Again, we need to remember that the issue is how geometry stands as a model of meaning -- in fact how mathematics stand as a model of meaning. We are looking now at topology.
In terms of our discussions we want to recognize Lynn's as well as Balmond's, van Berkel's, and Eisenman's contrast of topology with geometry, Cartesianism, and stasis. The question is what is the model of meaning given to us through topology? Well there are a number of things, but one of the most critical is that it shifts the ontology of archtiecture from a thing to an event. Look at the essays again and note the terms they use in relation to topology. Impossible to Avoid Geometry?
We should ask: what are some of the basic features, basic operations, that are internal to architecture?
By which I mean, when we are designing, are there specific things that we can’t avoid? – like drawing lines.
I would like to make a very simple proposition which is this: we can’t seem to avoid geometry. Now this might mean different things in different contexts. And so maybe there isn’t an ultimate foundation from which we could say, e.g., “This is how geometry is essential to architecture.” But at the very least, when we do something in architecture, that something seems to imply or involve geometry.
Geometry Lacking Space
(Let’s not get into this, just yet: Descartes development of an algebraic system for Geometry – geometry with symbols rather than figures. Now space has a metric, discrete and continuous, the grid comes later, but the coordinate system is the basis – whereas in Euclid, it is just continuous, metric, but continuous. Somewhere Panofsky writes about the problem of the infinite in the development of perspective)
By having you look at Euclid’s definitions, postulates, and common notions, I wasn’t considering that this would give us a foundation for geometry for architecture but rather, in its economy, we could look at the way in which geometry is defined from within mathematics. And so, when we consider the elements we are considering how they relate one to the other and so on and further that when we read them in a particular order they seem to be defined from the most simple elements to the most complex – that is, point, line, and plane, to metric relations between them, to figures and metric properties of those basic figures.
But notice that in the 23rd definition, Euclid seems to be saying something-- for the first time-- about geometry’s relation to space. One could say that he is already talking about space in the previous definitions. I’m not so sure. For one just because he is talking about the metric properties of figures, or the relationship between points and lines and planes, it doesn’t necessarily follow that he is pointing to geometry’s relation to space in general. Rather, it seems that whatever one constructs as ‘geometrical’ has these properties – for good. But in the 23rd definition, it seems as now those properties might also hold good in some fashion for something beyond the geometrical figure or element as such. By which I mean this: up until the 23rd definition we have something that counts as a general property for a geometrical thing – but it hasn’t yet been said what the space is around that thing only what that thing is as such. In other words: does geometry as such say anything about the nature of space (as opposed to geometrical figures)? If it does, it is ambiguous – the parallel definition simply says not what space is, but one of the conditions of space that holds good, for good. If two lines are intersected by a third and the interior angles are the same, then those two lines will never meet.
The fifth postulate, by contrast, is a demonstration that if those interior angles are not equal, those lines will meet on the side where the interior angle is less than 90.
[If you want to know something more about the history of his problem, see Bernard Cache, “A Plea for Euclid” on the server. The essay, among other things, is a critique of architect’s indiscriminant discussion of topology and digital design]
One of the points of this discussion is this: it is possible to talk about consistency in regards to geometrical propositions qua geometrical things, but not so easy at least to talk about space. Another point is to clarify for ourselves what we mean when we say that architecture is in some capacity always tied to geometrical propositions. That is maybe a more difficult discussion. In either case, let’s at least assume that without geometry, it is difficult to perform an architectural operation. But let’s also admit that an architectural proposition is not necessarily a geometrical one, and vice versa. The last point is that insofar as architecture needs geometry, then in some sense architecture is architecture insofar as it needs geometry. Without it, perhaps, it isn’t architecture. Perhaps.
Architecture, we might say, is always bound up with problems of spatial relations and it seems, or did seem for some time, that geometry was the only field of mathematics that seemed capable of providing general laws for the features of spatial organization (Descarte’s invention is still a continuation of geometry).
Thinking about space and continuity in other terms: Topology
Another field that became relevant recently was topology (for a history of topology see http://www-history.mcs.st-and.ac.uk/HistTopics/Topology_in_mathematics.html) . There are early indirect examples of this: Christopher Alexander used topological diagrams for his discussion of pattern languages. Hannes Meyer introduced a reconfiguration of topological urban space. Le Corbusier used a topological system to hybridize infrastructure (a sidewalk) and architecture in the Carpenter Center at Harvard. Baudrillard discussed the Centre Georges Pompidou by Foster and Piano as a topological system. Frederick Kiesler used a topological system in his Endless House, which he also discussed in his book Endless Space. (http://www.surrealismcentre.ac.uk/images/KIESLER.jpg and http://en.wikipedia.org/wiki/Frederick_Kiesler).
More recently, however, topology became a special field of interest especially during the 90s with the development of digital design (which, in general, really changed the formal domain of geometrical thinking in architecture through a new kind of plasticity, some of which was tied to animation techniques, others of which were related to the utilization of advanced geometrical control with nurbs curves and surfaces – Chu has I think correctly identified this as morpho-dynamism). Peter Eisenman, among others, invoked topology as a new way of thinking about spatial relations, and this is what should interest us (see for instance his essay on Rebstock on the server—topology is invoked indirectly through the reference to Rene Thom’s catastrophe theory, which is a combination of calculus and topology, the same way Descarte’s coordinate system is a combination of algebra and geometry). Jeff Kipnis suggested that topology was a way of removing the ideality of geometry – by which he meant, it was a means for rethinking architecture away from the traditional figure/ground relation. This is also what Eisenman had in mind and he and Gregg Lynn pioneered a new attitude toward space and geometry by introducing, thanks to John Rajchman, Gilles Deleuze’s discussion of the fold in A Thousand Plateaus.
Let’s now point out a few things about topology. First, topology is a distinct field of mathematics that emerged in the 18th century under the work of Leonard Euler. He produced at least two major insights into problems of spatial relation that significantly departed from geometrical concepts or spatial relation (for a history of this see: http://www-history.mcs.st-and.ac.uk/HistTopics/Topology_in_mathematics.html)
Definitions: topology “ignores individual differences among, say, figures bounded by closed curves , and treats them as a group that have certain invariants in common.” Barr 151. One way to get around this vagueness is to understand that topology is not as concerned with the specifics of a figure, but rather a mathematical logic that will relate an entire set of figures – and here the key word is set. “topology aims at the invariant in things; the the things have to be referred to somehow, if very genrally an, and the best way to referto things in this way, and yet retain the kidn of relationship that exist with topolotical invariants, is by treating them in groups, or sets. 163 Barr
(Important to note – see also the grammatical background to this issue in Euler who developed the formula for topological invariant given any polyhedron: Imre Lakatosh in Proofs and Refutations notes in a footnote that Euler made this discovery when he changed the terms that defined polyhedral from vertices lines, and faces to vertices, edges (acis), and faces.)
Topology as a different way of discussing space
“As a rule topologists confine the use of the word “continuous to processes, rather than spaces – a line is a 1-dimensional space – but if we want to use it for a line, then continuity relates the set of all the points on the line to the set of all the real numbers.” Barr 152.
What I would like to say is that topology offers a different way of talking about the properties of figures and of space. “In one sense it is the study of continuity: beginning with the continuity of space, or shapes, it generalizes, and then by analogy leads into other kinds of cointuity – and space as we usually understand it is left far behind.” And further: “A topologist is interested in those poerties of a thing that, while they are in a sense geometrical, are the most permanent – the ones that will surive disotrion and stretching.” And so from this point of view, one could also suggested that topology is interested in the kinds of and quality of connectedness. It may not seem important, but it not trivial that the two circles on a sphere, if they intersect, will have to do so twice, while on a torus, they only have to do so once. There is a topological distinction between a torus and a sphere, despite the fact that their surfaces both seem singular.
Another feature of topology that interests us is of course networks. But maybe the most important is that topological investigations leads to new kinds of surfaces and spatial organizations that we can’t consider or generate through geometrical thinking. Examples include the mobius strip and the Klein bottle. See, in particular chapter 2, new surfaces. And so also, with these new surfaces, new spaces and thus new spatial relations.
It should become a bit more clear now that when we are introducing the question of geometry and the question of topology in relation to architecture, we are looking at two different systems that are about spatial organization and where geometry seems limited in one sense, topology has advantages that allow us to think space in other terms that are equally meaningful, though the nature of precision is somewhat difficult and vague and this is perhaps because we keep trying to think of topology in geometrical terms.
