Delueze: Smooth and Striated

From the section of "Smooth & Striated" in which he discuss mathematics in the context of multiplicities:

"...it might be thought that the Number would belong exclusively to the other multiplicities, that it would accord them the scientific status non-metric multiplicities lack ... the number is the correlate of the metric: magnatudes can striated space only by reference to the numbers, and conversely, numbers are used to express increasingly complex relations between magnatudes, thus giving rise to ideal spaces reinforcing the situation and making it coextensive with all of matter. There is therefor a correlation within metric multiplicities between geometry and arithmetic, geometry and algebra, which is constitutive of major science..."

Although he goes on to provide exceptions, he explains, in relatively clear terms, how numbers can define magnitudes which define space.

In the first model (the technological model), Deleuze uses fabric as an example.
"...it is constituted by two kinds of parallel elemnets...there are vertical and horizontal elements, and the two intertwine, intersect perpendicularly...one is fixed the other mobile, passing above and beneath the fixed."

Based on the assimilation of fabrics, this math model allows us to determine which is smooth or striated. Felt, however, as the "anti fabric" "...has neither top nor bottom nor center, it does not assign fixed and mobile elements but rather distributes a continuous variation." There is no defined assimilation or intertwining like other various fabrics. Felt is like a continuous flow of meshed fibers, different, but continuous. "It implies no separation of threads..." It's a continuous variation revealing the principles of topology.

In the technological model, Deleuze discusses fabric, felt and quilts, using quilts or patchwork as an example of a mathematical model: "An amorphous collection of juxtaposed pieces that can be joined together in an infinite number of ways." (P. 476)

Deleuze argues that the patchwork is literally a Riemannian space, which is essentially a space made up of small domains. Explaining further, thanks to wikipedia: a Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, smooth manifolds with a Riemannian metric, i.e. with an inner product on the tangent space at each point which varies smoothly from point to point. This gives in particular local notions of angle, length of curves, surface area, and volume. From those some other global quantities can be derived by integrating local contributions.

Deleuze then uses the the smooth space of patchwork to demonstrate that the smooth does not mean that it is homogenous but rather "an amorphous, non-formal space prefiguring op(tical) art"

The Smooth and the Striated

“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.” (Page 484)

Numbers can be used to striate smooth space. The multiplicities of smooth space are difficult to measure or calculate without using numbers or units. However, dividing a thing into a smooth space limits the understanding of the characteristics of the smooth space.

Reading Schedule

Hello everyone, for the week following spring break this is our reading list
Please make sure that you are caught up on the reading.
I am giving you a short assignment in order to wrap things up in the first section on mathematical models and the tradition of Euclid, Descartes, the rupture with digital design and calculus based and topology based techniques.
Two pages, singles spaced: discuss the use of Cartesian and non-Cartesian models of mathematics in any of the contemporary architects (Lynn, UN Studio, FOA, etc.) and identtify how the transformation of models signals a new ontology for architecture.

Cecil Balmond, Informal (Munich; New York, 2002) 347-365
Ben van Berkel, Basically (for now) Three Topics, in C. Davidson ed., Anyhow (MIT Press: New Cambridge Mass., 1998) 88-93.
Alejandro Zaera Polo, FOA, Phylogenesis, passages.
Gilles Delueze, Fold: Leibniz and the baroque (Minneapolis, 1993)
Greg Lynn, “Blob Tectonics, Or why Tectonics is Square and Topology is Groovy, in Folds, Bodies, and Blobs (Originally published in Any Magazine, no. 14.)

DeLeuze uses the concept of fabric and the weave as a way to explain the organizational properties of the smooth and striated. Deleuze states that "A fabric presents in principle a certain number of characteristics that permit us to define it as a striated space" He then identifies these different qualities as being: the vertical and horizontal, fixed and mobile, the delimited, and having a top or bottom.
These princples give us a mathematical model in which we can asses weather the particular fabric is either smooth or striated by nature of its organization. Further, Deleuze states that felt uses a different set of organizational principles, and is in a sense an anti-fabric. "In which there is no top or bottom, and is infinite in scale and complexity." This continuous entanglement of felts microfibers creates a smooth space that is not bound by princlples of sepratation and intertwining(as in other fabrics), but instead is based on constant variation of connectivity and topology.

Deleuze uses a Mathematical model in his description of the sea. Through his distinction of striated and smooth space Deleuze identifies the ocean as a "smooth space par excellence." He notes the behavior of striated space is that "lines or trajectories tend to be [subordinate] to points," whereas the opposite is true in smooth space. This identification however disproves the properties in which smooth space can be determined since in his 'Maritime Model' Deleuze points out that "the striation of the sea was a result of navigation on the open water."

Using the ocean as an example of the geometric and topological mathematical models seen in smooth and striated space, Delueze physically realizes the intermingling relationship between the two systems.

mathematical model

Deleuze uses the mathematical model of three dimensional space to describe the manner in which the musical concepts of harmony and melody relate to each other. He states, "Returning to the opposition [between the smooth and the striated], the striated is that which intertwines fixed and variable elements, produces an order and succession of distinct forms, and organizes horizontal melodic lines and vertical harmonic planes." By equating harmony to a greater level of geometric complexity, he is drawing the distinction between the two elements such that he concludes harmony is the determination of melody plus some other elusive multiplicity. He is somehow trying to suggest that there is a greater level of conceptual depth within the construction of harmony than there is in melody. I chose to highlight this use of a mathematical model because I believe his analogy fails.

Harmony and melody are bound and ruled by the same lines and differ only in the mechanics in which they portray the same underlying complexity. Harmony portrays the (traidic) chord structure of a piece through the use of unison, melody through the use of sequence and arpeggiation. In melody interstitial tones are inserted not only to create stepwise motion along the scale, which may lead one to incorrectly conclude that melody and harmony are fundamentally different, but also as way to create delays or rhythmic way stations that allow the composer to hold off from using a tone which would establish the next chord of the harmonic progression before the chord infrastructure dictates. On average throughout a piece, these interstitial tones will appear more often on the off beat of the rhythmic meter, where as the three tones that make up a chord (or base triad of a seventh chord in jazz) will more often occupy the primary positions of the rhythmic structure. These primary tones make up the very same notes of the backing harmonies played by whatever music ensemble backs up the lead melody.

I would more view the concepts of tonal and rhythmic orders within a piece of music to be the multiplicities that act in different, often contradictory, ways which are the parameters whose intensities a composer will modulate to create the fundamental nature of their creation.

The Smooth And The Striated

"Fractals are aggregates whose number of dimensions is fractional rather than whole, or else whole but with continuous variation in direction." (Page 486)

The line with two end points and a center point starts out smooth and straight. By continuously adding triangles to the old and each new mid point, striated surfaces form. Eventually the striations become so dense, the surface eventually becomes smooth again.

The mathematical models that Eisenman is rejecting include the linear, static and deterministic models that dominate traditional architecture theory. He states that static conditions in traditional arch theory of figure/ground are not able to explain true complexity in the urban condition. He believes that a new reading, from another frame of reference, of figure/ground is needed to expose immanent and unseen conditions that allow for new urban structures.

Time of experience has changed, and a new time, conditioned by the discontinuity and autonomy of the sound bite, prevails. Eisenman states that architecture can't be bound by static conditions any longer, and the current situation requires that architecture deals with the Event. He suggests that the fold, which breaks the Cartesian order of space and allows for a temporal multiplicity of conditions, has the potential to reinterpret urbanism and allow for new events to unfold from the immanent conditions of existing urbanisim.

Eisenman is rejecting the mathematical models of Cartesian space and the "two static conditions of the object: figure and ground." He rejects them because they do not "explain the totality of urbanism."

He is replacing them with Thom's Catastrophe Theory because, in looking at the form of the fold, Cartesian order of space is broken up with a "possible new relationship between vertical and horizontal" in which form is continuous.

1: What are the mathematical model that he is rejecting?
I guess when he argues against figure/ground architectural thinking he could be talking about Cartesian coordinate system--however, Eisenman only mentions Cartesian coordinate system as only a description of some Leibniz' thinking as 'turning its back to Cartesian rationalism' because he suggested 'the smallest element in the labyrinth of the continuous is not the point but the fold.' Then goes on to talk about the fold as articulating a different kind of relationship between vertical and horizontal, or between figure and ground--something as revolutionary as breaking up Cartesian orders of space.

What is it about that continuous transition between ground and figure or vertical and horizontal that is so impossible in Cartesian space?

2: What is he replacing?
I suppose the answer you're looking for is topology since its been what we've been talking about. But in my limited understanding of topology, I get the impression that topology more about the relationship or connectivity of things rather than the elasticity of space as shown in the folded or bent Cartesian coordinate system that Eisenman obviously has interest in.

3: and why?
Searching in the space between figure and ground (what he calls fold) I think he hopes to find a bridge between Architecture and building--(The ideas associated with building and the manifestation of building). As an idea, its not a bad way to approach the problem but I still don't quite follow the how or the execution of these ideas within any processes yields the non-traditional, or non extruded architecture he envisions.

An extruded plan on a folded coordinate system is still topologically an extruded plan.

Eisenman is rejecting the model of Cartesian space and figure/ground contextualism. He is replacing it with the model of catastrophe theory. Catastrophe theory, characterized by sudden shifts in behavior arising from small changes in circumstances, can be used to give another reading or "frame" to the notion of figure/ground. Eisenman states how the idea of figure/ground is static, and dismisses the architectural "event". This concept of the fold allows for celebration of architectural events, can be used to re-frame context, and can allow new conditions to form that may have been repressed or ignored.

Eisenman rejects the model of Cartesian space due to the manner in which it discreetly divides space in a regimented uniform way, since that might in turn also imply that the matter represented therein should also maintain the same uniform consistency within its own fabric and construction. Space is framed within the Cartesian plane, and thereby susceptible to false contextualization. He rejects this model because it can not account for continual variation. He calls this concept the 'fold', as coined by Deleuze, in order to depict the sense of variation and inconsistency of density that can occur in space, whether that be material space, such as terrain, temporal space, such as the human experience of physical space, or some combination of the two, such as the urban fabric.

Eisenman rejects Cartesian rationalism and its notion of the point being the smallest element, and the Modernist interpretation of figure/ground, the linear ground slab as "the neutral datum, projecting its autonomy into the future."
"In the idea of the fold, form is seen not only as continuous but also articulating a possible new relationship between figure and ground, thereby breaking up the existing Cartesian order of space."
"continual variation of matter... characterized through the agency of the fold." "first defined culturally in the baroque"

Eisenman replaces traditional architectural theory with the notion of the fold as a "third condition", somewhere in between plan and section, figure and ground.

"the idea of the fold gives the traditional idea of an edge a dimension," opposed to an abrupt line the fold offers a volumetric dimension which Eisenman argues can be used as a re-framing of sorts, and applies it to social organizations and the urban environment.

Peter Eisenman is rejecting the traditional Cartesian rational model. He wants to replace this model with catastrophe theory that is more appropriate for the present day. Eisenman believes that architecture historically has been focusing on only space and place, but ignoring the idea of events. Architecture today needs to see the hidden possibilities in objects. Studying them gives new interpretations of existing organizations.

Eisenman rejects the rigid cartesian math model and figure/ground contextualisim. He states that there is a need for the possibility of reading figure/ground from another frame of reference in order to revel conditions that may have been repressed in the urban fabric. Eisenman supports the position of Leibniz as a non-cartesianist and adheres to Thom's and Deleuze's ideas of the fold. The concept of the fold allows the possibility of refocusing or reframing what is pre-existing in any site which denotes the fold as the aforementioned location of repressed conditions. Eisenman believes this is important because architecture can no longer be bound by conditions of space and place. Figure/ground contextualism ignored the idea of event which can be addressed through the catastrophe theory. The fold helps achieve this by revealing possible relationships between figure and ground, thus breaking up existing cartesian order.

Eisenmann suggests that architecture must not only deal with static conditions of space and place or figure/ground contextualism and that architecture must rather deal with the problem of event and time. He rejects the idea of the Cartesian coordinate system such as in the straight extrusion from a plan and replaces it with the idea of the fold as presented in René Thoms catastrophe theory. The idea of the fold affects both plan and section thus breaking up the Cartesian order of space and, as Eisenmann argues, since the fold is both figure and ground it can be used as a way of projecting new social organizations into an existing urban environment.

In folding in time, Eisenman rejects the mathematical model of the Cartesian rationalism. He explains that we have entered a new era of mediated environments and experiences in which past architectural expressions such as brick and mortar, structure and foundation, are no longer relevant. He replaces these ideas with Deleuze's and Thome's theory of "the fold" in which new relationships between vertical and horizontal, or figure and ground create a continuous space breaking up the Cartesian order of space and experience as it has been previously defined.

Eisenman, like Leibniz, is "turning his back on Cartesian rationalism," and instead focusing on objectile events as described by Gilles Deleuze and furthered by René Thom's catastrophe theory. In order to address historically ambiguous site or programmatic conditions in more complex and exhaustive ways, Eisenman argues that the catastrophic fold can represent a "non-dialectical third condition." This categorical differentiation suggests any architectural resolution is a distinct event, "already in place in the structure" of the site, context or condition, and thereby creates an architectural solution "of the past and the future," not merely a zeitgeist or static object.

1: What are the mathematical model that he is rejecting?
Geometry, Cartesian order of model, figure/ground contextualism

2: What is he replacing?
Topology, the event based interactive way of capturing space such as the catastrophe theory

3: and why?
Because architecture can no longer be bound by the static conditions of space and place. Traditional architectural theory which focuses on figure/ground contextualism has largely ignored the idea of the event. Now it is time to face with the problem of the event which the catastrophe theory can deal with.

Peter Eisenman states, " Today, a rock concert might be considered an architectural event..." The rock concert becomes an environment with lights, sounds, and movements. He tries to inform us that concerts are more than just music. Other factors like the lights and people create the complete experience. The same idea is applied to Figure and ground. These are two separate entities with one sitting on another. The fold is a volumetric edge which defines the space by framing the figure, rather than just a straight horizon line.

1) A syllogism is a logical argument. In two statements, call 'premises', terms are related in categorical terms. In the classic example;

All animals are mortal.
and; All humans are animals.
then; All humans are mortal.

"animal" is a categorie of "mortal". Also, "human" is a categorie of "animal". Thus, the conclusion of the syllogism is that "human" is a categorie of "mortal".

2) On page 3, Barr talks about a line being like a string, where it can be manipulated into many shapes without individual points that make up the line changing in order. Going further, the string could even concieved as an elastic band, where the distance between the points can possibly change.

Every arrangement of that string or elastic band has its own "geometry", but its "topology" is unchanging, provided the string isn't cut or attached back onto itself.

3) In many stages of design development, architecture is looked at in purely topological terms. For example, a program diagram may show the different building spaces using vague (bubble) shapes and lines that represent physical connections. This allows the designer to begin to plan the spaces without introducing specific geometry.

A syllogism is a logical argument and is thereby exactly what Aristotle is using to describe and organize his system of categories. Just as a syllogism is an equation of truths, the categories are simply equations for understanding and quantifying composition or structure.


The terms topology and geometry explored by Barr on pages 2 and 3 describe the permanence associated with topology versus the contingencies associated with topology. Furthermore, the strict parameters of geometry are identified as characteristically opposed to the understanding of topology.


Social and emotional affects and effects of architecture are not always direct results of physical geometry. Although light, air, orientation, acoustics and other environmental influences are often quantified using computational software, the attitudes and persuasions of a designer are difficult to describe or predict in pure geometry.

Syllogisms seem to work by relating things first in a more general way and then in a more specific way. The relationship of the general and specific supports the argument by either agreeing or denying the relationship.

Probably everything Barr says can probably be construed to somehow relate to geometry but before going to far into it, I found the discussion starting on page 5 about how something can still be alike even when its been taken apart and reassembled so long as the order is still the same. Not sure why it bothers me, perhaps its because I think of it in terms of geometric operations of stretching and for for cutting and reattaching there isn't such a neat geometric operation.

Again, probably everything can eventually be construed to relate geometry to architecture (at least the parts of it we like) but for the sake of discussion I think one area where it doesn't quite relate is when architects design an effect or environmental change without necessarily designing how it can happen.
If for example: if an architect may only be concerned with the temperature of the room insomuch that it is habitable by people, they wouldn't care about the geometric arrangement of the room, or mechanical systems (geometrically designed to perform) necessary to produce a specific temperature, just that the effect is achieved.

1. Syllogism is a kind of system which can be used to prove logics. Things have to be categorized when you use syllogism.

2. On pages 2~3, Barr mentioned about main geometric characteristics of the things which don’t change by distortion or stretching.

3. Geometry is not used in the concept, program diagram, adjacency diagram, style, and color.

1. How does syllogism relate to the categories?
Syllogism depends on Aristotle's categories by using systematic logic to determine relationships of things.
2. Where in Barr's text do you find something different than geometry?
Barr begins to describe topology as "a state of mind-and its own goal" and "space as we usually understand it is left far behind" once topology is understood. p2.
3. Where do we not use geometry in architecture?
Perhaps in virtual space when generating algorithms, but even so, the experience of these things are happening in a space dependent on geometry.

1) The syllogism is a manner in which common attributes of categories and sub-categories are parsed to form logical relationships in human language through the use of the transitive property for the purpose of discerning a greater fundamental understanding of an objects function, use or nature.

2) Just as Aristotle used the syllogism to create broader and broader categories within a membership by reducing its description to as few common characteristics and discarding unlike elements, topology seeks to find a more fundamental insight to the spatial relations between elements by reducing their description and relinquishing both measurable quantity and orientation that are so crucial to the representation of Euclidean geometry within Cartesian space. For example, on page 4 of the reading on topology, Barr states that a "line does not have to remain straight" in topology. In a strict geometric interpretation, where a line is defined by three points within a straight relationship, once this alignment is broken, the element would now have to be called a curve or suffer division into two independent 'straight' lines. In the topological sense, the definition of a line is reduced to "the quality of being continuously connected along itself."

3) The aesthetic appreciation of the visceral experience of architecture is not done by algorithm. Nor the budget.

1: How does the syllogism relate to the category?
Syllogism is a rule and system. Categories are produced by syllogism which is all the rules.

2: Topology where in Barr's text do you find something different than geometry?
(Page.2) ...the study of continuity: beginning with the continuity of space, or shapes, it generalize, and then by analogy leads into other kinds of continuity-and space as we usually understand it is left far behind.

3: where do we NOT use geometry in architecture?
Time, perception, space, change, continuity, distortion, stretching

1. Syllogism seems to be the logical argument which one is chosen over the other. Aristotle uses syllogism, when characterizing "man" in an appropriate way. We categorize man as a specific ethnicity or the action he has done, instead of just being a type of species.

2. Barr states on page 2 that typology started as a kind of geometry...one might say it is a state of mind - and is its own goal. If typology is perceived as a state of mind would it be considered "real"?

3. Architecture is not just the design of geometric forms, but also the study of programmatic layouts. In that case we do not use geometry in architecture when we research and formulate a strategy for program.

1. In order for syllogisms to affirm or deny, the subject must be absolute...they depend entirely upon categories to make distinct the relationship between the two terms.

2. One area of where Barr highlights a difference is on pgs 2-3. Here he points out that a topologist is interested in the "most permanent" properties of a thing. The properties that geometry is interested in are relative...contingent rather than permanent qualities. Ex: A straight line is must lies evenly with the points on itself.

3. The concept phase, or the entire area of architecture that happens before form-making can be carried out without the use of geometry.

1. How does the Syllogism relate to categories?
In last week’s class, we learned that one of the functions of categorization is to isolate a way of thinking that is systematic. Syllogisms are very systematic in the way that they use deductive reasoning, a tactful approach using logical consequences.
2. Where in Barr's text do you find something different than geometry?
Geometry is described as the study of figures in a space of a given number of dimensions and of a given type. On page 2, Barr says that in topology, “space as we usually know it is left far behind.” Wouldn’t the removal of space disqualify something as geometry?
3. Where do we not use geometry in architecture?
I’m going to have to agree with Andri and say that we are always using geometry in the design side of architecture.

1. Syllogism relate to categorism
Categories are based on distinct identities. I believe that catergories are aided by syllogism which reveals (intentionally and unintentionally) an array of finite characteristics and function.

2. Barr
What about the entire underlying notion that is based on viewing geometric shapes topologically according to their geometric definition. For example, on page 12 there is a pentagon and a topological pentagon. The only thing that makes these similar is the fact that they each have 5 verticies and 5 edges. Whether or not the lines are straight is irrelevant. That is not to be considered by the viewer. It's a differnt state of mind in a way.

3. No geometry in architecture
Other than the obvious negotiating and writing of contracts one could argue that the assembly of material schedules does not use geometry. Yes the physical materials relate to geometry but simply specifying a particular brand and assigning materials to their respective locations does not require geometry.

1. How does the Syllogism relate to categories?

Aristotle defines syllogism as "a discourse in which, certain things having been supposed, something different from the things' supposed results of necessity because these things are so." Meaning if all animals are mortal and all humans are animals then all humans are mortal.

2. Where in Barr's text do you find something different than geometry?

Since topology is so difficult to define, it can have a different meaning than just being a kind of geometry, Barr states that "one might almost say it is a state of mind" and that it has it's own goal.

3. Where do we not use geometry in architecture?

In design we are always using geometry in one way or another. The only place I can think of where we are not using geometry is when we are arguing with clients about issues such as time schedules and budget, although those arguments are evidently about geometrical shapes of some sort.

1. How does the Syllogism relate to categories?

My interpretation/understanding of Aristotle's definition of syllogism relates to the ideas of categories by its correlation to the idea of definable characteristics that in turn make it understood as specific functions. Similar to how Allen Turing created computation as a way to rationalize a specific set of commands to perform a range of functions, Aristotle's syllogism uses categorizes as a way to identify certain characteristics which can then form a more complex range of meanings.

2. Where in Barr's text do you find something different than geometry?

Throughout Barr's text it seems that the subject of geometry is a constant however on pg. 4 in which he explains the transformation of topology creating different functional uses of a given object, it becomes apparent that topological manipulations have the ability to alter what we recognize as a certain object(in this case a taurus to a coffee mug) while maintaining its fundamental organization.

3. Where do we not use geometry in architecture?

Perhaps when we focus on the use of the algorithm to perform a specific sequence of tasks, we are relying on the intelligence and looped system output of the equation to generate form rather than relying on particular geometric systems for their building potential.