Singularities

If I'm careful not to imply that this is mathematics, and clear that I don't use these terms in a mathematical sense, then I can borrow terms such as singularity and cusp to describe an aspect of the drawings that I'm calling scaling girih tilings. In these tilings it's interesting that there may be singularities — areas in which there either seem to be no solutions or an infinitley scaling point. Scaling tilings, or self-similar tile sets lend themselves to creating infinitely expanding, spiraling, and converging designs. But, they also require either trial and error or careful design to avoid dead end solutions.

In most of these drawings I use the scaling tile sets to vary density. That, and overall shape drive the design. As a design transitions from low to high density I can manage the tile selection through trial and error to avoid dead ends, or treat them as singularities. As smaller and smaller tiles are added, tile patches may fold in on themselves. It's possible to design tile sets and tilings that form infinitely converging singularities. [1] These are well behaved in the sense that we can add scaling tiles infinitely. If my diagrams were math then I would select only well behaved solutions, discarding those that can't be continued infinitely. Since this isn't math I'm just as likely to investigate dead ends, limits, or ill behaved singularities. Variety and options are preferred. Not that I'm mimicking nature, but natural examples of fractals are only approximately fractal, and limited when things fold in, come together, and can't overlap. For instance, ice crystals on a window pane grow until they collide.

So the diagrams below are a subset of the scaling girih tilings, and with these I've demonstrated a few singularities. I've found that with a certain tile set that includes a dart scaling tile, placing two decagons forces limiting cusps. Scaling continues everywhere there are pairs of equal sides at 216 degrees, as with the decagon. Two of these on a decagon accept three scaling darts, creating six more pairs of sides for the next scaled darts. But at some point between the original two decagons, I'm faced with a single side, and no possible solution. I can't add scaling tiles indefinitely.

Another type of singularity in these drawings is the type that is well behaved and infinite. As adjacent pairs of three scaling tiles converge around a cusp eventually they close, creating a bounded gap, maintaining the edge-to-edge property of the tiling, and creating new inner and outer edges where scaling tiles can be added infinitely. These are the singularities sought by mathematicians, and I encourage them here because they provide areas of smooth, symmetrical density change.

I also want to break up designs with ill behaved singularities. I can introduce seed tiles (imperfections, like a dopant) that create new singularities, some well behaved, and some not. I can also connect two decagons with other girih tiles, and generate singularities.

What if a scaling tiling can be extended indefinitely even as it generates singularities indefinitely? Is some original choice, such as two decagons connected by scaling tiles, necessarily going to generate both singularities and infinite boundaries? Or, will one or the other, infinitely scaling, or finally singular close the boundary?

Reference
1. Fathauer, Robert W. (2000). "Self-similar Tilings Based on Prototiles Constructed from Segments of Regular Polygons," presented at the Bridges Conference (July 28-30, 2000, Southwestern College, Winfield, Kansas).