Fractals, Chaos, and All That

During the Baloney years we did a lot of cool cerebral stuff that I’m still hooked on, including Mandelbrot zooming, Conway’s Game of Life in 3D, iterated function systems, etc. Recently I helped crowdfund the Mandelmap Poster. However, despite their beauty (dare I call it stark?), there always seems to be something harsh and sterile in fractal images, with anything soft or biological missing or contrived. I have the feeling that to represent physical reality with this kind of math, each point in space and time should dictate not only the seed value for the iteration, but the formula as well. At any given scale and location I imagine you’d have a richer, smoother, looser landscape of possibilities, while as you zoomed in or out, particular self-similar shapes would persist for only a few orders of magnitude, giving prominence to different emergent ontologies (e.g. “Seahorse Valley”, “Main Cardioid”, “Spleenwort Fern”) at different scales. In my grandiose fantasies, an alternative scheme for representing physical reality could be developed from this kind of perspective.

Take iterated affine transforms sensu Barnsley. The idea would be that there might be hundreds or thousands (perhaps an infinity!) of transforms in a given mapping, but rather than apply them all according to a fixed set of probabilities, you’d vary the probabilities as a function of coordinate location or iteration count or something, with some transforms dropping out entirely and new ones coming in to replace them. Perhaps emotionally satisfying images could emerge from this kind of conceptual expansion of fractal algorithms. But can that kind of art inform science? In a sense, the universe is already an iterated function system or cellular automaton, with the next state of each Plank-voxel being computed by some mapping that we currently understand as obeying the Standard Model and/or Relativity. Perhaps some kind of deep computing project could identify patterns in images or datasets generated by my approach that resemble patterns in physics, thus revealing some kind of basis settings for further exploration.

It turns out that just as I was beginning to draft this post, Wolfram and colleagues released work that supplants his A New Kind of Science, purporting to potentially contain the seeds to a unification of relativity and quantum mechanics. I read ANKoS when it came out, and I would say I have (actually, already had, having been familiar with some of Wolfram’s published work in the area and done my own computational experiments) a pretty good understanding of the material, but this new work definitely supersedes it. ANKoS wrings out just about everything that could be interesting about one specific class of ultra-simple cellular automaton, including the tantalizing notion that, because Turing-complete computation can emerge from such simple abstract constructs, simple physical systems could potentially accidentally implement them, leading to an inevitable evolution of complex algorithms (e.g. life itself) from utterly basic substrates and inputs. The new work recapitulates much of ANKoS, but starts with even simpler constructs – so simple that to model even his simple cellular automata, a rather complex arrangement needs to be implemented using the new parts list. Wolfram et al. invite us to help, SETI-at-home style, by buying a copy of Mathematica and running his group’s free notebooks. I’m darned busy right now, so I am meticulously avoiding this sort of distraction. “Know thyself”: I am quite easily addicted to or distracted by dopamine micro-reward providers like certain kinds of computer games and puzzle-like recreations. “Nothing to excess”: best for me to simply not dip my toes in that stream, as the slightest exposure could well be too much, and I have Responsibilities.

My aim, were I to venture into this realm once again, would be to seek parallels to the Taylor-Couette flow demonstration that inspired some of David Bohm’s work, (skip to 13:25 if you’d rather not watch the entire video) and which he begins algebraic development of with Basil Hiley in The Undivided Universe. I would be unable to resist attempting to cast everything I did into some kind of coordinate system based on n-dimensional aperiodic tilings (essentially, derivatives of Penrose Tilings), and attempting to link philosophy (is it pointless to consider determinism vs. non-determinism?) to basic physics. A fool’s errand, perhaps, but a grandiose one. I am nothing if not grandiose. I won’t claim to not be a fool.