One drop of ink placed in still water results in a Viscous Finger Fractal. Detail from a work by Theresa Byrnes.

This is a sort of art critique of Theresa Byrnes' recent work, in a geeky sort of phenomenalist way. By that word (phenomenalist) I mean a technique that makes use of a particular phenomena, mastering it, and using it to express further. I have a fascination with science and physics and numerical modeling, so the study is also exploring details of the phenomena of viscous fingering using the APL computer language.

I have set out to explore the process that results in these fantastic images. The ink on paper work detailed above appears at first glance to be a subset of the Mandelbrot set, but this is certainly neither intentional nor phenomenal. The mechanism at work making the fractals is Viscous Fingering, which results when the boundary between a lower and a higher viscosity fluid advances into the higher viscosity fluid. Typical lab models use air into oil.

		Self similarity suggests the mechanisim of sympathetic magic, a physics of fractal causality. If infintessimal details of the cosmic structure are of the same design as those infinitely greater, and that is the rule of it, then to work with the infintessimal is to work with the cosmic. Chapter, book, or letter, one still studies the text, and that is enough. To re-arrange your life, re-arrange your book shelf, your kitchen, the art on your walls. Out of the noosphere new thought is forming, preciptating out of the fractal fluid of our questing group minds onto the canvas and paper and words of artists.
The Mandlebrot set	Theresa Byrnes 05 Ink on Paper, 5" X 5" detail

The actual fractal in this ink drop process may be in the boundary between the ink solvent and the water, and may be quite a bit more detailed than the final image can show. I presume the ink pigment settles out from solution and onto the paper below at a certain rate, that the image forms over a period of time similar to the time it takes the viscous fingers to form. The resulting image truncates at the point where the time the fractal has existed is less than the time it takes the ink to make an impression on the paper.

Diffusion Limited Aggregation Fractals result when particles diffuse until they hit the surface of the growing fractal. This is a process that is very easy to model on a computer. Essentially one line of APL code (admittedly a compact language) describes the algorithm as 'Wander around until you Touch the fractal or step over the Fence. Stick there. Do that over and over.'

Now I got interested in the water just outside the forming fractal fingers of ink. If this were Diffusion Limited Aggregation, there would be a concentration gradient of the diffusing particles going up the veins of the fractal.
The left simulation above is a typical DLA fractal. On the right, instead of recording each point that sticks to the fractal (the :w term) I record every point tested. If a pixel is tested more than once, its color just gets brighter. So the brightness of the violet indicates how much 'attention' the algorithm has spent at that point. Due to using the WOS (Walk On Spheres) computational shortcut, it is almost the opposite of the concentration, a sort of homeopathic intensity gradient. In homeopathy, the strength of a potion increases with each dilution. Interesting, but not what I want.
There is an image of the contours of the concentration gradient, published in Physics Today. As I am assuming the phenomenon is Viscous Fingering, these gradients would be representing the hydrostatic pressure of the water.

My simulation uses a method called 'Walk On Sphere' to avoid simulating large parts of the Brownian Diffusion. I do not need to know *how* my test particle gets to some point on a boundary, just that it got there. This makes the fractal grow faster. But in order to simulate the concentration contours, I need to know the probability my test particle has been in any particular spot in the water.

I know where I Am, I know where I Was: where in the world have I been?

After each step in the 'Walk On Sphere' method, I want to color the inside of the circle (I am a flatlander, I use circles where the real physicists use spheres) according to the probability of where the particle has been, leading up to the point where it escaped the sphere. This is a very strange inverse problem that I could not solve. So I simulated that, too! I just ran a lot (1150) of Brownian random walks out to a circle (flatlander) 300 units large, and added them all together. The image to the left is a false color representation of the result. Code for making it is just above the image. The image presents contour levels of the probability a particle diffusing from the center to the point on the bottom of the circle has been in any other part of the circle. Here is the false color algoritm. It is a variation on 'byte reversal': Here, one maps to red128, two maps to red64, three to red192 .. 256 maps to green128, 257 to red128+green128 That is not exactly just right, so I shuffle the reversed red, green, and blue bytes: Now one maps to red128, two maps to green128, three to red&green128 ... I like byte reversal, but that explicit [24 16 8 23 ...] bit is kinda heavy handed. Uncertainty is a property of physical objects at the quantum level; atoms and electrons and such. Observations that similar properties of uncertainty can apply to thoughts, ideas, and mundane synchronicity suggest that these things also might be some kind of quantum particle. Useful (at cocktail parties or motivational sessions) aphorisms arise from it all: "the uncertainty of a message is inversely related to how exactly it is delivered." Or: "sometimes it takes too long to explain that the downbeat is on its way and you need to put down your smoke, pick up the drum stick, and hit the drum." This exit problem has some hints of it. Something is observed some time in the past to be far away. It is ignored for a while, we hope not so long that it has been able to sneak up on us. It is observed again. In between that, it has not been observed, it exists as a sort of wave function of possibility. The boundaries are a little different in the WOS, though. The second observation is not at some point in time, but at the boundaries of a sort of fence, a circle at a certain distance from the first observation; a 'drag net' or containment perimeter. You might argue that the Something is in fact under observation all the time, because through some kind of analytic continuation, knowing everything at all points on the perimeter allows deducing everything in the interior. I do not know. bb

OK, I like that picture. I have yet to get an expression out of it to use filling in the simulated concentrations; that is for some future work.

Lichtenberg figures,
a frame from Teslamania.com
Here is another metaphor relating art creation and fractals. Here, the creative act is the point where all accumulated energy cascades into a lightning bolt, burning its mark onto the artwork. It is as if the art work is at the exit point of the artist's 'matrix', where all that is pent up within it has gushed out, etching a channel as it flows.