Alt.Fractals

Andy Lomas: DLA fractal art

Andy Lomas makes computer simulations of things that look a little like limescale deposits in water-pipes ... but really pretty limescale deposits. They're another example of the DLA (Diffusion Limited Aggregation) fractals mentioned earlier (like my iced-up fridge!), but his are computer-simulated at extremely high resolution, and can contain between around fifty to a hundred million individual particles.

• Andy Lomas' profile on CGSociety

More unusually, he's also done high-resolution plots of the tracks that particles took to reach those points, which gives a sort of inverse "ferny"-looking or lichen-like pattern. You can see some of his "Images of Aggregation" and "Images of Flow" art on his website.

An example of "viscous fingering" between two glass plates

This is a “finger fractal” that I noticed on the pavement a few days ago, embedded in somebody's basement skylight window. The effect is sometimes known as “viscous fingering”, and it happens when you glue two plates of glass or perspex together, and then slowly prise the plates apart at one edge before the glue is properly set.

Air penetrates between the plates, but the thick surface of the glue clings to the glass and doesn't want to retract. Eventually a weak point in the wall “fails”, and the glue behind the tip of the inclusion finds it easier to retract than the glue at the sides, and a finger of air extends into the glue.

These sorts of inclusions tend not to meet up and join – in fact they seem to avoid each other and maintain a critical distance – so presumably a region of glue that has a lot of “edge” (anchored to the glass by curved meniscus surfaces on multiple sides) is more strongly connected to the glass, and more difficult to get rid of. Once an air finger penetrates within a certain radius of another inclusion or edge, it seems to be easier for further penetration to happen somewhere else, so when a finger starts getting too near to another edge, or the plate separattion within the wedge reaches a critical point, the penetrating finger's progress "stalls", and a new finger breaks through the perimeter somewhere else. What we end up with is a branching system of inclusions, and a branching network of remaining glue, interleaved. 

We don't usually think of glue as being a "clever" material, and yet here it is, unwittingly helping to create complex, self-regulating branching designs that look more like the results of some sort of encroaching lifeform's growth pattern.

This is a physical example of something known as a DLA (Diffusion Limited Aggregation) fractal. I found it in my fridge.


So what does DLA mean? The "aggregation" part means "collection of things stuck together"  (in this case, water molecules from the air freezing on contact with the existing ice, and "aggregating" to produce a clump). The "diffusion-limited" part means that the aggregation process is hampered in some regions by the statistical difficulty of new material being able to wander into those regions without hitting another overhanging sticky-zone before it can get there.

In two dimensions, DLA fractals tend to create long, straggly, self-avoiding strands that look like the roots of a small plant (book, figure 36.7, p182). In three dimensions it produces lumpy cauliflower-like clumps of the type found in my fridge.

Once one piece of ice starts to protrude more than the others, it gets to snag more passing material than the other regions, and grows faster. And as it grows, it "shades" the more low-lying regions, and grabs the material that would otherwise have gotten to them, so they find it progressively more difficult to catch up. It's a positive-feedback process -- small initial random imbalances in height get exaggerated, and you end up with a pronounced pattern of overhanging clumps and deep valleys. 

The reason why it counts as a fractal (rather than just an example of low-pass-filtered noise) is because once you have a nice big blob of a clump, any small peaks or protrusions on top of that start to shield the lower-lying regions, and the differentiation process continues, giving clumps-on-clumps-on-clumps, and clefts-on-clumps, and clefts-in-clefts-in-clefts.

The little pale brown speck is probably a breadcrumb.