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?
