On Fractal Geometry

Fractals are an example of a discovery from what claims to be a very new kind of science — a science of feedback, turbulence and emergence — which harnesses an old but ubiquitous species of chaos. A fractal can informally be defined as a recursively self-similar figure. They are not extrapolated from a geometric logic based on units. Rather fractals are constructed on the basis of an infinite program, such that the result ‘overflows’ any unitary dimension, and so cannot be said to (only) occupy ‘regular’ plane geometries (of two, three, four dimensions, or indeed any whole number of dimensions at all.)

Fractals are freaks, geometric monstrosities, pure chaos — and yet, a strange order persists behind their intricate complexity, a drunken infinity, a sublimely-tangled mess of dimensions. Fractals are a movement-image, but exactly what dynamism does a fractal capture? Not just a fracturing of the one, for fractals also construct their own plane of consistency and originate metrical regimes.

In fact, fractal space ‘twists’ itself up in order to explode a decentralized self-similarity. Shapes in fractal space occupy a dimension in-between ‘unified’ dimensions (a fractional dimension.) They are, in an extraordinary sense, more original than Cartesian coordinates, even though their representations take the form of a chromatic distortion and reconfiguration of our ordinary geometric space.

In other words don’t think of a fractal as wrapping ‘flat’ space around an infinite point, somehow ‘rupturing’ it. Think of the rupture as pre-existing, and that within the rupture, all kinds of spaces are fulminating — flat, smooth, open spaces; curved, broken, and highly structured spaces; complex, self-similar, recursive spaces; even stranger kinds of spaces yet.

Again, the most regrettable misunderstanding would be to think that fractals re-instantiate some kind of logic of unity. On the contrary, the idea of a fractal, in its own way, represents the closure of a certain mathematico-metaphysical era. In their infinite construction, we somehow produce a radical non-decision about the one (or ‘continuity’) which opens an infinity of new dimensional spaces, as though just on the ‘boundaries’ or interstices of ‘ordinary’ natural spaces.

Fractals are not ‘degenerations’ of a primary unification, but a primordial distortion or infinite self-transformation which precedes the logical ‘coordination’ of space. They open up unities — in a sense they are lacerations, deterritorialized assemblages glittering with radical discontinuities — all of whose tangled interconnections nonetheless seem to find a curious resolution at higher structural levels.

The visual approximation of a fractal is an infinitely recursive, self-similar figure ‘compressed’ onto a flat plane. (We can also easily produce 3-dimensional fractal shapes, representing the evolution of fractal sets or families over time.) Really, a fractal is nothing more than a Mobius-like transformation of the coordinates of space, taken as an algorithm and repeated ad infinitum.

It’s not hard to understand, but it is really kind of bizarre to think that they actually ‘exist.’

For example, holograms have many interesting properties related to their fractal structure. Far from reducing everything to a single viewpoint, a hologram records information from all perspectives, and maps this information onto the holographic surface. This surface is as though it were covered with eyes, each capturing the scene from one angle. Thus each tiny ‘micro-image’ itself still contains the ‘whole’ scene from a single perspective. Breaking apart a hologram only diminishes fidelity. Instead of destroying pieces of the scene, we merely lose clarity (we’re losing bits of ‘perspective’.)

However, it is still an open question whether the boundaries of certain fractals are continuous (like the Julia or Mandelbrot sets.) I believe this question will remain open for a long time — though I think a certain ‘fractal algebra’ may emerge which enables us to re-integrate these exotic inter-dimensional topologies. [Topological Geometrodynamics, for instance, another candidate for a rehabilitation of quantum mechanics and general relativity, operates in many ways upon special spaces with embedded fractal structures.] These boundaries appear to be continuous, but how would we ever know if there was a ‘tiny’ break?

Fractals are ‘broken’ in such a strange and original sense such that our concepts of continuity and discontinuity no longer apply. They express a completely natural kind of symmetry. Evolution is recursive. Fractals represent a ‘brokenness’ only in relation to our ordinary conception of symmetry — whose classical image therefore has to be disarticulated, decentralized — in short, “fractalized.” Not only the natural world, but human social systems, science, literature, knowledge itself has a fractal structure. Which in a sense is merely to say there is no guiding One lurking behind fragmentary events, but a second-order univocity — already social and ethical interconnection, above and beyond tension, conflict, disorder.