the wherewithal

Just a quick addition to the previous reflections on topology. I have just been rereading Brian Massumi’s Parables for the Virtual, and his distinction between Euclidean and non-Euclidean topological forms seemed particularly appropriate to the differences between networked and recursive topologies. The following passage, from a section titled ‘Notes on terminology’, certainly helped me clarify my thinking. He writes (the full chapter is also available on his website - I have also added a few relevant links):

“Topology” and “non-Euclidean” are not synonyms. Although most topologies are non-Euclidean, there are Euclidean topologies. A Möbius strip or a Klein bottle are Euclidean figures, of one and two dimensions respectively. The distinction that is most relevant here is between topological transformation and static geometric figure: between the process of arriving at a form through continuous deformation, and the determinate form arrived at when the process stops. An infinite number of static figures may be extracted from a single topological transformation. The transformation is a kind of superfigure that is defined not by invariant formal properties, but by continuity of transformation. For example, a torus and a coffee-cup belong to the same topological figure because one can be deformed into the other without cutting. Anything left standing when the deformation is stopped at any moment, in its passage through any point in-between, also belongs to their shared figure. The overall topological figure is continuous and multiple. As a transformation, it is defined by vectors rather than coordinate points. A vector is transpositional: a moving-through points. Because of its vectorial nature, the geometry of the topological superfigure cannot be separated from its duration. The figure is what runs-through an infinity of static figures. It is not itself determinate, but determinable. Each static figure stands for its determination, but does not exhaust it. The overall figure exceeds any of its discrete stations, and even all of them taken together as an infinite set. This is because between any two points in Euclidean space, no matter how close, lies another definable point. The transformation joining the points in the same superfigure always falls between Euclidean points. It recedes, continuously, into the between.¹