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.

Readings

The assignment is the assignment. Whatever comments want to be thrown out is fine, but do the assignment. Daniel, thanks for the post - I'm not sure either Durand or Boulle are doing topology. But that's an interesting point. Consider again we are asking a question about how architects use mathematics and are we clear on that? Look at how Vitruvius, Durand, and Boullee idealize geometry in different way, look at how they use the term symmetry. They are all talking about geometry. In vitruvius there is an interesting contradiction and there is one between Boullee and Durand, to all: the comments posted here are comments meant to address the assignments and the readings primarily. Do that first, them maybe the rant will have value. Maybe. Ok, so Daniel, good, go back and look up the word symmetry - how do they use it? (And yes, with respect to Durand, he is anticipating something about computation, but we'll take a closer look at that later.). Thanks p
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initial reaction to the readings

The Durand reading starts to imply a topological investigation of the orders of columns while questioning the initial reasoning behind the forms. Boullee starts to explore the topology of solids (whether he thought of it in those terms or not), pulling out their attributes of regularity, symmetry, and variety. Both writings start to touch on ideas that we've discussed, but beyond that I'm not sure the relevance aside from "hey look, these guys kinda got it, maybe."

The Value of a system..

I previously commented on the value of Topology, or more specifically, the lack of value in a system so myopic. I can't help but feel like I started a war of sorts between geometry and topology. I found myself defending geometry, or at least its relevance, while it was systematically challenged by the class discussion. In this war, I noticed an interesting relationship between these two factions: you can't dis one without using it. It was like Delanda's class where he kept using Ratified Generalities in order to show how worthless ratified generalities were. Was I taking crazy pills in class or did I stumble upon an insight: geometry and topology are complimentary, or rather, one surely supports the other in a tower of knowledge. If we remove geometry, topology collapses.

Topology will not get us anywhere without geometry. That bottle flying through the air follows a perfect arch, also created when you squash a circle. If you remove that arch, you are left with relationships. Relationships have no meaning without corresponding subjects.

Is anybody else picking up on this quasi-meaningless battle that I just invented?

If I could address Universality: To me, the study of anything universal is in the psychology realm. Freud, Jung perhaps can teach us something, but does architecture
or even mathematics belong there? I would say no. We bought mathematical universals in "Contact" with Jodie Foster, and left it there. Does a certain part of the brain correspond to feelings, sure. Can you condition a monkey to use a computer, why not? But does red, white, and blue make one feel patriotic? Do buildings with the Golden Section calm one's senses? If you dig a hole, dig another hole next to it, then put them together, do you get two holes? No. Not here, and certainly not everywhere.

Do we have any psychology majors out there? I wouldn't imagine we do, under my personal stereotypes. They search for commonalities, truths, universals. I happen to be an Anthropology major where we almost inherently search for differences. We find and place value in plurality, human interactions, in relationships.. And here I circle back again. This may not be as clear to you as it is to me, but perhaps topology is the study of intricacies that are too human (for lack of better term) to be universal. What if psychology is to geometry what anthropology is to topology? I don't care that you salivate when you smell beef. I am absolutely fascinated over avuncular disparities in Mongolian nomads (anyone want to read my dissertation?) To be sure, Peter may flick his nose at a circle or square and yet spend weeks infatuating over mugs and donuts. So maybe we don't have to dismiss geometry or even assign values to anything. Maybe we can agree that it is simply more interesting to step past universals and into something more contingent, more organic, more ...?

Am I taking crazy pills?

functions

Today we discussed a couple of issues. We are considering what topology allows us to consider as a logic of organization difficult if not impossible to conceive with just geometrical concepts and tools. One way of looking at this is ontologically -- and that only means the way in which something is said to exist. If i asked you to describe this water bottle, this seems non-controversial: it has such and such geometrical properties, such and such measurements, and so on. But now i throw the water bottle and ask you to describe it mathematically. All of sudden geometry seems insufficient and we need another mathematical tool to describe the arc of its movement. As Eric pointed out, that would imply the introduction of time, the change of position, etc. It would imply calculus. In order to mathematically describe the bottle being thrown, we’d have to introduce the mathematics of calculus. But note: there are also two different ontologies here. The thing qua thing -- that is the object as such as static. Then there is the thing qua event -- the bottle moving through the air. Renee: Good point, once we draw the trajectory we reintroduce geometry, we reintroduce a geometrical artifact, a curve – but also note that the curve would not have the curvature without calculus. This also plays into our discussion of diagrams. The curve is a description of the behavior, it is a diagram. The curve doesn’t necessarily describe the arc of movement, though it could – what it describes the rate of change, as long as it is continous. Here’s an interesting question: if I draw the bottle in two dimensions using geometry am I making a diagram? Ok, so, what I am asking you to consider is the nature of a diagram and how it can be used to describe the behavior of events. If we consider Plato’s ontology in relation to the Meno and the Phaedo, the highest form of existence is really ideas and forms: Eidos/Form. The discussion of Equality in the Phaedo, and Beauty and the rest in Meno are discussion about Universal Principals which we know only as pure souls but which we have the capacity to recollect. As humans we acquire knowledge through experience, but this is contingent – not universal. The mathematical demonstration in Plato then belongs to a kind of Ideality – things like geometry are Universal facts, they are not contingent on our experience: they transcend it. Mathematics as always been thought of in these terms and architecture has continuously purchased Uinversal principals of meaning through using such models. This is partially what I mean by a model of meaning. Le Corbusier’s use of the golden ratio and the modular are such examples. Ok, now consider the example from Bentham’s Panopticon and Foucault’s discussion of it as a function of a function. It is a diagram of relations of force. The incredible thing is that it doesn’t have any particular formation as, say geometrical object or space – it is a way of networking and creating space. Deleuze thought that this was important: a diagram is not a thing, but a series of relations through which things come into being. They are more event-related. And this is also why he is critical of the Platonist tradition and offers an important reading of ontology through Stoic ontology and the way in which it privilages events over things or ideas. So, one way to consider the problem of what is topology is to consider it as having a different form of existence than objects. This does not define it, but helps clarify and distinguish it from geometry. When you eat and apple there is an apple and then there are a series of processes that convert that apple into something else. We can consider these functions – could we say the functions map the apple on to other functions, nourishment, energy, etc? Well, at least we could say the entire digestive process of which there a numerous different events is not really a visible process – its visibility is not the same as seeing the apple. But we could diagram those functions and consider their topological properties – that is their forms of continuity. And this might now help – we can’t really see as diagram as a thing, but rather as something that relates, that networks a series of functions. So it is not a picture of a thing so much as a state of affairs and in that sense has to be abstract. When Choisy discovered the principle of asymmetry in the Acropolis and introduced into architectural notation for the first time a vector describing the arc and movement of the spectator he was introducing a new ontological concept in architecture – the experience of walking through the site, not just seeing the building as such, and this is what accounted for the odd juxtaposition of the buildings. Le Corbusier took this diagram and turned it into the architectural promenade of which there are now countless variations of which Koolhaas’s work is just one. Here is a case in which a notion of organization had to take on features of topology (that is continuity of functions movement, building, space, time, position, etc.) through the logic of the diagram. What this introcued to architecture was the concept of the event. Tschumi’s work is principally based on this. It is principally diagrammatic.

Biology

Foucault makes an important reference to Cuvier's taxonomic reorganization of species according to a topology of functions. Note that the section in The Order of Things has to do with Modernity and an epistemological shift in which knowledge and discourse (how we see and what we say) move from a representational logic to an anlytical one. That is, that which constitutes the order of the world is no longer understandable on the level of representational comparisons.
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Question

Under what conditions does a mathematical model become explicit in architecture?
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sweeping up

We also need to understand -- at least confront a very basic and very complicated problem: architecture's use of mathematics. Is it so simple that we just use mathematics? I mean, is it clear how we use it? Ok among the texts i am uploading for you, and you'll have email note about the ftp site, is Plato's Phaedo. Look at passage 75 for this coming week and read maybe a bit before and after. Consider the use of "Equal." Could the term, and the concepts to which it applies be replaced with, say, Triangle, or Square? Think also of the following: is there something beyond the world of flux? In otherwords, is all knowldge experiential? How do we have knowledge of things like geometry? And now ask yourself whether the fact that a triangle always has 180 degrees when you add the interior angles is a fact of our experience or something transcendent of that? And what is the implication? All of this has to do with the Platonic theory of Forms, what these share with ideality and mathematics and the strange problems we have in architecture when we talk about the use of mathematics. in the following week we will look at Vitruvius, Boullee and Durand to see that problem in relief

a stammering

Reading Euclid (and remember, he is establishing principals about what geometry is as a set of unique laws, these are the things that absolutely, not contingently define geometry) we find that we proceed from the simple to the complex to the totality of the basis of geometry. We construct spatial figures according to the same principles.When we get to the idea od "space" however, we have neither a figure not an easily graspable handle on What it is.It has a queer ontology. I see a point, i know what that is, a line, no problem, a plane, easy. but "space"?It can only be described indirectly (one might argue that is not part of Euclid's goal -- perhaps not). (And yet it was thought that the system was complete. Bracket and suspend for a moment the idea of the Cartesian coordinate system and the idea of a generalizable rule for describing space as an infinite grid or non-euclidean space. For now it is helpful just to keep in mind the fact that although Euclid doesn't describe specific numbers for things like angles, his system absolutely relies on the notion of discrete metric properties.They are fixed. So, in short, coherent and clear understanding of elements and figures and their basic essential properties, but not so space. Topology, on the other hand is all about space, or spaces, or manifolds, and not so much about figures and their discrete metric properties. And to that extent, it is difficult to grasp topology as a contructional system. It isn't clear what it means to construct a topological Thing (unless it is the wild behavior of surfaces, or intricat knots). So here, in topology, the problem is quite the opposite of geometry: the specific properties of topological spaces (like the torus, the mobius strip, the Klein bottle, and knots) all seem to be ontologically clear while the geometrical features are ontologically vague. As someone pointed out, the fact that the coffee cup and the torus are the same flies in the face of our geometerical intuition. Right: metrically, and geometrically speaking they aren't at all alike. But topologically they are. They are homeomorphic -- they can be mapped on to each otehr.As Barr says, topology cares about those things which remain after strertching and distortion - it cares about those things that remain after distortion, it cares about those things which are invariant. And so it is rather indifferent to the shape of the thing. And this is frustrating for architects who live in the world of forms. topologists like ants live in the world of surfaces/manifolds and networks (as well as coffee cups and donuts). Where the systems topology and geometry overlap: they both utilize points, lines, and surfaces.Only in case of topology, there are all kinds of surfaces all of which are also spaces.So toplogy gives us a wide range of different kinds of spaces each of which has unique properties (the inverse of geometry) but no definite figures or shapes or forms.What topology allows us to do, then, is create kinds of space that were impossible with geometry (a klein bottle has no edges, no inside, no outside.But then, how is this?>What does "create" mean?In a sense, it means that we can take something like a plane, cut, distort, and reattach it to itself and generate a complex space. And if we fail to see the advantage of that as a diagram for architectural consideration, then we're not really paying much attention to architecture either. Topologically challenging figures and networks are from our point of view dynamic, constantly changing not static -- topology offers different paradigms of organization which we can't conceive of geometrically (and don't argue that we still need geometry to build the thing -- i know that, i'm no idiot. The latin root pli means to fold -- complicate, to make the experience more complicated not necessarily confusing, though maybe, but certainly more challenging. You'll see this when we read Eisenman, Balmond, Ben van Berkel, Alejandro Zaera Polo and others in the comming weeks. Topology introduces intensities, transofrmations along spaces that are actually continuous. Geometry has fixed positions, static conditions, rigid distinctions. But now, here's the real problem, at least for architects. The topology from which we began to draw inspiration during the 90s in our infantile slobbering over complex surfaces as we were given digital 3d modeling tools is diagrammatic. From a topologists point of view, diagrams help deliver a certain mathematical intuition for public understanding -- but the rigor of topology has nothing to do with those figures. As architects, that's pretty much all we understand. Topological figures are diagrammatic. Not computational. And the point of this seminar, among other things, is to see topology in relation to computation and algorithm -- to find the space for a new argument about topology that goes beyond the diagram (which is a pictorial imposition of a topological figure onto a geometrical one, which is not bad, but it is simple-minded -- its the wrong kind of mapping). A few others points: remember that Euler derived topology from geometry, from the analysis of polyhedra and the grammatical transformation of side to edge (point side plane to vertice edge face). Wittgenstein would say that this is a transformation of signs according to a new paradigm and in a sense that is important for it shows us that mathematics invents systems -- not arbitrarily, of course, but with internal consistency and that is part of its creativity. Two important texts, Imre Lakatos's Proofs and Refutations and Bernard Cache's A plea for Euclid. Experiment for the relation between geometry, topology, and computation.< Version 1. take two points and draw a line between them. now draw a third point and draw a line from that point to some point on the first line. make a fourth point, and draw a line to somewhere on the first or second line (or draw a line between the first and second lines, etc. Now take those exact same points and instead of drawing a line, use a pieces of string. take a string, suspend it between two points. take another string, attach it to the first and then to an outside point. keep adding strings until you have about ten. what happens to the strings as you add them consecutively?
Version 2. make a series of twenty random points. make three copies of each set of points. in the first one define a rule by which to connect three points, for the second, change the behavior of the rule, for the third change it again.

What aspect of all of this is geometrical, what topological, and what computational? And finally, if you know Gaudi, what aspect of this system is architectural?

I find it Ironic..

To quote from today's discussion, "Topology allows us to talk about objects, such as the Klein Bottle, that geometry does not allow." I find it ironic that topology can bestow any system of discussion while it precludes the capacity to talk about the difference between a mug and a donut. We cannot talk about the difference because the only thing we can see, topologically speaking, is the quantity and quality of drawn circles crossing. For the discussion to have gone as far as it did, we need more substance than a few circles crossing. What truly does topology see? Allow us to see? Allow us to talk about?

How does it allow us to talk about the Klein Bottle? The Klein Bottle, topologically, takes the same surface as, say, a plate. You can follow the surface contiguously from the inside and outside, one solid surface. Therefore (borrowing logic from Euclid) a plate is a Klein Bottle. Or rather, topology gives us the tools to talk about the Klein Bottle, but not to distinguish it from a plate!?

Unless topology does give us a tool set, a conversation, a method of describing differences between a donut and a mug, then we cannot talk about the differences between a plate and a Klein bottle and therefore lose all intricacies and any awareness the Klein bottle may have given..

late for class...

Why we do it..

I remember 5th grade when the teacher was showing us what she promised to be the most amazing thing in the world. It was almost a magic trick the way this particular paper machine functioned. It didn't have an inside or an outside. It was a milestone in mathematics, physics, and perhaps even philosophy. And yet, nothing could disappoint more than a flimsy piece of paper twisted then taped back together again. Lame.

It was not revolutionary in my world. I even think I totally understand the mystery, and yet it was truly lackluster, pedestrian, banal.

As a fifth grader gawks at a strip of paper, so do civilians at Rem Coolhouse. What does it mean? What does it really do? Outside of a few esoteric circles, nothing. Insert here the economist, George Stigler: ordinary people don't have the time or energy necessary to appreciate/understand the world around them. When it comes down to it, do these things really matter? Of course they do! (so says the architect) And thus, it becomes our duty not only to produce things that matter, we now have to supply the relevance as a sort of operator's manual. A building that blends inside and outside is good for the soul because _____.



We could legalize stimulating drugs, that would make these ideas much more appealing to most people, sure. However, I think we are just going to have to please ourselves. We will go to Pratt, push the envelopes, erect some crazy designs then submit them to an unappreciative world. I don't really see any other way to do it. And yet, isn't that the best way?

Other Studies of Topology

After having gone back and read the article associated with the diagram I posted I realize that the author, Yasushi Kajikawa, was a contemporary/collaborator of Buckminster Fuller and was researching "Synergetic Topology." Perhaps the field of Synergetics could be a side discussion in class, but basically the periodic table of the polyhedra that Kajikawa produced demonstrates that any Platonic or Archimedean polyhedron can be collapsed (folded) into three basic shapes: tetrahedron, octahedron, icosahedron.
Discussion of the donut and the coffee mug reminded me that the Poincaré conjecture had been in the news recently (2006) as the only one of the seven Millennium Prize Problems to be solved. The conjecture (now theorem) is slightly beyond my grasp, but a New Yorker article (Manifold Destiny) describes the general concepts:
"From a topologist’s perspective, there is no difference between a bagel and a coffee cup with a handle. Each has a single hole and can be manipulated to resemble the other without being torn or cut. Poincaré used the term 'manifold' to describe such an abstract topological space. The simplest possible two-dimensional manifold is the surface of a soccer ball, which, to a topologist, is a sphere—even when it is stomped on, stretched, or crumpled. The proof that an object is a so-called two-sphere, since it can take on any number of shapes, is that it is 'simply connected,' meaning that no holes puncture it. Unlike a soccer ball, a bagel is not a true sphere. If you tie a slipknot around a soccer ball, you can easily pull the slipknot closed by sliding it along the surface of the ball. But if you tie a slipknot around a bagel through the hole in its middle you cannot pull the slipknot closed without tearing the bagel."
Another use of polyhedra, more recently in the news, was originally studied by Lord Kelvin. From a New York Times article (A Problem of Bubbles Frames an Olympic Design):
"Lord Kelvin studied foams to try to understand the 'ether,' the medium through which he and others thought light propagated. In his work he wondered what would be the most efficient foam — how space could be partitioned into cells of equal volume that would have the least surface area.
True bubbles were not the answer, of course, because there would be gaps between the spheres. The answer Kelvin came up with used 14-sided polyhedrons."
In 1993 the Weaire-Phelan solution proved that two polyhedrons of equal volume, one of 14 sides and one of 12, that nest together in groups of eight was the configuration with the least surface area. The fact that the polyhedral matrix was sliced at an angle and used to construct the geometry of a space frame for the roof and wall structure of the Water Cube doesn't make the design topological, but does create a visually interesting yet repetitive pattern.

How have other disciplines started to address these questions of topology, and where on the timeline have other arenas approached a complication of inside and outside?

As an idea, I follow the conversation architecture seems to be having, and fully appreciate a sensibility of categorizing things in a paradigmatically different way. I do wonder, though, if in terms of architecture, we’ve been selectively ignoring scale as a part of the dialogue.

Does the scale of the body versus the scale of a building mean that fashion has been dealing with topology much much before the late 1990s? Or, at the scale of a product, are the blurring of lines more productive—the Klein bottle, unlike the D+S Eyebeam project, doesn’t have to address where to place the thermal break.

Are we being cavalier to want the wall-to-floor-to-façade condition for our design proposals? Is our desire to complicate things flippant—a greedy hope we can claim a vocabulary because we like the way it sounds, the way it makes us sound (how evolved of us to think of a square as so much more than Cartesian)—or is it actually a catalyst of creativity, a way of giving ourselves permission to question even those fundamentals which we thought to be givens?

Transitions in the Topology of Polyhedra

This isn't my response (yet) but I have an article I found a while ago which (despite the title) is basically a history of the study of polyhedra through 1993. I don't think I can upload the .pdf file, but I can e-mail it if people are interested. It's pretty short and has interesting diagrams.
I mention it because includes the invariant Euler equation we touched upon last week:
V (vertices) + F (faces) = E (edges) + 2.

1 introduction - intensity

What does topology change about our notion of "space?" (and remember mathematics has areas, "geometry," "algebra," etc. -- are these areas discovered? did they Naturally exist?). Read the text by Euclid, then read Barr's text on topology. Many of the "elements" are the same. But now consider, in what sense can we talk about "space" in Euclid? Therein lies one of the distinctions. For the moment, consider today's lecture on the Topological continuity between different polyhedra. The Klein bottle looks like any other object insofar as it seems to have things like boundaries. But in fact, when I asked to compare it to, say, a box, it became clear that one of the distinctions is that it doesn't have an inside and an outside -- it is continuous along its surface. Why is this of any relevance for architecture? Well, among other things, this kind of observation allows us to reconsider the notion of an architectural envelope and begin to PLAY with traditional architectural properties of inside and outside. A key concept is obviously the notion of folding spaces. We'll get to that a bit more later in relation to Deleuze, but for now it is a way of "complicating" space and thus making it more dynamic, more intense. As Rajchman points out, complicate is elborated from a Latin root pli, to fold.
When we look at many of the projects during the '90s by diller/scofidio, Lynn, Spueybroek, UnStudio, Koolhaas, etc., we see there are a series of these attempts at folding: folding of the inside and outside, folding of site and building, folding of ground and envelope. This changes the way in which we can experience and understand spatial relations in architecture -- they make it potentially more dynamic.
As I continue to write on this, keep in mind the last part of the lecture: topology has, among other things, enabled architecture to shift from the concept of the grid and metric space and bounded form to networks and folds. The latter introduce not limits and positions, but rather transitions and therefore intensities.