Thursday, 24 February 2011
The "Delta" is is one of my new favoritest fractals. It was difficult to discover because it didin't have any obvious 3D siblings. It kinda seems to be a one-off. I haven't found any record of it being documented before, I used it on page 161 of the book (anf referred to it as just the "Delta" fractal), but it really needs a more distinctive and search-engine-friendly name than just "Delta" ... so now, unless anyone can find evidence of prior work, I think I'm going to start referring to it as the Baird Delta. :)
The shape's building-block is a solid with six identical triangular faces. You can then cut away material to produce three smaller rotated copies of the original (and repeat), or stack three of the rotated blocks together to produce a larger copy. Similar things happen with the Sierpinski Pyramid, but this is a leetle bit more subtle in that the component blocks (and the main shape) aren't quite regular polyhedra, they have to be twisted perpendicularly at each iteration, and the shape doesn't seem to have an obvious two-dimensional fractal counterpart. This makes the shape more difficult to visualise and more difficult to stumble across. The first time that you see the shape, it probably takes a bit of squinting before you realise that the three corner-pieces are identical smaller angled copies of the whole thing, rotated from the original baseline by 90 degrees, and rotated with respect to their siblings by 120 degrees.
As you iterate, the area of each face gets progressively nibbled away until it's effectively a Koch Curve (although its a differently-angled version of the curve to the one that gets used for the Koch Snowflake).
All in all, a cool shape.
Thursday, 17 February 2011
|An example of "viscous fingering" between two glass plates|
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.