Jan 27 blog: Contrasts Between Topology and Geometry

Euclid’s three categories of geometry are definitions, postulates and common notions. Definitions serve to describe the most basic geometric elements and the figures that can be constructed from them. The postulates describe specific relationships that exist among geometric figures, and the Common Notions are basic statements of equality.

Euclid seems to primarily concern himself with space that can be fully rationalized and physically measured. His notion of space is somewhat limited in that it only allows for the creation of space in two dimensions. Space is conceived of, and described by the vocabulary set forth in the definitions, postulates and common notions. He does not establish a means of describing space that exists outside his language of points, lines, planes and figures.

Barr defines geometry as the study of mathematical space. He defines topology as a type of geometry but states that it also encompasses other fields. The two differ in that geometric spaces must, by definition, be visually expressive. This is not necessary for a topological space. Typology provides the means of expression for figures that would, in geometry, be otherwise indescribable. It deals with rules of continuity rather than rules of form.

It becomes easy to see the differences between geometric and topological spaces when looking the example of the geometric and topological pentagon (pg. 12, fig. 14). A pentagon is defined geometrically as a polygon having five sides and five interior angles. However it talks nothing of the number of faces, vertices and edges. The topological pentagon is not a geometric pentagon because it does not have five interior angles, but it still can be classified as a pentagon because it retains the same number of faces, vertices and edges.

Eisenman rejects the traditional ideas of Cartesian geometry and instead favors the idea of the fold rather than the point. He is interested in the way that the fold can change and manipulate the existing relationships of the horizontal and vertical, and figure and ground. He favors the ideas of Deleuze and objectile, which imply continual variation through the fold. He is interested in its affect on the object / event relationship.

He also is interested in the ways that the fold can be applied to Thom’s catastrophe theory. He explains that in catastrophe theory, a grain of sand that appears to cause a landslide is really not the cause at all, but rather the cause exists within the conditions of the entire structure. The fold in relation to architecture is similar in that it can serve as the unseen force that explains abrupt changes in form as well as in urban conditions. He sees enormous potential in the fold as a way to reinterpret and reframe what exists as well as connect the old and the new. He sees it both as a potential formal device as well as the way to transform architecture and urbanism from static objects to meaningful events.