There is a topology of objects. In mathematics topology is sometimes referred to as “rubber sheet geometry”. What is studied here is not the fixed metric properties of a shape– such as the degrees of each angle of a triangle –but rather the permutations a shape can undergo through operations of bending, stretching, twisting, and folding. For example, a triangle can be turned into a square by folding over one of its vertices. In topological language, triangles and squares are thus topologically equivalent because there is a series of operations through which the one can be transformed into the other. Likewise, an equilateral triangle can be turned into an right triangle by pulling and stretching one of its vertices. Topology is a dynamic geometry that thinks shapes in terms of movements and transformations.
The claim that objects are topological is the claim that they are subject to permutations that transform both their phenotype and their qualities. As I have argued in The Democracy of Objects, every object is characterized by an endo-structure and the exo-relations it entertains to other objects. The endo-structure of an object is its internal structure, coupled with the way its potentialities or attractors are related to one another (recall that I argue that the being of an object is not defined by it’s qualities but by its potentials or capacities, by what the object can do. The qualities of an object such as its shape, its color, its density, etc, are actions on the part of objects; events that the object produces as a result of what it does.
By contrast, the exo-relations of an object are its relations (if it has any) to other objects in the world. I have called these fields of relations to other objects “regimes of attraction”. If I refer to them in this way because the manner in which one object interacts with another generally leads to the genesis of new qualities in the object. For example, when the air conditioning is pumping in my classroom my skin prickles. The relation of my body to the air conditioner and the dense air brings about the quality of prickled skin.
Regimes of attraction define the topology of objects or the permutations of which that object is capable. Take the example of insects. Why aren’t there beetles the size of a car or a house? Is it there genetics? Well that’s certainly part of it. However, the regime of attraction in which beetles exist on the planet earth also plays a tremendous role in size of insects. Exoskeletons are wonderful forms of protection against other predators, but they’re also costly. They’re costly because they’re heavy. On the planet earth (the regime of attraction in which beetles exist), the specific gravity of the earth, coupled with atmospheric pressure, available food resources, and the amount of oxygen in the atmosphere define a very abstract space of possibility for critters with exoskeletons. If the insect is too large, gravity and the pressure of the atmosphere require too much energy to operate the exo-skeleton. The cost of a large exo-skeleton outweighs the benefits that accrue from its large size. And, of course, within the regime of attraction defined by the planet earth does not pre-determine what types of exo-skeletons will appear and what they’re shape will be; it only sets particular constraints on the size that exo-skeletons can take and how the different parts of the exo-skeleton must be cobbled together. Just as there can be an infinite number of absolutely unique triangles within the space defined by being a three-angled figure, there will be an infinite number of possible exo-skeletons within these constraints.
The important point is that where the regime of attraction is changed, the space of possibility also changes. If, somehow, Mars were terraformed– something I think is unlikely, despite my love of Kim Stanley Robinson, because the planet lacks an electro-magnetic field –and beetles were raised on this planet, the topological possibilities open to the size of the beetle would change. If this is so, then it’s because the mass of Mars is about half the size of Earth, changing the nature of the constraints. Something similar happened on the planet earth. Many millions of years ago there were centipedes that were six feet long and dragonflies with two foot wing spans. Why were insects this large possible then but we don’t encounter any of them now? Part of the reason was because the earth had much more oxygen in the atmosphere at this time (about 30%). That additional oxygen allowed critters to burn energy much more efficiently which, in turn, allowed them to feasibly grow to larger sizes.
This last point about oxygen is important for it shows that topology of an object is not simply defined by things like gravity and pressure, but has chemical and organic components as well. For example, people sometimes wonder if humans have undergone significant genetic mutations in the last couple hundred years because we’re so much taller than we were before. This is unlikely. Rather, what’s changed is the diet of people: we now have far more milk and meat available as regular portions of our diet. As a result, this allows potentials of our endo-structure to be actualized that couldn’t be actualized in previous regimes of attraction defined by different systems of production and distribution.
The topology of objects should be thought in the case of every type of object. There will thus be a topology of cities, a topology of nations, corporations, revolutionary groups, individual lives, and so on. The study of regimes of attraction is crucial to understanding questions of change because in many circumstances these regimes can themselves be transformed allowing for hitherto unknown and unimagined potentials to be unleashed in the world. Often it’s not a belief that keeps certain oppressive forms of life and existence in place, but a regime of attraction. Getting at that regime of attraction is a condition for change.