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 āoverflowsā any unitary dimension, and so cannot be said to (only) occupy āregularā 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 ātwistsā itself up in order to explode a decentralized self-similarity. Shapes in fractal space occupy a dimension in-between āunifiedā 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 donāt think of a fractal as wrapping āflatā space around an infinite point, somehow ārupturingā 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 ācontinuityā) which opens an infinity of new dimensional spaces, as though just on the āboundariesā or interstices of āordinaryā natural spaces.

Fractals are not ādegenerationsā of a primary unification, but a primordial distortion or infinite self-transformation which precedes the logical ācoordinationā 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 ācompressedā 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.

Itās not hard to understand, but it is really kind of bizarre to think that they actually āexist.ā

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 āmicro-imageā itself still contains the āwholeā scene from a single perspective. Breaking apart a hologram only diminishes fidelity. Instead of destroying pieces of the scene, we merely lose clarity (weāre losing bits of āperspectiveā.)

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 āfractal algebraā 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 ātinyā break?

Fractals are ābrokenā 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 ābrokennessā only in relation to our ordinary conception of symmetry — whose classical image therefore has to be disarticulated, decentralized — in short, āfractalized.ā 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.

Fascinating piece. I always get the impression that fractals give me a glimpse into the deeper structures of everything we see around us.

As you say, it’s bizarre to even consider them to actually exist.

I will agree! indeed very fascinating stuff, I wish we could find a way to “understand” them better. I remember years ago there was an attempt to use them for image compression but I haven’t heard any news. I puzzle myself very often with them… Any suggestions for any news, sites, etc about fractals? (Topological Geometrodynamics is news for me)

R.I.P. Benoit

The connected of the Mandelbrot set I think was actually known (to be true, no tiny separate islands) at the time this was written — so much for that prediction š

I’m trying to know how to integrate quantum mechanics with relativity using a fractal view of reiteration. I believe that our reality lies in between the 3rd and 4th dimension. I’m still thinking how to define a ”full” 4th dimension with time.