Contrasts between topology and geometry – preliminary investigations

Euclid’s definition of space relies on the construction of points, lines and planes. These elements, when used in tandem, produce objects that are two-dimensionally grounded. By dividing his theories into definitions, postulates, and common notations, he is able to categorically define the objects constructed, explain their self-evident truths as related to mathematics and discuss the self-evident truths that are not specific to mathematics. By using Euclid’s definitions, postulates, and common notations, one is able to create a flat object, that in its nature is concerned with its form.

One of the major distinctions that Barr addresses when defining space is that topology isn’t focused on a form or shape so much as how it is put together. In Cartesian geometry the focus is on the aesthetic appeal of the object, which is defined by a multi-dimensional gird system. This system was passed down through the generations as a golden chalice. However, Euler’s Law radically changed the world’s perception of space placing emphasis on continuity as opposed to object appeal. Barr notes “a topologist is interested in those properties of a thing, that while they are in a sense geometrical, are the most permanent- the ones that will survive distortion and stretching.”

The specific properties of a thing that will survive distortion and stretching are its faces, edges and vertices. Through Euler’s Law, topological invariants can be used to bend geometric polyhedra from an aesthetic driven form into a continuous, self informed object. Through a series of scripted rules, topological geometry is capable of producing outcomes unimagined by Cartesian geometry. Space is no longer thought of as an inside/outside if/then statement, instead it is looked at as defined by faces, edges, and vertices.

Peter Eisenman takes it upon himself to outright reject Euclidian and Cartesian geometry for the likes of topology and the Catastrophe Theory. Through the research and analysis of Deleuze and Leibniz, Eisenman has equipped himself with an arsenal of weaponry that allowed him to reject previous Euclidian and Cartesian geometry for a more ‘event’ based form. Derived from Deleuze’s idea of continuous variation through a fold, Eisenmann proposes using these philosophical ideas as architectural matter. The concept of the fold allows for mediation between previous figure/ground strategies. The fold is neither figure nor ground yet it is capable of reconstituting both. Eisenman felt that the fold would allow for “an opportunity to reassess the entire idea of a static urbanism that deals with objects rather than events.”

On 2009_01_20

algorithmus conflictus @ pratt institute

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