These readings really took me back in time. A time where Architecture was perfect (Vitruvius), where the follies of the mind were simplified by true natural beauty (Boullee), and then Durand. It seems to me like Boullee and Vitruvius had many things in common: they were both utopian, they both built this road to architectural superlatives. How they used mathematics is less obvious. Their system of dialectics parallels geometry, a sort of progression from simple to complex, yet this was not necessarily laid down heavy in the readings.
Boullee used geometry to legitimatize architecture. Meanwhile, it's difficult to follow anyone who states, "Weary of the mute sterility of irregular volumes.." When he begins the next sentence with, "An irregular volume is composed of a multitude of planes," Maya quickly comes to mind, as do NURBS surfaces, and it becomes clear that we have a much varied respect for the perfection of shapes. Math, I guess, is a tool for determining symmetry which leads to order which finishes with clarity.
Durand is sort of the grey goose. He tackles the five orders of architecture, Vitruvius & Boullee as well, by challenging established principles and at times by simply saying they are wrong (love the footnotes). Architecture to Durand is not imitation based on nature. Instead, he focuses on fitness and economy. He states, "The more symetrical, regular, and simple the building is, the less costly it becomes." The dispositions of the architect come together to make a good building. Durand doesn't use math the same way the others did. He uses math more passively while he critiques previous authors. I get a hint of Maya as well when he states, "Furthermore, is not such a model even more defective than the copy," and remember all the times I've tried to duplicate or copy things in my scene and everything gets screwy. Is he saying that non-geometric shapes have a history as opposed to definite location in space?
I realize I am just brushing the surface. The question, how do they use math, is not as clear as it was in Barr or Euclid. Knowing the discussion leans towards topology, we can only insert ideas and extract pieces of understanding.
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Good, Durand is definitely challegning previous models of meaning, which he finds irrelevant - but he doesn't just dismiss them, he does his work and shows why there are wrong and then concludes that the models are a complete mess, so there is no point in "copying" them. Passive is a good point: he is using a kind of excell spreadsheet logic and this points to a new horizon. We don't have to say he is right and Boullee is wrong -- that's not what you imply. But rather he is shifting the terms regarding how we can use a model, or a model of meaning, like the principal of symmetry. its pure, its simple, and its very powerful. keep in mind, he was writing primarily for a new kind of engineering training -- to make engineers quick architects. But eventually, this became a standard in architectural education. voila!
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