And finally

So this is what we have in Vitruvius - things like models of meaning, the figure of the human, ideal proportions, harmony, etc. They all fit nicely along a chain. Symmetry is part of the foundation. Why should anything be symmetrical? And so in the first part we have these idealizations of math and geometry. So many. They all go together in the name of perfection. But then at the end, there is something else. He asks us to consider distorting the perfect geometry so that it will appear perfect when we see it. And that is because even if a line is straight, if it is long, it will appear curved - so let's correct that. But now thye question is whther this conflicts with the first use of mathematics and geometry? Is it the same kind of use?

And Durand, who was a student of Boullee, will say that symmetry is important because it is economical. This is a different model than Boullee surely. Everyone I think understood that. Your comments were clear.

(Sure, some might argue whether symmetry is a geometrical or mathematical term - I would say so, but you can debate it)

And now, finally, the point of these readings, at least one of them. Was to distinguish and get clear that mathematics and geometry just do offer us models of meaning. And we use those models in various ways. But also, a model is a kind of idealization - things ought to be this way, this is how we should understand the nature of things, etc.

And that is quite different than an instrumental use which says, in order to measure the length of this or that piece do the following . . .

So, we use mathematics (including geometry and toplogy) in ideal and instrumental ways. Only that we often find them in conflict. For reference see my discussion with Alejandro in Log 3.

So, for toplogy, I want you to see it as something architects have offered up as a model of meaning. And see what kind of model it is.

Only that we needed to have a grasp of what that means in architecture and the three readings were a way of getting to that problem.

In his essay, a plea for Euclid, Cache discusses this problem of the toplogical model. Things do have to be built in Euclidean space. But that doesn't necessarily devalue the usefulness of the topological model. It just makes us critical in an insightful way. Not negative, just insightful.

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What is a model?

So we use models. How do these function? What do we do with them? A model of meaning gives us a reference for how we ought to understand the nature of things. Think of Lacan's diagram of the self. Think of Freud's mystic writing pad. Think of Kepler's model of the universe, or Descarte's. Now think of how architects have used various models of meaning from mathematics. That's all I'm asking you to do. Just see how they use it. And then look for contradictions. Not in order to confute them, but to recognize that is one of the things we do. It just is.



Try going to studio with a mayline, or a fist full of watercolor markers and tell your instructor "I'm going to do it this way, hell with Maya.". Try modeling your project in just cubes of foam and say "Hell with curves and nurbs"



Tell me what the response is.



Now. Ask your instructor: "But really, what is a surface as opposed to a plane? What is a curved surface and what is a spline?". Or if they are using grids, ask them about those. Ask why you have to conceive of geometry in the way they are asking you to. Just ask.



It should be an interesting conversation.



And maybe they'll give you models of meaning. Maybe

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Whar is space?

Or more precisely where?



No, the readings don't really touch on that issue. But its an interesting question. Let me put it this way; what enables us to talk about space? I mean, where do we point to and what do we use TO talk about it? Someone might say; Well, I just see it here, its all around me. And I walk through it, and so on. And that might be perfectly fine. And we might accept that. But what does it mean "I see . . ."? In what sense is that automatically meaningful? How do I know by what you say, that we see the same things. And now you might resort to physiology, and psychology, or some other discipline.



But would that be enough? I mean, would that be sufficient for architecture? Would psychology or physiology or anthropology give us the authority to say what it means to see space? To give us a definition? And what about mathematics or philosophy? Each discipline, each author might give us a model, a model of meaning to make clear what space is, what it means to KNOW it.



Can you have, for example, I private language that only you understand? This comes from Wittgenstein. Would it be a language?



Could you say what space is only because you experience it? Would that be the foundation of your knowledge? Then would it be possible to say WE have a concept of space?



Do we "know" the earth is round? That the sun is the center of this solar system? Etc.





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divine being?

Interesting that none of the readings really talk much about “SPACE”. Boulee discusses at length the transcendent quality of masterful use of geometry and the reliance on nature as the source of all inspiration (I’m not necessarily buying all of this), but he never really discusses any quality of space. Also there seems to be missing a criticality. Boulee simply accepts geometries, specifically the sphere, as a pure and perfect form. Does this mean, as architects, that if we simply erect spheres we are creating in the order of the “Divine Being”. Personally I see some controversy with spheres and question their use in an architectural context. Surely there is something more to architecture than “geometrical perfection”. Is this really how we perceive space and value buildings; simply by their “clarity, regularity and symmetry”??? Boy that building was….perfectly symmetrical…it really took my breath away. “Symmetry is pleasing because it is the image of clarity and because the mind, which is always seeking understanding, easily accepts and grasps all that is symmetrical”-Montesquieu. “Weary of the mute sterility of irregular volumes, I proceeded to study regular volumes” How banal and dismissive this sounds. Aren’t we, in fact, finding out new phenomena in neural science, self-organization and complexity? Our understanding of “the mind” is changing and so should our use of it. Boulee seems naive and outdated in his simplistic acceptance of geometrical forms as a model for meaning. Hasn’t science, technology and modernity moved us a little further than this?
Durand strikes a little closer to home for me. The ideal form in architecture does not seem to be based on the primitive hut and the human body was not the proportional system employed to design Greek columns. Good. “FITNESS” Good. This does not mean that architecture does not have meaning. It just indicates that its meaning might not be derived through imitation and use of symbols. Where Boulee tends to talk about Math as a way for our work to gain meaning, Durand talks of math in terms of metrics and dimensions not in terms of transcending meaning. Durand is looking beyond implied meaning into the nuts and bolts. He looks at (1)the objects that architecture uses (2) the combination of these elements and (3) the alliance of these combinations in a composition of a specific building. There is nothing here about implied legitimacy because of the choice of objects, Durand would not reel over the sphere the way Boulee has. For Durand the choice of elements is only 1 part of the equation.Opinion-We are searching for meaning in our work and in our use of geometry- I would like to think that our understanding of the world and ourselves has mutated and hopefully matured over the last several millennium. I would think, intuitively, that we have more to learn and it is only through critical use and study of math that we will develop more and deeper meaning. Or maybe we can just keep making cenotaphs and placing temple fronts on banks without question, confident in its antiquated and implied meaning.

v+b+d

of note on Vitruvius :: Quite obvious is his insistence on both symmetry and proportion. But he’s not talking about proportion as a truth in a purely mathematical way. He writes about columns perceived as proportional from the angle of the viewer. Which means his in an interpreted mathematics, not a universal reality inherent in the geometric proportion itself, but one which respects, and maybe even needs, a context.

for Boulee :: He’s the strongest voice for symmetry, with a nod to proportion but not as priority. And while I can’t claim to know how far the comparison would carry, Boulee’s talk of the impossibility of pure invention of form (the comments regarding Perault’s argument), did remind me of the way Plato treats the term equality in Phaedos. While Durand clearly is the grander champion of economy, Boulee does, if indirectly, borrow Durand’s soapbox for a moment to reinforce his own by arguing that symmetry is that which the eye easily understands. I don’t think it too far a stretch to see this as an example of what Durand calls the “love of comfort and dislike of all exertion.”

Durand :: Instead of starting his categories with the geometric associations Vitruvius and Boulee both marry to symmetry and proportion, Durand seems much more interested in topological relationships. His call for “fitness” defines the formal requirements of a building by their behaviors, their events, their program*, instead of the categories of columns by height and feet as proportional measuring sticks.

*I can’t help but think of Koolhaas, particularly the TED talk by Josh Prince-Ramus on their Lousiville project :: http://www.ted.com/index.php/talks/joshua_prince_ramus_on_seattle_s_library.html

vitruvius - boullee - durand

Vitruvius definitely uses mathematics to idealize architecture as it relates to natural form, namely the human body. Proportion and symmetry come from the relationships of the body (digits, palms, feet, etc.). He also brings in numerology, the sacredness and/or perfectness of certain numbers (also based on nature and the body). He lays out applications of math in the thickness and spacing of columns, based on proportions (still based on nature). Value is applied to "perfect" symmetry.

Boullee also cites nature as the generator of perfection, but in terms of regularity, symmetry, and variety. Proportion is the combination of these three elements, once more relating to the "human organism." Very much along the same lines as Vitruvius, proportions and harmony are derived from nature and symmetry is "the image of order and perfection."

As Eric pointed out, Durand steps away from (and even criticizes) the reliance on natural form as inspiration, using mathematics in a more practical way. As an architect (and builder) he is more concerned with how regularity of form is more easily constructable, and thus more economical. He also uses mathematical reasoning in arranging building elements to establish a richer experience.

Reaction to Readings

These readings really took me back in time. A time where Architecture was perfect (Vitruvius), where the follies of the mind were simplified by true natural beauty (Boullee), and then Durand. It seems to me like Boullee and Vitruvius had many things in common: they were both utopian, they both built this road to architectural superlatives. How they used mathematics is less obvious. Their system of dialectics parallels geometry, a sort of progression from simple to complex, yet this was not necessarily laid down heavy in the readings.

Boullee used geometry to legitimatize architecture. Meanwhile, it's difficult to follow anyone who states, "Weary of the mute sterility of irregular volumes.." When he begins the next sentence with, "An irregular volume is composed of a multitude of planes," Maya quickly comes to mind, as do NURBS surfaces, and it becomes clear that we have a much varied respect for the perfection of shapes. Math, I guess, is a tool for determining symmetry which leads to order which finishes with clarity.

Durand is sort of the grey goose. He tackles the five orders of architecture, Vitruvius & Boullee as well, by challenging established principles and at times by simply saying they are wrong (love the footnotes). Architecture to Durand is not imitation based on nature. Instead, he focuses on fitness and economy. He states, "The more symetrical, regular, and simple the building is, the less costly it becomes." The dispositions of the architect come together to make a good building. Durand doesn't use math the same way the others did. He uses math more passively while he critiques previous authors. I get a hint of Maya as well when he states, "Furthermore, is not such a model even more defective than the copy," and remember all the times I've tried to duplicate or copy things in my scene and everything gets screwy. Is he saying that non-geometric shapes have a history as opposed to definite location in space?

I realize I am just brushing the surface. The question, how do they use math, is not as clear as it was in Barr or Euclid. Knowing the discussion leans towards topology, we can only insert ideas and extract pieces of understanding.