided into zones of intensity that determine which zones of stability are available in state space: as long as the frontiers of the intensive zones are not crossed, the phenomenon being diagrammed retains its current distribution of singularities. There is therefore a similarity between phase diagrams and control space, but while in the former the n –1 entities mark empirically determined limits, in the latter the critical transitions are determined by the differential relations themselves, and thus represent a more intimate relation between a map and that which is being mapped.

Gilles Deleuze quickly realised the metaphysical importance of these developments in mathematics, referring to the possibility spaces formed by the conjunction of state and control spaces as ‘multiplicities’ or ‘Ideas’, both terms being somewhat misleading. The latter suggests something Platonic, while the former is the term used by mathematicians to refer to the differential or topological manifolds themselves, that is, to non-metric spaces in general. But the example that Deleuze gives clearly shows that he is not thinking about non-metric spaces, but about spaces in which the dimensions have been assigned to variables and which are therefore spaces of possibilities, like the space of possible colours. As he writes:

An Idea is an n-dimensional, continuous, defined multiplicity. Color – or rather, the Idea of color – is a three dimensional multiplicity. By dimensions, we mean the variables . . . upon which a phenomenon depends; by continuity, we mean the set of relations between changes in these variables . . .; by definition, we mean the elements reciprocally determined by these relations, elements which cannot change unless the multiplicity changes its order and its metric. When and under what conditions should we speak of a multiplicity? There are three conditions which together allow us to define the moment when an Idea emerges; 1) The elements of the multiplicity must have neither sensible form nor conceptual signification . . . They are not even actually existent, but inseparable from a potential or a virtuality . . . 2) These elements must in effect be determined, but reciprocally, by reciprocal relations that allow no independence whatsoever to subsist . . . In all cases the multiplicity is intrinsically defined, without external reference or recourse to a uniform space in which it would be submerged . . . 3) A multiple ideal connection, a differential relation, must be actualized in diverse spatio-temporal relationships, at the same time that its elements are actually incarnated in a variety of terms and forms. The Idea is thus defined as a structure.19

In Chapter 7 we will explore in more detail all the different mathematical concepts used in this quotation, as well as the question of how we must think about mathematics so that its results can be used within assemblage theory. At this point all that matters is to emphasise that despite the proliferation of different terms, we are dealing with just one kind of component of an assemblage: the term ‘diagram’ as it is being used here is synonymous with ‘multiplicity’ or ‘Idea’. All three can be defined as the structure of a possibility space, a structure given by topological invariants like dimensionality, connectivity, and distribution of singularities.20 On the other hand, the concept of a diagram adds a properly metaphysical component to the scientific concept of state space: all its components (variables, parameters, invariant structure, differential relations) must be given the ontological status of a virtuality, that is, they must be considered real but not actual. The ontological gloss added by Deleuze can then be summarised like this:

The reality of the virtual consists of the differential elements and relations along with the singular points which correspond to them. The reality of the virtual is structure. We must avoid giving the elements and relations that form a structure an actuality which they do not have, and withdrawing from them a reality which they have.21

Now that we are in possession of a definition of a diagram, and that its ontological status relative to the actual components of an assemblage has been elucidated, we can attempt to tackle the much harder problem of conceptualising the cosmic space formed by all diagrams, what we referred to above as ‘the plane of consistency’. We said above that this problem can be solved by performing a limiting operation in thought: we begin with an actual assemblage in which both parameters are at the highest value, the most territorialised and coded version, that is, a stratum; then we make the first change, turning the coding parameter to zero, the decoded stratum becoming an assemblage (in the original sense of the term); we then find the most deterritorialised component, its diagram; and finally, we detach the diagram, liberating it from its last links to the actual world, and bring it into relations of exteriority with other detached diagrams. What we get at this limit is the plane of consistency, the mark of which is that the distinction between the material and the expressive (content and expression) is no longer discernible, and all diagrams enter into continuous variation. Deleuze and Guattari always have this entire sequence in mind when framing definitions and explanations. Thus, they write that the opposition between strata and assemblages is

entirely relative. Just as milieus swing between a stratum state and a movement of destratification, assemblages swing between a territorial closure that tends to restratify them and a deterritorialising movement that on the contrary connects them with the Cosmos. Thus it is not surprising that the distinction we were seeking was not between assemblages and something else but between the two limits of any possible assemblage, in other words, between the system of strata and the plane of consistency. We should not forget that the strata rigidify and are organized on the plane of consistency, and that the plane of consistency is at work and is constructed in the strata, in both cases piece by piece, blow by blow, operation by operation.22

Let’s now follow the opposite movement, starting in the plane of consistency and increasing the value of territorialisation (and eventually coding) until we arrive at the strata. As before, let’s first gather some concepts that have been tested elsewhere to use as aids in this reversed journey. In particular, another element of Felix Klein’s classification of geometries can serve as a point of departure. As we saw, his schema is based on the existence of a symmetry-breaking cascade through which metric geometries are generated by non-metric ones. Accompanying this process of ‘becoming metric’ there is a parallel process of figure differentiation: as we move down the cascade, more and more figures become distinct, and vice versa – as we move up they tend to blend into one another. Thus, in Euclidean geometry small and large circles, small and large ellipses, small and large parabolas, are all different figures. At the next level, the level of affine geometry, circles of all sizes are the same, as are ellipses and parabolas of all sizes. One more level up, the level of projective geometry, all conic sections are one and the same figure. This reduction in the degree of differentiation of geometrical figures is explained by the fact that if a figure can be transformed into another using only transformations in a group, then the two figures are one and the same. Affine geometry has in its group the scaling transformation, so the size of a given conic section is not relevant to establishing its identity. Similarly, in projective geometry tilting the screen on which a circle is projected transforms it into an ellipse, and moving the screen so that part of the ellipse is now outside it yields a parabola. In other words, all conic sections are inter-convertible using transformations in the projective group, so they are all one and the same figure.23 The folding and stretching transformations available in topology take us beyond this: all closed figures (triangles, squares, pentagons, circles) are inter-transformable so they are all the same. This suggests that a symmetry-breaking cascade represents a process of progressive differentiation that takes a few ‘elastic’ topological figures and through successive broken symmetries generates all the different metric figures and their rigid segments.

How can we adapt this geometric version of progressive differentiation to the birth of discrete or discontinuous things and events? We can start by using it as a metaphor and slowly removing the analogical content until we reach a more literal version. Let’s begin with the simplest example, a topological diagram defined by a few dimensions and a single point singularity. This diagram may be divergently actualised into a variety of objects that have only one thing in common: the process that creates and maintains their identity has a tendency towards a steady state, which in many cases maximises or minimises the value of a given property. Thus, a soap bubble acquires its spherical shape by minimising surface tension, a crystal of table salt acquires its cubic shape by minimising bonding energy, a light ray acquires its geodesic shape by minimising travelling time. Thus, much as a topological closed shape in Klein’s classification can differentiate into a large variety of metric polygons and conic sections, a topological point in a diagram forming part of the cosmic plane can differentiate into a large variety of actual shapes. There is, of course, no reason to stick only to shapes. When we say that ‘water seeks the lowest level’, what we mean is that a body of water on a planet subject to gravity will have a tendency to minimise gravitational potential, and this makes it another actualisation of the same topological point. In fact, as we saw in the previous chapter, all the phenomena in classical physics can be modelled by a single master equation, a Hamiltonian, the solution of which is a minimum (of the difference between kinetic and potential energy), making all of them relatives of bubbles and crystals. The only problem with this analogy is that the actual shapes are generated in a single step, so to speak, as the molecules in the soap film or the sodium and chlorine atoms seek the minimising state actualising the topological point. In other words, they are differentiated but not progressively so. So a better analogy would be a pluripotent cell. As the embryological process that transforms a fertilised egg into a multicellular organism proceeds, a single cell progressively differentiates into bone and muscle, nerve and blood, and over two hundred more different cell types. Each of the final cell types is locked into one state, its diagram possessing a steady-state attractor performing the locking, but the pluripotent cell itself has a diagram with two hundred point singularities. In this case, progressive differentiation occurs as a series of symmetry-breaking bifurcations transforms the original diagram by successive removal of singularities until the final diagrams are generated.24

The first step in removing the metaphorical content is to get a better idea of how the diagrams should be assembled to form the top level of the symmetry-breaking cascade. Each diagram has a different number of dimensions, because the latter represent the number of relevant variables (the number of ways a phenomenon is free to change) and simple inorganic objects can change in only a few ways, while living creatures can change in hundreds of ways. To retain its immanent status the plane of consistency cannot be a space with an extra dimension over and above those of its diagrams. On the contrary, it must always be n –1. One possible solution is to think of it as the result of a transformation, like a slicing or an intersection, which when applied to an actual object always yields a lower dimensional slice: a point if we slice a line; a line if we slice a surface; a surface if we slice a volume. Thus, Deleuze and Guattari argue that ‘Far from reducing the multiplicities’ number of dimensions to two, the plane of consistency cuts across them all, intersects them in order to bring into coexistence any number of multiplicities, with any number of dimensions. The plane of consistency is the intersection of all concrete forms.’25 No doubt, speaking of such a cosmic intersection is still partly metaphorical, but it is now linked to the requirement of keeping the plane from becoming transcendent and establishing only relations of exteriority between diagrams or multiplicities.

Next, we must replace the levels of the cascade mediating between the top and the bottom levels. First we need to make an analogy between metric and non-metric geometries, on the one hand, and intensive and extensive properties, on the other. Properties like length, area, and volume, which remain invariant under the rigid group of transformations, are also extensive properties, divisible or segmentable. The divisibility of extensive properties is important because at the lowest level we must locate entities that are discrete and discontinuous, many bounded by an outer surface that separates them from other discrete entities. Above this lowest layer of qualified extensities we can place several intensive levels, inhabited by entities defined both by their distance from thermodynamic equilibrium, as well as by how permeable their bounding surfaces are. Equilibrium defines the final point of a series of energy transformations at which all useful energy has been exhausted and has acquired a homogeneous form: heat. No processes can occur at equilibrium, so not even the most rigid strata can be identified with it. The first step away from this terminal state is constituted by phenomena that are isolated, that is, phenomena in which there are no flows of energy or matter across the outer boundaries, and that have a tendency to move towards a state of maximum entropy. The next level contains phenomena that are closed, in flows of energy but not of matter across the outer surface, and that have a tendency towards a state of minimum free energy. The level above that contains phenomena that are open, traversed by low-intensity flows of both matter and energy, and that have a tendency to be in whatever state has a minimum entropy production.26 In all three cases the final state is stationary, corresponding to a point singularity. Finally, above these three levels there are the variety of phenomena studied by far-from-equilibrium thermodynamics, in which the flows of matter and energy are of high intensity and the end states need not be stationary nor unique, with a wider variety of singularities available to govern processes: steady-state, periodic, and chaotic singularities.27

This construction, using Felix Klein’s classification as a scaffold then hanging pieces of conceptual machinery from it, yields only a picture. It is a useful image since it contains all the different elements of the ontology of assemblages – a topological continuum that becomes progressively more rigidly segmented, passing through several stages of decreasingly less supple segmentation – but it is still just an image. To go beyond a mere image we need to consider in detail specific cases of actualisation, a task we will postpone until the next chapter. At this point we can hold on to the scaffold as a transitory structure to be discarded later, to clarify how other concepts related to assemblage theory hang together. In the first chapter we said that Deleuze and Guattari use the terms ‘molecular’ and ‘molar’ as synonyms for ‘micro’ and ‘macro’. This was only partially correct. In its original usage the term ‘molar’ referred to a body of matter as a whole, while the term ‘molecular’ referred to the invisible components of that body. This distinction between parts and whole is preserved in the philosophical usage, but it is relativised to scale. In particular, the term ‘molecular’ is used for any population of components, not just the molecules that chemists study. But to the relativised original meaning Deleuze and Guattari add the qualification that a molar entity be characterised both by its extensive properties, such as the volume of the body of matter, as well as by its intensive properties measured at equilibrium. In other words, molar entities are located by the authors at the bottom of the symmetry-breaking cascade. If we examined one of these highly territorialised molar entities, its temperature or its pressure could be explained by showing the activity of its molecular population: temperature is generated as the kinetic energy possessed by molecules in motion averages itself out, and pressure as the momentum of molecules colliding against the walls of a container averages itself out. Thus, even in the most rigid molar entity there is activity at the molecular level, where flows are driven by gradients, and qualitative changes occur at critical thresholds. Hence, the authors locate molecular segments at the intermediate levels of the cascade. They also distinguish segments located just below the topological continuum, communicating with it and carrying what is left of the other segments with them. If we replaced ‘segment’ by ‘line’ we should be able to read the following quotation as expressing a very literal thought:

Whether we are individuals or groups, we are made up of lines and these lines are very varied in nature. The first kind of line which forms us is segmentary – of rigid segmentarity: family-profession; job-holiday; family-and then school-and then the army-and then the factory-and then retirement . . . In short, all kinds of clearly defined segments, in all kinds of directions, which cut us up in all senses, packets of segmentarized lines. At the same time, we have lines of segmentarity which are much more supple, as it were molecular. It is not that they are more intimate or personal, they run through society and groups as much as individuals. But rather than molar lines with segments, they are molecular fluxes with thresholds or quanta . . . Many things happen on this second line – becomings, micro-becomings, which don’t even have the same rhythm as ‘our’ history . . . At the same time, again, there is a third kind of line, which is even more strange: as if something carried us away, across our segments, but also across our thresholds, towards a destination that is unknown, not foreseeable, not pre-existent . . . the line of flight and of the greatest gradient . . .28

Let’s follow Deleuze and Guattari as they map these lines at different scales. At the largest human scale we find the assemblages constituted by entire cultures in interaction, such as the assemblage that emerged on the European continent during the fall of the Roman Empire: the Roman cities, their institutions, and their geometrically organised military camps and rigid phalanxes are mapped with molar lines; the movements of the nomads from the steppes, their highly flexible and mobile armies, and their highly destabilising effects on sedentary settlements are mapped with lines of flight; and finally, the migrant barbarian tribes that were caught in the middle, and were pushed by the nomad wave against the empire, are assigned molecular lines.29 At a smaller scale we find the assemblage of urban and rural settlements (and the organisations exercising authority in those settlements) composing an archaic empire, like the Egyptian Empire. In this case, the authors’ map shows that the semi-autonomous agricultural villages at the periphery of the empire possess a molecular segmentarity, while the central state apparatus in its urban capital displays the most rigid molar segmentarity.30 A line of flight in this case could be illustrated with a mobilised state army that, returning triumphant from a faraway military victory, resists being demobilised, threatening the very stability and identity of the state apparatus.

Since in all cases we are dealing with assemblages of assemblages, each of the components in these examples must, in turn, be mapped. Thus, a bureaucratic organisation – a single component of a state apparatus – can be broken down into its molar segments: its separate offices, tight schedules, rigid task assignments, and written regulations. But a closer look will reveal the personal and professional networks that its staff tend to form, networks that display a more molecular form of segmentarity.31 The third kind of line would be exemplified by long-distance transportation and communication technologies that decentralise control, allowing the molecular segments to be mobilised to reform the organisation, or otherwise change its identity. Moving on to the smallest scale that is significant for social explanation, the scale at which the persons who staff a bureaucratic organisation operate, we can also create maps of their bodies and minds, considered as molar aggregates of sub-personal components: ‘Take aggregates of the perception or feeling type: their molar organisation, their rigid segmentarity, does not preclude the existence of an entire world of unconscious micro-percepts, unconscious affects, fine segmentations that grasp or experience different things, are distributed and operate differently.’32 An example of a line of flight at this scale is provided by psychedelic substances, chemicals that liberate those micro-percepts, accelerate their escape from a molar subjectivity, and produce a state of delirium, changing in the process the identity of the person, even if only temporarily. It is important to emphasise that everything that can be said by analysing assemblages in terms of segments or lines can also be expressed by setting the territorialisation parameter of each assemblage in a nested set to the appropriate value: each composing line could be treated as itself an assemblage, its molar or molecular status defined by its parameter settings.

Let’s return to the concept of an assemblage’s diagram. As already argued, the diagram captures the structure of the space of possibilities associated with an assemblage’s variable components, as well as the structure of the space of possible parameter values. These coupled possibility spaces are much like the state space and control space used in the geometric approach to the study of differential equations. One advantage of mapping historical processes using these concepts is that once we understand that the possibilities open to an actual assemblage have a certain virtual structure, we do not have to think about primitive societies and their urban counterparts as stages of development of humanity. Some forms of social organisation may indeed have appeared earlier than others – hunter-gatherers certainly existed before any central state apparatus – but that succession occurred only in actual time. In virtual time both coexisted, the latter being a possibility already prefigured in the former. It is ‘precisely because these processes are variables of coexistence that [they can be] the object of a social topology . . .’33 Thus, hunter-gatherers and their molecular segmentarity already contained in their associated possibility space a line of flight prefiguring a state apparatus, a line of flight that simultaneously offered an opportunity to escape from their current identity, as well as the risk of becoming rigidly segmented by the emergence of centralised authority. Hence, Deleuze and Guattari characterise primitive societies by the mechanisms of prevention and anticipation with which they resist this possibility. An example of these is the burning of all surplus food in ceremonial rituals to prevent the formation of a reservoir that a central authority could use to promote a division of labour, thereby forcing primitives to cross the town-threshold and the state-threshold.34 Many other insights emerge from these metaphysical maps, although in some cases they are muddled by an imprecise segmentation of social reality. We have already had opportunity to criticise a social ontology composed of individuals, groups, and the social field, but we also explored solutions to this problem that preserve and extend Deleuze and Guattari’s insights.

There is one more difficulty to overcome, one that has nothing to do with segmentation but with the ideally unsegmented continuum. Of the three kinds of lines used in these maps, the hardest to conceptualise is the third. On the one hand, a relative line of flight is simply a movement of deterritorialisation taking place in an assemblage, a movement that can be captured by changing the setting of the territorialisation parameter. On the other, a line of flight can become absolute and perform a very different role, accelerating the escape from molarity to the limit, taking with it detached diagrams to produce the plane of consistency.35 Earlier in this chapter, we followed the conceptual movement needed to capture this plane in thought. Starting with a highly territorialised and coded assemblage, we changed the values of the parameters, then progressively dismantled it, until we reached the critical threshold at which diagrams become detached from actual objects and enter into relations of exteriority among themselves. (The cosmic plane is also referred to as the ‘plane of exteriority’.)36 But Deleuze does not confine this movement to thought alone. On the contrary, he argues that there is an objective movement running in the opposite direction to that of actualisation, a movement he refers to as counter-actualisation.37 The need to postulate such a counter-process derives directly from the requirement that all transcendent entities and spaces be excluded from a materialist ontology. In particular, we cannot simply postulate the existence of an ideally continuous cosmic plane (or of ideal surfaces) but must account for its production and maintenance. Otherwise the plane will be nothing but a Platonic heaven in which essences have ceased to be metric (‘sphericity’) and have become topological. Thus, we need a mechanism for the production and reproduction of immanent surfaces, or more exactly, of hyper-surfaces, since these can have any number of dimensions:

Many movements, with a fragile and delicate mechanism, intersect: that by means of which bodies, states of affairs, and mixtures, considered in their depth, succeed or fail in the production of ideal surfaces; and conversely, that by means of which the events of the surface are actualised in the present of bodies (in accordance with complex rules) by imprisoning their singularities within the limits of worlds, individuals, and persons.38

Absolute lines of flight are components of this mechanism for the production of ideally continuous surfaces. But here we run into a difficulty. Any process of production must occur in time, that is, the series of events that compose the process must actually occur. The time in which assemblages are born, live, and die is the present time, and the present belongs to the actual world. Thus, the liberation of singularities and the assembly of detached diagrams that produces the plane of consistency must take place in another temporality, one without any presently occurring events. Earlier we used the contrast between metric and non-metric spaces as a guide to conceptualise the relation between virtual and actual spaces. Would it be possible to conceive of a non-metric time proper to virtual spaces? If a metric space is defined by rigid lengths that are measurable and divisible, then chronometric time must be thought of as a form of temporality defined by rigid durations that are the measurable presents of actual entities, from the longest cosmic or geological presents to the shortest sub-atomic ones. But just as lengths and areas are meaningless in topology, a non-metric temporality would be one in which the notion of a stretch of time with a measurable duration is meaningless. Only singularities should be used to think about this virtual time: the minimum thinkable continuous time and the maximum thinkable continuous time; a present without any duration whatsoever that is unlimitedly stretched in the past and future directions simultaneously, so that nothing ever actually happens but everything just happened and is about to happen.39

When speculating about actualisation we can use ideas from a diverse set of scientific and mathematical fields to sharpen our intuitions, but when investigating counter-actualisation we are on our own. This is, therefore, the most properly philosophical part of this ontology.40 Lines of flight that do not merely escape from one actual configuration of the material and the expressive but that reach ‘escape velocity’, leaving behind the actual world altogether while taking away with them the most deterritorialised component of assemblages, their diagrams, are indeed highly speculative entities. Assemblage theory can be developed without including these controversial lines, as we did in previous chapters. Is there anything we can do to constrain speculation and guide our exploration of the plane of consistency? This would have to be something beyond merely following the process of counter-actualisation conceptually, something allowing us to follow it phenomenologically, by treating our minds as intensive spaces, with their own flows and thresholds. In addition to gradients of electrical potential, our brains are affected by chemical gradients, like that of concentration of neurotransmitters. Tools to manipulate these intensities do exist, in the form of a growing variety of psychoactive chemicals that can be deployed to go beyond the actual world, and produce at least a descriptive phenomenology of the virtual. These chemicals produce phenomena that are public (since anyone can have the experience), reproducible, and remarkable, and that should therefore be collectively explorable in a systematic way, just like any other laboratory phenomenon. The difficulties encountered in trying to capture the experience in linguistic terms are more than made up for by being able to experience a time without present, topological time, directly, without the mediation of concepts. As Deleuze writes:

The point of sensory distortion is often to grasp intensity independently of extensity or prior to the qualities in which it is developed. A pedagogy of the senses . . . is directed towards this aim. Pharmacodynamic experiences or physical experiences such as vertigo approach the same result: they reveal to us that difference in itself, that depth in itself or that intensity in itself at the original moment at which it is neither qualified nor extended. At this point, the harrowing character of intensity, however weak, restores its true meaning: not the anticipation of perception but the proper limit of sensibility . . .41

Notes