In Vitruvius one od the first principles is of course symmetry and it is connected with a number of paradigmatic uses of mathematics in architecture, such as perfect geometrical figures (circle and square), ideal ratios, harmony and proportion. These elements to repeat hang together as if on a chain (they are NECESSARILY connected). The importance of course is a metaphysical lesson, not a contingent value; symmetry and it attendant terms point to Nature and man's relation in an ideal essentialist manner. So, the term is abstract and ideal but nonetheless a powerful notion of the idea of perfection in architecture. The ideality of this is that a) it is a gemoetrical notion embodies in the circle and the square and b) timeless and univseral (in a platonic sense). That then is one kind of invocation of mathematics and geometry - their uses MEANS a certain formal coherence in architecture. There is another discussion of the use of mathematics and geometry in architecture which comes at the end and that involves entasis, wherein although a temple might actually use ideal mathematics its actual appearance will not cohere with the perfection of the idea since our eyes and physical material existence are not perfect.
Thus a different use of mathematics enters which we can call instrumental and which in no way we can isolate as a principal since it doesn't of necessity lead to any Principal but merely corrects our experience of it. It is just a way to get a job done.
Are these two uses in conflict? Are they coherently aligned? Doesn't the introduction of an instrumental use of mathematics call into question the ideality of mathematics in architecture?
At any rate we may want to remember that Gothic architecture which shares so much with the history of Northern European "barbarian" art was dismissed throughout hsitory because it failed to achieve the quality of geometrical purity found in the GrecoRoman tradition. (Interestingly, Leibniz's curve was often alligned with the barbaric curve of the gothic arch). It was seen as vulgar, a sentiment which Eisenman still shares today because it was more of a material contingent system (think here of Gaudi"s funicular chains as an example-they output geometry as an effect of gravity on the network of chains). The Gothic system actually uses much in the plan dimension of the ideality of mathematics from the grecoroman tradition, but not in elevation and section. There all geometry derives from experimentation.
At any rate, back to models of meaning. When we read Boullee he talks about symmetry in many senses consisten with Vitruvius - and of course he is entirely aware of this ideality. At the same time as a models of perfection he is modernizing it in relation to a theory of sensation as and I pointed out this is what accounts for the notion of irregularity. Regardless, at stake again is symmetry and geometry as a model of meaning. When we get to Durand, that model entirely shifts. And to make a long story short, Durand kind of inflates the ideal and the instrumental by saying that sysmmetry is an economic function.
Ok, so what does all this imply? Well, for one when we use a geopmtrical model in the history of architecture we are often introducing a model of meaning. Mathematics and geometry are not simply tools. Ok, we get that. We alrwady know that. For another, it clarifies the distinction between a principlaed use and a contingent use. And finally, that the meaning of the model is just in fact how it is used.
Now we are and have been asking about toplogy as in so many different ways different from geometry and looking at what possibilities it holds for architecture.
We've looked at it in a kind technical sense and now with Deleuze we are looking at how it emerged as a model of meaning.
Firsyt let us note that Deleuze and Bernard Cache stand as essential references in the emergence of digital design (the first iteration of computational architecture that allowed for complexity). Without Deleuze no Greg Lynn or Eisenman, for example. But in order to see what this means in theuir work let us first take a glimpse at Deleuze'
The first thing we can note, right of the bat is the idea of the fold, and the opposition between the curve of Leibniz who introduces a new mathematical system and Descartes's whose system is still tied to a notion of stasis. Despite my poor skills at teaching mathematics I think you got the point: calculus deals with change and variation, it deals with motion and time, and it deals with therefor events.
We will discuss this more on Tuesday next week but meanwhile please finish your examination of both texts and write an entry for the blog about a specific moment in which Deleuze is using mathematics as a model of meaning and try and write something explicit about what that model is mathematically ( eg if he is talking about curves or vectors or whatevr)
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