I read "nomad" and got excited. Then I came to "religion," "war machine," and was on the edge of my seat. When I read Mongol, I jumped for joy like no Deluezing reader ever has. All these issues are right up my alley and not so common in architecture. By the end of the article, however, I was at the back of my chair trying as hard as I could to figure out how he was using mathematics.

Deleuze jumps around a lot in this text. I give him immediate credit for inserting social issues but they are so watered down in all the analogies they become distracting. He compares the striation to geology, organisms, fabrics, human anatomy, composers and more all while insisting on Greek language lessons. He seems to be munching through ideas like breakfast cereal.

He is either covering way too much ground or he just can't nail down what he is trying to say. (I will recant this statement after a few more readings.)

Deleuze had some very interesting tangents. For instance, "composers do not hear; they have close-range hearing, whereas listeners hear from a distance. Even writers write with short-term memory, whereas readers are assumed to be endowed with long-term memory." Woa. He gets into some great ideas pertaining to scale, point of view, orientation. His notion of the sea also paints an eloquent picture of complexity in common notions of striation. He really has me when he talks about bearings and fabric and yet I get confused again when he brings up the Mongols and nomads. I think a discussion of this reading would do me a lot of good.

I'll read through it a few more times and get back..

Comment for Adam

The response is interesting and in light of one Deleuze's books logique de sens, not far off target in terms of Deleuze's multiplicity. Deleuze for instance was very interested in Lewis Carroll's work on mathematics and his book Alice in Wonderland. I like the notion of a kind of delirium. I'm not sure though delusional is something he is offering as an alternative form of thinking. Difficult as the texts are, they still offer one interesting insight into our problem: namely, how far can the use of a mathematical model travel?
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delusionalDeluze

And that is to say I'm not quite sure if he makes me more delusional than his text is or the other way around. Than again, delusions, or thought disturbances are to be expected when math (the concrete science) moves into calculus, and is used to describe limits and attempts to describe infinity (the unknown, the untouchable, the elusive, the angst evoking). is Deluze delusional talking about the "Soul of point, line, surface, or the fold"? He sure is eloquent in the logic of his model. He is reaching all the way to Greek roots when he makes a move from the imperfect physical point "neither atom, nor Cartesian point - elastic point-fold," to idealistic mathematical point "rigorous without being exact". He follows the logic with Euclidean strictness and set of basic parameters where point describes extremities of lines. Though his model is broader, including vectors, magnitudes, movement, force, and direction. so far so good. But wait, now that seems to transform the point into "the metaphysical point," or the soul, or the subject. Now while I can follow each little step, this general move is something I still have to digest. how exactly did we move form the physical to ideal to now "a point of view" or "a point of inclusion"?

I find both texts dualistically very stimulating and discouraging. But my mind too ventures off while reading, sometime stimulated by the multitude of ideas and possibilities he opens (especially in the smooth & striated), and other times just finding it difficult to follow the condensation of his lifes' work seemingly neatly packed in every few lines.

deleuze, mathematical model

Ugh.

Ok, I made it through most of this and was able to absorb a very small amount. I often found my mind wandering while my eyes ran over the words. However, I think I found a section that deals with what Peter was asking us to look for (not that I found it particularly clear).

In The Folds In The Soul, Deleuze gets heavy into mathematical language in talking about "inflection," or the fundamental component (the atom) of a fold. He breaks it down into three types, vectorial (symmetrical/orthogonal/tangential), projective (defined by hidden parameters), and infitely variable (like fractals).

Ok, I think I'm following.

He then goes into irrational numbers, talks about limits (yay Calculus!), and then I get a bit lost again.

"When mathematics assumes variation as its objective, the notion of function tends to be extracted, but the notion of objective also changes and becomes functional."

I'm looking forward to reading someone else.

Models of meaning

Our discussion last Tuesday examined a few issues at stake between Vitruvius, Boullee, and Durand. Keep in mind that Vitruvius stands in the history of Western arch as the first systematic exposition of arch Principals. And please also keep in my mind what is at stake in the articulation of Principals and what a principal is in the notion of a system. Its first major use comes of course in the work of Aristotle who used it to systematize philosophy.

In Vitruvius one od the first principles is of course symmetry and it is connected with a number of paradigmatic uses of mathematics in architecture, such as perfect geometrical figures (circle and square), ideal ratios, harmony and proportion. These elements to repeat hang together as if on a chain (they are NECESSARILY connected). The importance of course is a metaphysical lesson, not a contingent value; symmetry and it attendant terms point to Nature and man's relation in an ideal essentialist manner. So, the term is abstract and ideal but nonetheless a powerful notion of the idea of perfection in architecture. The ideality of this is that a) it is a gemoetrical notion embodies in the circle and the square and b) timeless and univseral (in a platonic sense). That then is one kind of invocation of mathematics and geometry - their uses MEANS a certain formal coherence in architecture. There is another discussion of the use of mathematics and geometry in architecture which comes at the end and that involves entasis, wherein although a temple might actually use ideal mathematics its actual appearance will not cohere with the perfection of the idea since our eyes and physical material existence are not perfect.

Thus a different use of mathematics enters which we can call instrumental and which in no way we can isolate as a principal since it doesn't of necessity lead to any Principal but merely corrects our experience of it. It is just a way to get a job done.

Are these two uses in conflict? Are they coherently aligned? Doesn't the introduction of an instrumental use of mathematics call into question the ideality of mathematics in architecture?

At any rate we may want to remember that Gothic architecture which shares so much with the history of Northern European "barbarian" art was dismissed throughout hsitory because it failed to achieve the quality of geometrical purity found in the GrecoRoman tradition. (Interestingly, Leibniz's curve was often alligned with the barbaric curve of the gothic arch). It was seen as vulgar, a sentiment which Eisenman still shares today because it was more of a material contingent system (think here of Gaudi"s funicular chains as an example-they output geometry as an effect of gravity on the network of chains). The Gothic system actually uses much in the plan dimension of the ideality of mathematics from the grecoroman tradition, but not in elevation and section. There all geometry derives from experimentation.

At any rate, back to models of meaning. When we read Boullee he talks about symmetry in many senses consisten with Vitruvius - and of course he is entirely aware of this ideality. At the same time as a models of perfection he is modernizing it in relation to a theory of sensation as and I pointed out this is what accounts for the notion of irregularity. Regardless, at stake again is symmetry and geometry as a model of meaning. When we get to Durand, that model entirely shifts. And to make a long story short, Durand kind of inflates the ideal and the instrumental by saying that sysmmetry is an economic function.

Ok, so what does all this imply? Well, for one when we use a geopmtrical model in the history of architecture we are often introducing a model of meaning. Mathematics and geometry are not simply tools. Ok, we get that. We alrwady know that. For another, it clarifies the distinction between a principlaed use and a contingent use. And finally, that the meaning of the model is just in fact how it is used.

Now we are and have been asking about toplogy as in so many different ways different from geometry and looking at what possibilities it holds for architecture.

We've looked at it in a kind technical sense and now with Deleuze we are looking at how it emerged as a model of meaning.

Firsyt let us note that Deleuze and Bernard Cache stand as essential references in the emergence of digital design (the first iteration of computational architecture that allowed for complexity). Without Deleuze no Greg Lynn or Eisenman, for example. But in order to see what this means in theuir work let us first take a glimpse at Deleuze'

The first thing we can note, right of the bat is the idea of the fold, and the opposition between the curve of Leibniz who introduces a new mathematical system and Descartes's whose system is still tied to a notion of stasis. Despite my poor skills at teaching mathematics I think you got the point: calculus deals with change and variation, it deals with motion and time, and it deals with therefor events.

We will discuss this more on Tuesday next week but meanwhile please finish your examination of both texts and write an entry for the blog about a specific moment in which Deleuze is using mathematics as a model of meaning and try and write something explicit about what that model is mathematically ( eg if he is talking about curves or vectors or whatevr)
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if it ain't baroque...

Only when Deleuze is summarizing things (at the end) do I have any sort of a vague notion of what was contained in the preceding 30 pages. So let's discuss the end where he lays out six points relating to the fold and my attempt at a translation of the various representations of the fold contained therein:

1. The fold: as a condition of continuity and infinity.
2. The inside and the outside: a separation and division.
3. The high and the low: a dialectic condition.
4. The unfold: not the act, but rather the lack of a fold.
5. Textures: a multiplicity of folds.
6. The paradigm: the framework within which the fold is considered.

In this case, Leibniz and the Baroque are the framework of choice for his entire discussion. I think the above needs to be expanded upon considerably, but it's a beginning.

better late...

I'm not cheating because I haven't yet read what anyone else posted, but I'll still issue my comments on the general question (as I recall) from last class, "What does mathematics have to do with architecture?" in relation to the viewpoints presented in the readings. A pseudo-summary first...

Vitruvius:
A codified characterization and description of the temple form derived from his study of Greek architecture. It is significant that Chapter 1 is titled "First Principles of Symmetry." Also significant is the “foot” which becomes a base unit of measure, derived from the proportions of the human body which, as it has come from Nature, is perfect.


Boullee:
He is recounting a conversation which, presumably, he was present at. The question posed for discussion is, “Are the basic principles of architecture derived from Nature?” People relate to the “human condition” thus anything that is symmetrical and proportional is pleasing to the eye.


Durand:
He summarizes the works of Vitruvius and Laugier (imitation of the human body and the hut, respectively) then politely disagrees with their conclusions. “Fitness and economy,” are the principles that must be met. Economy is further characterized by symmetry, regularity and simplicity.

It would seem there is something about symmetry, whether it is a means to an end (Durand) or it is the means (everyone else). There is an instinctive beauty that is universally perceived in that which is symmetrical. In our “universe” we also relate most readily to Nature (with a capital “N”) where symmetry is ever-present. In mathematics symmetry is a basic property of pure geometrical shapes (i.e. the circle, the square, etc.). It is not much of a leap to presume a correlation between such geometry and nature with symmetry as a common thread.


Here's my favorite quote:
“If we imagine a Palace with an off-centre front projection, with no symmetry and with windows set at varying intervals and different heights, the overall impression would be one of confusion and it is certain that to our eyes such a building would be both hideous and intolerable.” -Boullee
Take that Frank Gehry.