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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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.

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?