Circumscribe the wagons

I think it is safe to say that I am engaging with math by other means.

Monotonal, simple traversal circle immersions in a plane,” is a very specific mathematical description for my subjects over the last nearly two decades of work. The “in a plane” part refers to the looping forms in their final states representing strictly 2 dimensional configurations. “Circle immersions” are basically circles that fold over themselves without creating any sharp angles or kinks. “Simple traversal” means that the intersections created through immersion are limited to one segment intersecting another only once at any given point, with the intersections all being direct crossings rather than tangents. (Think of a tangent as the way a wheel intersects the road – touching but not crossing.) And “monotonal” means that the curve acuity can vary, but never so much that it changes the curve's direction from clockwise to counterclockwise or vice versa. So, no ‘S’ curves. Before I ever get to considering ‘S’ curves, I would first need to decide to incorporate concurrent intersections, then tangents. Either one of those would dramatically change things, and I’m not ready for that yet. The profligate and indiscriminate introduction of new uncontrolled variables is not to be taken lightly.

A simple way to think about these forms is to take a rubber band and add a twist, making a figure eight or infinity symbol. Then fold it down to drop the one loop inside the other loop. That’s a first order circle immersion of the simplest type with the fewest variables to contend with. Only, it isn’t very simple at all. Because there’s a combinatorial explosion of isotopically distinct permutations that gets more confounding with the addition of each new intersection. Weirdness shows up very early on in the proceedings. It’s so fascinating that I am spending a lot of time doing the most challenging math I’ve done since my undergraduate astronomy class at CSU. Turns out I might have stumbled upon a problem in discrete mathematics for which there is not yet a published solution set. And since I am well aware that I’m missing some critical prerequisites, I have solicited guidance from, and am currently working with humans more qualified than myself to understand this highly constrained problem space. And we’re having a blast.

Ah… the lives of artists!

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