Notes on: Deleuze, G and Guattari, F. ( 2004) A Thousand Plateaus.London: Continuum. Chapter 14 The Smooth and the Striated

As implied, smooth space is continuous and uninterrupted whereas striated space has lines, grids or structures imposed on it. However, in actual examples, the two are usually mixed, sometimes in a dynamic way, so that striated spaces can be re-smoothed and vice-versa ( if anything, the trend is to striate more and more smooth spaces). Examples can be found in textiles (assembled fabrics like felt or woven fabric), music (strict tempo or uneven intervals) or the sea (striated eventually by charts and mapping systems like longitude). In each case, social implications follow –eg weaving implies settlement [and surpluses, a division of labour etc], cities rather than wildernesses or deserts. Naturally, some of this joins up with what has been said before – eg smooth space is like the bwo. Travel on a smooth space is rhizomatic.

[This chapter is an interesting development of the more abstract and general argument in Difference and Repetition about the intensive as the presupposition or ground of the extensive.When I read that account,I was particularly interested in how the intensive became extensive. Deleuze talks of realization, actualization, differenTiation and differenCiation (my capitals to make sure I distinguish the two), working in the intensive/virtual and the extensive/actual respectively. DeLanda helps us understand these processes by  giving some familiar thermodynamic examples, where forces and attractors can produce actualization, or change their states within an overall system, like gases being turned into  liquids or solids as temperatures and pressures change. Even so, I found Diff and Rep very 'philosophical', about principles rather than actual cases. This chapter seems to start with, or move quickly onto, actual cases as mixtures, with the abstract distinctions less central,and remaining as a kind of important but potential component. It works better that way round I think, for me at least.]

In more detail: 

Smooth space is nomad space, where the war machine develops, and striated space is sedentary and occupied by the state apparatus.  However, it is not a simple opposition that is interesting, but a more complex difference, a mixture.  The two spaces 'exist only in mixture' (524), and they are constantly turning into each other.  However, we still need an abstract  de jure distinction to explain the actual form that mixtures take de facto.  There is no simple reversibility, but different movements to explain how one turns into the other, which we can see in the various models:

The technological model.  The first example is the fabric, striated by vertical and horizontal elements, one which is fixed and the other mobile.  We might think of these as '"supple solids"'.  Such a striated space is closed on one side [the width of the fabric], and textiles can also have a top and bottom, for example by placing the knots between the threads on one side.  Weaving was even taken as 'the paradigm for "royal science"' (525), by Plato, who saw a model for the state governing people.  Felt is different from woven fabric, however because threads are not separated or intertwined, just entangled at the micro level.  We have a smooth aggregate but not a homogenous one.  There are no fixed and mobile elements, no tops or bottoms, no boundaries.  Felt was invented by nomads, enabling us to draw nice structural analogies: weaving helps sedentaries make clothes 'to annex the body and exterior space' extending to [indexing] the interior of the house.  Nomadic weaving [seems to depend on the felt as a model?] and clothing and houses are indexed to the outside.

There are interlacings between the two types represented by felt and woven fabric [nice third terms] like knitting where the needles alternately play the role of warp and woof, compared to crochet which 'draws an open space in all directions' although it still has a centre.  We can compare embroidery and patchwork, central motifs with 'piece by piece construction' respectively.  We would have to modify the notion a bit, because central motifs can produce tremendous variability and complexity, while patchwork can come to resemble embroidery: however, there is no centre, although there may be recurrent elements [a source is cited for the history of the quilt in American society, and its aesthetic evolution or trajectories, including its importance for women's groups.] Patchwork reminds us that smooth spaces do not have to be homogeneous.  It prefigured op art with its 'amorphous, non formal space'(526).

In music, Boulez worked with smooth and striated space, explaining abstract distinctions as well as concrete mixes.  At its simplest, smooth space time is occupied without counting, offering nonmetric multiplicities and 'directional' spaces not dimensional ones.  The difference can be seen in terms of a break between the regular and undetermined and the standardized.  Frequencies can be distributed [according to official notation], or 'statistically without breaks' (527).  There is a 'modular' principle to regulate the standardized, which can be straight or curved, even or irregular.  The statistical distribution has no break, however, although it might still be equal or 'more or less rare or dense'.  It might still have intervals, however, as intermezzi.  We can see smooth as Nomos and striated as Logos [these terms have several meanings when they are opposed as we saw, and will see below].  Boulez was interested in how the two types of space communicated, melded together, corresponded, how the octave can be replaced by non octave scales for example which might spiral, how musical texture can be created without 'fixed and homogeneous values', the sonic equivalent of op art.  At bottom, striated produces order in succession, for example in 'horizontal melodic lines and vertical harmonic planes'(528), while the smooth offers us continuous variation, continuous development of form, the fusion of harmony and melody, a diagonal across the vertical and horizontal [see chapter 10].

The maritime model shows how lines or trajectories are subordinated to points, whereas in smooth space, 'the points are subordinated to the [line or] trajectory'.  We can see this in the habitation patterns of nomads, where 'the stop follows from the trajectory'.  In smooth space the line is a vector 'not a dimension or metric determination'.  Changes in direction might be directed by the journey itself, but more often by the 'variability of the goal or point to be attained' [anthropological material is being implicitly claimed here?]. 'Smooth space is filled by events or haecceities', rather than well formed and perceived things.  It features affects rather than properties.  It is haptic rather than optical.[an interesting Google definition of the haptic says it is not only a matter of touch and proprioception, but that "haptic feedback devices create the illusion of substance and force within the virtual world" for computer nerds -- you feel the vibration in your control device as you drive down a road etc].  Striated has organized matter, but in smooth, materials 'signal forces and serve as symptoms for them'.  One is intensive and the other extensive, one has distances, the other measures, 'Spatium instead of Extensio', a BWO instead of an organism.

The sea is an interesting example, 'a smooth space par excellence' yet one which was rapidly striated by navigation, following the notion of bearings and the map.  1440 seems to have been a decisive year, although others see a more extended period of struggle to striate.  Early forms included navigation before longitude, based originally on the characteristics of the seas, then an early astronomical system using latitude, then a system which picked up particular characteristics of the Indian and Atlantic oceans to draw 'straight and curved spaces'.  Although commercial trading cities participated, states were necessary to complete it at the global level and impose 'a dimensionality that subordinated directionality'(529).  The striation of the sea led to similar attempts to striate the desert, the air, the stratosphere.  However,  the sea still retained some characteristics of smooth space, however with conceptions like 'the "fleet in Being"' (530), and then the  'perpetual motion of the strategic submarine...  a neonomadism': here, the smooth characteristics are only for 'the purpose of controlling striated space more completely', since the smooth can be deterritorialized more easily [for strategic purposes].  The coming of military automation indicates the same characteristics, where images on a screen are deterritorialized compared to natural objects.  These examples show the 'diabolical powers of organization' that can colonize the smooth.  By contrast, the conquest of the striated by the smooth or holey spaces are better seen as 'parries' to this 'worldwide organization'[a lot of work is done here by Virillo].

Back to abstract definitions: the key issue is the relation between the point and the line, whether the line is between two points, striated space, or the point is between two lines, smooth space.  The lines also differ according to whether they are dimensional with closed intervals, or directional with open intervals.  There are differences of surface as well, whether they are closed and allocated to specific places, or distributed according to frequencies, for example frequencies of crossings.  Again we can see this as a difference between Logos and Nomos.  When it comes to locating the abstract notions however there are difficulties: although sedentary cultivators are obviously different to nomadic animal raisers, peasants can participate in both.  We can also see that nomos as open space can be contrasted not only to the Logos of cultivation, but to the polis as well.  One definition of Bedouinism includes cultivators as well as nomadic animal raisers, and again Bedouin are contrasted to town dwellers.  This is because it has been the town that has always invented agriculture, spreading out to organized farming and imposing striated space on, say, the transhumant.  So the opposition that results between farmers and nomads requires 'a detour through the town as a force of striation' (531).  There may be a deeper distinction, between spaces affected by towns and those are unaffected by them.  We might also wish to oppose the sea to the city, and see the latter as the striation force.  However cities also involve an attack on the town, offering a more striated version, but also a particular combination of the smooth and the holey in its shanty towns and patch works, 'an explosive misery secreted by the city'.

So in each case simple opposition produces complications and alternations, although these 'basically confirm the distinction, precisely because they bring dissymmetrical movements into play' [so nothing falsifies as usual -- the very difficulties only show how right the original premise was, like Christianity ].  It is enough to say that there are two kinds of voyage, according to the combination of line and space [and 'Goethe travel and Kleist travel', an unplayable reference, unless you have read the earlier chapter on becoming].  This is also the difference between tree and rhizome travel.  However everything intermingles: 'because the differences are not objective'[in the sense of not fixed or natural?]. So we can live in a striated way in the desert, and be an urban nomad, as was Henry Miller in [his novel about] Clichy [and also see De Certeau on walking in the city].  Fitzgerald said the same, that we can have voyages without long distance travel.  Drug users have such voyages.  The Toynbee quote crops up again about nomads not really wanting to move but rather holding a smooth space.

'To think is to voyage' (532), trying to be in a different way in space or for space, voyaging smoothly and thinking smoothly.  Again, there are often intersections or reversals [the example is Wim Wenders' film Kings of the Road].  It is not a matter of returning to old forms of navigation or ancient nomadic ways of life, but organizing a confrontation between the striated and the smooth in all sorts of ways.

Multiplicities are described again, first as a mathematical concept devised by a certain Riemann [DeLanda explains]. [Apparently the concept arose from certain problems of measurement, the difficulties of comparing things in some media, such as the difference in pairs of sonic tones –metric and non-metric differences, distances and lengths]. We then get reminded about Bergson who contrasted duration as a different kind of multiplicity from [objective clock time?] – something ‘qualitative, fusional, continuous, the other numerical and homogeneous, discrete’ (534). I've had a bash at Bergson, via Deleuze on Bergsonism here

Again, in more detail:

 The mathematical model.  [The first part of this is manageable, largely thanks to chapters one and two of DeLanda] but the next section is formidable, and presumably requires some advanced knowledge of topology.  If you ever want to test anyone who claims to have read this chapter, ask them about the section on the physical model, 538 -40].  Riemann was one of the first to decide that the multiple was a noun, with a multiplicity.  This immediately led to 'the end of dialectics' (533), and instead, people started to study multiplicities, their typology and topology.  Riemann worked with N dimensions in the multiplicity, with 'n determinations', some independent of the situation and some dependent on it.  [The example says the magnitude of a vertical line can be compared to the magnitude of a horizontal line  - - presumably, in Cartesian space as on a graph, where X and Y values vary with each other].  This is a metric multiplicity that can striated [gridded in the case of a graph], and we are talking about determinations as magnitudes.  However, when we are looking at the relation between sounds, it is different because we cannot simply compare sounds in terms of their combined intensity and pitch [which are non metric measures, or at least not ratio].  Here, it is possible to make some progress if one sound is part of [envelops]  the other, or if we confine ourselves to ordinals, saying that one is smaller than the other, but not being able to go any further.  In this case, multiplicities are not metric.  They cannot be directly striated, although they can be indirectly striated, although 'they always resist'.  They still have properties that we can describe with rigour, but they are 'anexact'.  Instead of magnitudes, we have to think in terms of distances [Meinong and Russell are credited with this].  

By distances, we mean something which is non metric, [so we are departing from the commonsense terms here, and invoking something like 'proximity'] although we can analyze them in terms of a kind of series of determinations [something like an accumulation of distance?].  However, a distance, like any intensive variable 'cannot divide without changing its nature each time'(533).  The examples here are temperatures or speeds, which are measured on scales that do not allow us to sum two smaller temperatures or speeds to get a larger one.  So we can think of distance in terms of an 'overall set of ordered differences' [as in the sum of different proximities - - first close, then more distant and so on, as in an elliptical orbit.  However, it is normal in common sense to simply metricate distances and then take averages?  We cheerfully ignore the intensive qualities of the variable, just as we do with student assessment, where we add grades for all sorts of different things to produce an average grade].  In an example, the movement of a horse can be divided into gallop, trot and walk, but these are changes in nature, and it we could not just sum them.  In other words, distance implies 'a process of continuous variation' unlike metric multiplicities where we have regular variations [?] and we can plot constants and variables on them.  [Again, this might be a difference in the abstract, but practical measurement simply ignores these differences and metricates everything?]

Bergson is important here as well.  In Time and Free Will, duration is one of these intensive multiplicities which cannot be metricated or turned into magnitudes [of course,  we do this all the time in practice by imposing clock time] Duration can be divided, but again each division changes its nature [the example quoted is one of famous paradoxes about Achilles and the tortoise --it's a paradox only because we are trying to metricate the speed of the running by turning it into discrete steps. I've never really understood this -- until recently, when I realized that in Zeno's model Achilles could only ever take steps the size of the last one taken by the tortoise , so of course he never catches up. As Bergson says, real running is never like that].  Bergson contrasts this to metricated multiplicities, which extend in an homogenous fashion.  The two different kinds of multiplicity were important in the confrontation between Bergson and Einstein [which again I have never really understood, although the gist of it, apparently, is that even relative time is clock time, so Einstein is still working within metric multiplicities and ignoring the non metric qualities of time. A subtler  point is that Einstein is claiming a classic metaphysical stance in declaring himself able to compare moments widely separated by space in order to judge them as similar, so the debate is really between two cosmologies, not 'real time' and mere 'subjective time' which is how it is usually handled. There is a lengthy discussion by Latour here and some brief notes on that discussion here].  For Bergson, 'matter goes back and forth between the two: sometimes it is already enveloped in qualitative multiplicity, sometimes already developed in a metric "schema" that draws it out of itself' (534).

Much of the discussion in the book is about the differences between two types of multiplicity, including the distinction between the arborescent and the rhizomatic, the mass and the pack, striated and smooth space, and its metamorphosis. 

Here is a typical chunk of prose:

We have on numerous occasions encountered all kinds of differences between two types of multiplicities: metric and nonmetric; extensive and qualitative; centered and acentered; arborescent and rhizomatic; numerical and flat; dimensional and directional; of masses and of packs; of magnitude and of distance; of breaks and of frequency; striated and smooth. Not only is that which peoples a smooth space a multiplicity that changes in nature when it divides-—such as tribes in the desert: constantly modified distances, packs that are always undergoing metamorphosis— but smooth space itself, desert, steppe, sea, or ice, is a multiplicity of this type, nonmetric, acentered, directional, etc. [that is, it describes the 'container' that is space itself, better than seeing space as a sphere, plane etc] . Now it might be thought that the Number would belong exclusively to the other multiplicities, that it would accord them the scientific status nonmetric multiplicities lack. But this is only partially true. it is true that the number is the correlate of the metric: magnitudes can striate space only by reference to numbers, and conversely, numbers are used to express increasingly complex relations between magnitudes, thus giving rise to ideal spaces reinforcing the striation and making it coextensive with all of matter. There is therefore a correlation within metric multiplicities between geometry and arithmetic, geometry and algebra, which is constitutive of major science (the most profound authors in this respect are those who have seen that the number, even in its simplest forms, is exclusively cardinal in character, and the unit exclusively divisible). It could be said on the other hand that nonmetric multiplicities or the multiplicities of smooth space pertain only to a minor geometry that is purely operative and qualitative, in which calculation is necessarily very limited, and the local operations of which are not even capable of a general translatability or a homogeneous system of location [ and this is Riemann's notion of big space being made up of contiguous local spaces which can be quite different -- it had a number of scientific implications too, including relativity] (534—5)

Here is my gloss:

Let us discuss number again [referring back to the chapter on the nomad].  Numbers can be applied to non metric multiplicities and being able to apply a number is an important aspect of striation; it's also true that numbers are used to express relations in an increasing complex way, producing 'ideal spaces', which might now be 'extensive with all of the matter'. Metric multiplicity is connected with geometry and algebra and the way that turns into major science, leaving nonmetric multiplicities having to operate with a minor geometry, a qualitative one, where 'calculation is necessarily very limited', and generalization impossible.  However, this 'nearly illiterate, ametric geometry' somehow confers independence on the number, permitting it to be applied in major geometry [is this to argue that numbers look objective and independent of political projects to striate, because they are not universally applicable?].  In smooth space, numbers refer to distribution, and when we divide, we divide the nature of the unit each time. The units represent distances not magnitudes: this is the 'ordinal, directional, and nomadic, articulated number, the numbering number' (535).  [I think we're getting to the punch line here - - 'we may say of every multiplicity that it is already a number, and still a unit.  But the number and the unit, and even the way in which the unit divides, are different in each case'].  Minor science is a source of creativity for major science, by reminding it constantly of 'matter, singularity, variation, intuitionist geometry and the numbering number'.

We have discussed the way in which smooth spaces can feature 'enveloping' distances, but there is 'a second, more important, aspect', where we cannot compare two determinations.  We are back to Riemann spaces as a series of patches, nonhomogeneous, defined in an odd way ['the expression that defines the square of the distance between two infinitely proximate points'], which apparently permits us to locate local points, but not the spaces in relation to each other.  However, these can be linked 'in an infinite number of ways', a form of juxtaposition not attachment.  We have no need to metricate, to consider, for example frequencies or accumulations of patches of space.  We cannot metricate, apparently.  There are connections, like 'rhythmic values'.  There is heterogeneity.  This also helps us define 'smooth space in general' (536) [note that our heroes admit that the relations and values' 'can be translated into a metric space'].  Again, smooth space can be described as a nomos.

Dissymmetry is involved when crossing from smooth to striated and vice versa [again it DeLanda is very helpful with this notion of symmetry, which turns on the shape of objects and their regularity.  He goes on to describe the transitions from virtual to actual in terms of symmetry breaking thresholds, heading to less and less symmetrical versions].  The characteristics we have discussed of minor and major geometry, metrics and nonmetrics are necessary to translate from one kind of space to the other, or rather to translate their data.  Translating is not simple.  It used to be done by translating movement into space traversed [see the discussion in Cinema 1], but this is flawed, Bergson argued.  Translation obviously involves 'subjugating, overcoding, metriczing' smooth space, and this neutralizes it but [somehow] allows it to propagate, extend and renew.  'Major science has a perpetual need for the inspiration of the minor', and minor science must also 'confront and conform to the highest scientific requirements'.  As examples, intensities can be translated into extensive quantities, multiples of distance into systems of magnitude [through logarithms, for example --the mysteries of the logarithmic scale in path analysis, which I used to know about once: Wikipedia to the rescue again --

A logarithmic scale is a nonlinear scale used when there is a large range of quantities. Common uses include the earthquake strength, sound loudness, light intensity, and pH of solutions. It is based on orders of magnitude, rather than a standard linear scale, so each mark on the scale is the previous mark multiplied by a value.]

In another example, Riemann's notion of patches of smooth space can be joined to euclidean notions [something to do with imposing parallel vectors to striate it -- pass].  Overall, 'nothing is ever done with', and smooth and striated space constantly interact with each other. However, ‘all progress is made by and in striated space, but all becoming occurs in smooth space’ (537) [we can see why below] .

Some other mathematicians have tried to define smooth spaces, including Mandelbrot with his notion of the fractal: 'aggregates whose number of dimensions is fractional rather than a whole, or else whole but with continuous variation in direction'.  One example is given in a diagram, where we take a line, replace its central third by the angle of an equilateral triangle, then do the same for each side of the triangle and so on.  Apparently, the line overall would be infinite, and have 'a dimension greater than one but less than a surface' [which requires two dimensions] (537) [I don't see how that dimension can be more than one but less than two --because the line does not enclose a figure? But then, I don't understand fractal geometry. Wikipedia might help :

The fractal dimension of a curve can be explained intuitively thinking of a fractal line as an object too detailed to be one-dimensional, but too simple to be two-dimensional.^.. Therefore its dimension might best be described not by its usual topological dimension of 1 but by its fractal dimension, which in this case is a number between one and two...[think of] a basic concept of change in detail with change in scale...a curve with fractal dimension very near to 1, say 1.10, behaves quite like an ordinary line, but a curve with fractal dimension 1.9 winds convolutedly through space very nearly like a surface...fractal curves have complexity in the form of self-similarity and detail that ordinary lines lack.. a fractal line measured at first to be one length, when remeasured using a new stick scaled by 1/3 of the old may not be the expected 3 but instead 4 times as many scaled sticks long.

The example on 538 says we calculate the dimension of that particular Mandelbrot line by dividing log 4 by log 3 -- giving 1.2. This seems to be one way, out of several, of estimating the dimension of more complex lines. [It would fit the actual example, where a line originally thought to be 3 units long at a particular scale, turns out to have an equilateral triangle in its middle third when we have scaled it up . Instead of 3 segments we now have 4 of equal length,and when we scale it up even further, each of those segments turn out to have equilateral triangles in the middle third --and so on. So every apparently 3-unit segment is a 4-segment unit. The division of 4 by 3 is some way to indicate this. Presumably we have to convert our normal measures into a log scale for the usual reasons as above. Apologies to any mathematicians]

We can also do this with the example of the cube  into which holes are drilled, then each hole is surrounded by holes, and then those holes are surrounded and so on.  The resulting figure, the Sierpenski sponge [illustrated on 538] is between a volume and a surface , and we can see this as a mathematical model to show the relation [affinity] between a free space and a holey space.  Models of brownian motion and turbulence are also fractals, and can help us define a fuzzy aggregate.  [There is a slightly simpler account of Sierpenski triangles here]

[As usual, whether our heroes really knew all this or whether they were content to rely on some texts they had browsed is unclear. How it all fits the philosophical notion of smooth space is also unclear. Is the fractal really an rigorous example of a smooth space in mathematical terms?  Is this their example of what a smooth space is or did Mandelbrot and the others agree? Are these examples of peculiar spaces really meant to illustrate smooth space or perhaps just to shake our normal conceptions of {Euclidian} space?]

The main issue is that we can have a general rigorous and mathematical determination for smooth space that adequately accounts for its differences from and relations to striated space. [Which seems like it is the strong version of the claim]   Thus: (1) a striated or metric aggregate has a whole number of dimensions and constant directions; (2) non metric or smooth space involves the 'construction of a line with a fractional number of dimensions greater than one', or a 'surface with a fractional number of dimensions greater than two'; (3) the fractional dimensions indicates 'a properly directional space (with continuous variation in direction, and without tangent) [part of the formal definition but why is it important here?] '; (4) a smooth space therefore has dimensions lower than that [any real or realized in 2 or 3 dimension Euclidian space] 'which moves through it or is inscribed in it; in this sense it is a flat multiplicity, for example, a line that fills a plane without ceasing to be a line'; (5) smooth space tends to be identified with that which occupies it, with the same power, with the same 'anexact yet rigorous form of the numbering or nonwhole number (occupy without counting) -- [counting with integers that is]'; (6) a smooth space of this kind can be seen as 'an accumulation of proximities' constituting zones of indiscernibility -- this is why smooth spaces are proper to becoming, [because they are themselves poised between a line and surface, a volume and a surface - my gloss].

[Seems like a structural analogy here -- fractal space is odd and so is smooth space -- up until point 6 {maybe point 5 too?} . This is where Riemann is finally adjudged to be able to explain space and its odd structures best?]

The physical model [equally baffling].  So far we have implied a certain model of striation, such as a case where two series of parallels intersect perpendicularly, and the verticals are fixed or constant while the horizontals are variables.  This is so for weaving, music, and grids of latitude and longitude.  Tight striations produce homogenous spaces, so homogeneity does not belong to smooth space, but to striated: however when striations work to produce perfect homogeneity 'it is apt to reimpart smooth space'(539) [almost to permit a smoothing superimposed on the homogeneous].  Otherwise, the smooth relates to heterogeneity, felt or patchwork,  rhythm rather than harmony-melody, Riemann rather than Euclid.  It is 'a continuous variation that exceeds any [regular] distribution of constants and variables, the freeing of a line that does not pass between two points, the formation of a plane that does not proceed by parallel and perpendicular lines'.

We can pursue the relation between the homogeneous and the striated by thinking of an imaginary physics.  First you striate space with parallel gravitational verticals, and note that the resultant of these are parallels producing a point inside a body occupying the spaces, the centre of gravity.  We can then change the direction of parallel forces, rotating them until, say, they become perpendicular to the original direction: the centre of gravity does not change.  The implication is that 'gravity is a particular case of a universal attraction following straight lines, or biunivocal relations between two bodies' [the former was Newton's view, apparently] .  We can then define work as 'a force - displacement relation in a certain direction'.  We have then described 'increasingly perfect striated space', striated vertically and horizontally and indeed in every direction, 'subordinated to points' [that is running through a point at the centre of gravity?].  The Greeks foresaw this and described perfectly homogenous space in similar ways.  We can see double striations as an adequate model of the state apparatus, with the vertical lines dominating empire and the 'isotropic apparatus' [Wikipedia defines Isotropy as uniformity in all orientations] defining the city state.  In this way, we can use geometry to understand the state, since it 'lies at the crossroads of a physics problem'[we have to see politics as fundamentally a physics problem first, though?].

However, problems arise when more than two bodies are considered [since they interact in nonlinear ways?].  Space can escape from striation.  One way involves 'declination...the infinitely small deviation between a gravitational vertical and the arc of a circle to which the vertical is tangent'.  Another way involves developing spirals or vortexes which can occupy all the points of the space, but not in a way which involves the striation of parallels, unlike laminar distributions (540).  Between the declination and the vortex stretches a possible smooth space 'whose element is declination and which is peopled by a spiral', the declination providing the minimum and the spiral the excessive possibilities.  Apparently, much of this is argued by Serres [French reference page 644], who said the Greeks also saw links between atoms and hydraulics, since 'the ancient atom' was always seen as essentially a matter of course and flow.  The Greeks also had an noneuclidean geometry and a physics interested in matter that was not solid or lamellar [SIC].  [Bless wikipedia again for this:

Lamellar structures or microstructures are composed of fine, alternating layers of different materials in the form of lamellae. They are often observed in cases where a phase transformation front moves quickly, leaving behind two solid products, as in rapid cooling of eutectic (such as solder) or eutectoid (such as pearlite) systems.]

This helped Greek maths and physics describe things that look rather like a war machine rather than a state apparatus - 'the physics of packs, turbulences,  catastrophes and epidemics'.

There is a distinction between the free action in smooth space and work in the striated space, and this distinction is also developed in the 19th century.  A physical conception of work as a matter of weights, heights, forces and displacements became allied to a socio economic concept of labour power or abstract labour, something homogenous and abstract which could be applied to all work and which could be metricated.  This produced a link between physics and sociology, the first providing some sort of mechanical currency for work, and the latter providing an economic standard.  This clearly affected the wage regime: 'physics had never been more social' (541) because there was a convergence between attempting to define and metricate things like the value of a force of lift and pull exerted 'by a standard-man'.  As a result, every activity could be translated into work and free action could be disciplined, or at least relegated to mere leisure, defined against work.  This work model became a fundamental part of the state apparatuses.  Public work came to support the notion of standard man, not the usual example of pin manufacturing [in Adam Smith], but in public construction, the organization of armies including industrial production of weapons.  We can also see this as an appropriation of the war machine, capturing it by subjugating it to the work model [sounds like Foucault].  The war machine was perhaps the first thing to be striated in this way, and where free action in smooth spaces was first conquered.  The concept of labour that developed was also always associated with surplus and stockpiling, and the disciplining of free action, 'the nullification of smooth spaces'. If there is no state and no surplus labour there will be no work model either.  Instead we will have 'the continuous variation of free action', passing from action to song to speech to enterprise: only 'rare peak moments' will resemble work.

This analysis might explain some of the curious historical findings that, for example black people and Indians did not understand work, (and Indians didn't understand slavery either).  We can see such societies as 'societies of free action and smooth space that have no use for a work factor' (542), not societies that value laziness, or that have no proper laws.  Continuous variation of non-economic activity also featured 'a rigour and cruelty all its own', such as abandoning people who could not travel.  It is also the case that surplus labour appeared in archaic and ancient forms as tribute or corvée.  Here we can see 'the concept of labour...at its clearest', since much striation is required [based on political, social/cultural and religious practices] .  In capitalism, surplus labour gets more and more like ordinary labour: in the older societies, normal labour was strictly separated from surplus labour by time.  Marx would argue [maybe in Grundrisse, Notebook VII?]  that this indicates that surplus value is no longer localisable in capitalism, and went on to suggest that machines themselves could produce surplus value [in that strange bit about the general intellect?], and that the circulation of capital itself would blur the distinction between variable and constant capital.  All labour would involve surplus labour, but labour itself would not be the only source of surplus.  Instead of explicit striations of space and time, human regulation becomes a matter of 'a generalised "machine enslavement"', to which all contribute even if they do not actually do work.  Capitalist calculation no longer requires just calculating quantities of labour, but is more complex and qualitative and includes infrastructure, the media, 'every semiotic system' (543).  In this, it has 'reconstituted a sort of smooth space' There is still naked striation, mostly exercised by the state, but in integrated world capitalism, a new smooth space is produced, and capitalism is increasingly based on machinic components not human labour.  The multinationals produce this deterritorialized smooth space, breaking with the classical striations.  As new forms of turnover develop, and the circulation of capital increases, the old distinctions between constant and variable capital become increasingly relative: the real difference is 'between striated capital and smooth capital', and how the former can produce the latter in new, including global, ways.

[I think this is the best discussion of Marx in the whole book, and underpins the 'society of control' model]

The aesthetic model: nomad art.  Nomad art has been defined in terms of some general characteristics.  First '"close range " vision'; second tactile or haptic space not optical.  Haptic involves more than tactile, and also implies that vision imparts more than just a visual sensation.  These two are coupled together, according to someone called Riegl and others, like Worringer. [see wikipedia, as ever --both, apparently, saw the 'urge to abstraction' as a part of some inherent 'artistic impulse'. The entry explains that textiles and art might be linked, but adds little to the notion of the haptic].  Borrowing this work helps us see that the smooth involves close vision and haptic space, whereas the striated relates to a more distant vision and a more optical space.  Again we have to remember that the two are interrelated.  Painting can become a close range even if viewed from far away, and Cezanne among others talked about losing your self in the objects to be painted.  Subsequent striations can be imposed, or perhaps a cycle between striations and smoothness.  Musicians also seem to have some idea of close range hearing unlike the listening audience, and writers write with short term memory, while assuming long-term memory in the reader.

Haptic smooth space has its orientations 'in continuous variation; it operates step-by-step' (544), as when we navigate deserts.  We always feel we are 'on' such a space, never able to see it from another perspective.  The landmarks change according to local conditions including temporary vegetation.  There is no convenient visual model enabling regularization of points of reference [until we invented maps, you clots].  Fully qualified nomads, unlike observers, contain 'tactile relations among themselves', and these are structured in an interesting way [an interesting note on page 644 [those pages discuss Riegland Worringer too] quotes a certain Chatelet in attempting to use the concept of Riemannian space with its associated connection to monads, which can pursue paths 'step-by-step by local relations'.  In a smart retort, the one that ends Deleuze's book on Leibniz, we need to replace these monads with nomadology]:

[There have been quite a few Châtelets it seems. The index says this Châtelet is François Châtelet, but he is someone different. In looking up stuff in Wikipedia I also came across an amazing female philosopher and polymath, one Émilie Du Châtelet]

There is no ambient space containing a multiplicity, rather a pattern of proceeding 'according to ordered differentiation that give rise to intrinsic variations in the division of a single distance' [a bit like the way I drive over familiar roads, from point to point instead of seeing the journey as a line embedded in some objective space]. We can see these conceptions in nomad art [or at least in 'the most famous works', 545], with twisted animals floating in the air, the ground changing direction, limbs pointing in odd directions, challenging the normal perception based on '"monadological" points of view', a special kind of animality requiring some direct contact with the mind.  Striated space has all the opposite qualities, and therefore 'it is less easy to evaluate the creative potentials'.

We should not think of the opposition as a matter of global and local, because the global is relative [too vague to be striated?]  whereas the local is absolute.  [Then back to perception and the difference between haptic and optical .  Another example of where we badly need an editor] - in the former, there is no easy horizon, background or perspective, no easy division into intermediary and far distances - 'like Eskimos space' [citing a novel I think].  Arab architecture also produces a space that begins near and low, with 'the light and the aery below and the solid and heavy above', reversing the normal notion of gravity and direction.  This is an illustration of the 'nomadic absolute...a local integration moving from part to part and constituting smooth space in an infinite succession of linkages and changes in direction.  It is an absolute that is one with becoming itself, with process'. We see the absolute as local first because that is where it is manifested.  There is also a global striated space, something optical, long distance vision, and this can be seen as 'the relative global', the horizon or background, something which encompasses or englobes, and against which we can distinguish forms.  The absolute might appear as a definite centre, repelling any other attempt to integrate.  In this way, striation appears in the smooth, which serves as the horizon, something to ground elements in the foreground.  It can also serve as representing 'the loathesome [SIC] deep', (546) something smooth beyond life.  We can see it with the great imperial religions, who need a smooth space on which they can impose a law which domesticates the absolute.

The art critics cited above, including  Riegl, have analyzed Egyptian art in this way.  It has a definite horizon or background, reduces space to the plane, 'enclosing individuality and withdrawing it from change'.  Pyramids indicate how every plane can become a surface against the immobile desert.  In Greek art, however an optical space separates itself out, merging background and form, conquering depth, working with volumes, organizing perspectives and introducing reliefs.  'The optical makes that striation tighter and more perfect', not necessarily from the point of view of the artist.  It becomes imperial.  When empires are threatened by barbarians, new forms emerge like 'Gothic art in the broadest sense', but there are also nomadic elements - for example the goths and the huns linked to the east and the north, but represented neither.  Other empires have their own nomads, with their own specificity and their own '"will" to art' (547), although this is often overlooked: again it shows us the crucial role of the intermediary or the interval as something substantial, a becoming: 'it invents a "becoming artist"'.

So we have subordinated the differences between haptic and optical, close and distant vision to the 'primordial duality between the smooth and the striated'.  We usually find mixtures, however, where the haptic can be used to striate as well as to produce smooth space.  It is more that these functions 'presuppose the smooth'.  Similarly, the optical can also reinstate the smooth, 'liberating light and modulating colour'.  Again, however it is important to get the distinctions between spaces before we can understand [pure] distinctions between haptic and optical [so again we are playing the transcendental deduction card, taking people's work on the haptic and the optical and arguing that they really need to be translated into some other terms.  In this case, possibly embarrassingly, it is a duality, however, not the usual multiplicity].

We can identify an additional quality of lines, which may be abstract or concrete.  Again art critics [Worringer in this case] use these terms, seeing the abstract line as the beginning of art, the first expression of artistic will, appearing first in the imperial Egyptian form with its rectilinear depictions.  However, D and G argue that the best manifestation actually is in the '"gothic or Northern line"' (548) [it is a bit gothic come to think of it - apologies to non Londoners, the Northern Line is a subway route. I was making a joke. Sorry.], something nomadic.  The Egyptian rectilinear line 'is negatively motivated by anxiety' and attempts to depict instead 'the constancy and eternity of an In-Itself', but the nomad line 'has a multiple orientation and passes between points', abstract in a different sense.  It is 'positively motivated by the smooth space it draws'.  This abstract line 'is the affect of smooth spaces not a feeling of anxiety that calls forth striation'.  Even 'prehistoric savage and childish' art break with representation as imitation of nature, requiring no particular will to art.  The line can be even more abstract if there is no writing: when writing develops it takes charge of abstraction and downgrades the artistic line.  If there is no writing, only the artistic line provides an outlet for the power of abstraction.  Empires always use the abstract line to produce concrete figures, striations.  Abstract lines proceed Empires, however and can be seen as 'part of the originality or irreducibility of nomad art' (549).

It is not that the abstract is 'directly opposed to the figurative' [the figurative here meaning representations of actual normal figures like humans and animals, and not 'the figural' as some more abstract depiction?].  There can be a figurative line equally motivated by the will to art, but particular characteristics of lines produce the figurative.  [We have already seen how] simple forms of striation involve systems of verticals and horizontals running through points.  Here, the line defines a contour of striated space, so it is already representative.  Lines that do not follow these contours, do not go from one point  to another but pass between them, always avoiding horizontals, verticals and even diagonals, that constantly changes direction, does not divide outsides from insides, forms from backgrounds, or beginnings from ends, are truly abstract, 'and describe[...] a smooth space'.  There is still expression, but not in a 'stable and symmetrical form', grounded in a series of points and lines.  However it still possesses 'material traits of expression' which have effects, for example repetition rather than symmetry [symmetry here limits repetition and maintains dominating central points].  Free action on the other hand 'unleashes the power of repetition as a machinic force' [I haven't got time to check, but I wonder if this is contradicting what Deleuze says about repetition as limited in its creativity in Difference and Repetition].  Free action is disjointed and decentres, 'disjointed polytheism'.  This is a different form of expression than the one we are used to which depend on grids and organized matter.

Worringer also wants to contrast the abstract to the organic, the latter inspiring a feeling 'that unites representation with a subject'[the actual term is empathy], (550) and sees processes at work in art which correspond to these 'natural organic tendencies'.  However, there is no simple opposition between the geometrical and the organic, and the Greek organic line is clearly linked to the Egyptian geometrical one.  The organic is still symmetrical, corresponding to rectilinear coordinates.  The organic body is still 'prolonged by straight lines' into the distance.  It is not surprising that it awards primacy to human beings or their faces.  Human expression itself is both organic, and a way of relating organisms to metric space.  Abstract or nomadic art belongs to free action, 'inorganic yet alive, and all the more or alive by being inorganic'[French wit].  It is neither geometrical nor organic.  It replaces mechanical relations with intuition.  Heads 'unravel and coil into ribbons' spiral, zigzag and snake expressing flow, unconfined by organism, representing life which organisms only divert, ' a powerful life without organs, [guess what's coming up -- yes, the BWO] a Body that is all the more alive for having no organs', representing 'everything that passes between organisms'.  Even though critics have established other distinctions within nomadic art, 'in the end everyone agrees that it is a question of a single will or a single becoming'.  It can express animality as inorganic  or 'supraorganic' (551), and this is what makes it 'combine so well with abstraction'.  It is not tied either to organisms'or two geometry, but represents a 'vital force', and it is this that 'draws' smooth space.  The abstract line is the affect of smooth space', and organic representation is the 'feeling presiding over striated space'.  This distinction between abstract and organic lines again operate at a more general than haptic/optic divisions.  Ultimately what counts as abstract in modern art is this 'line of variable direction that describes no counter and delimits no form'

For this reason, because we want to be parsimonious, and because we want to keep our own preferred categories intact]...

Do not multiply models.[fucking rich coming from these obsessives]  We are well aware that there are many others: a ludic model, which would compare games according to their type of space and found game theory on different principles (for example, the smooth space of Go versus the striated space of chess); and a noological model concerned not with thought contents (ideology) but with the form, manner or mode, and function of thought, according to the mental space it draws and from the point of view of a general theory of thought, a thinking of thought. And so on. Moreover, there are still other kinds of space that should be taken into account, for example, holey space and the way it communicates with the smooth and the striated in different ways. What interests us in operations of striation and smoothing are  precisely the passages or combinations: how the forces at work within space continually striate it, and how in the course of its striation it develops other forces and emits new smooth spaces. Even the most striated city gives rise to smooth spaces: to live in the city as a nomad, or as a cave dweller. Movements, speed and slowness, are sometimes enough to reconstruct a smooth space. Of course, smooth spaces are not in themselves liberatory. But the struggle is changed or displaced in them, and life reconstitutes its stakes, confronts new obstacles, invents new paces, switches adversaries. Never believe that a smooth space will suffice to save us.