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.