The Curvature of Reflections

Writing my last post I truly believed that splitting reflections into folds would allow me to solve the finals kinks of reflective gravity within days or weeks. However, when trying to fit the parallelogram fold with previous models to not explain the directions of motion of the reflected particles (of energy, or molecules, depending on what scale one is interested in), I’m getting constantly stuck .  The reason for this is rather simple. I can no longer rely on rough estimates. Any handwaving will result in almost immediate collapse into chaos.

So, instead of banging my head against the wall, I decided to go for a totally different approach: could I explain the shape of the Saint Hannes Knot with reflections? To people who’ve read my blogs, and possibly to others as well, this seems like a much harder problem to solve. Its core shape is an Archimedean spiral of two turns and an infinitesimally small height to radius ratio, that is bent around the y axis by 360 degrees. This means that the spiral curves around a sphere with a rather unintuitive curvature.

Or more specifically, before today, I didn’t even really understand what its curvature was. Even yesterday I thought that the curves encircling the surface of the spere would be curving around the center of the sphere. The simplified reason for why I thought so was that I hadn’t really thought about the issue.

However, when I tried to explain the shape with circular arc, simplified into circles, the shape that emerged was quite different than the vague idea that I had had in my head. From the sphere, eight easily distinguishable curves could be observed, with gradual transitions from one curve to the next.

This is what the eight curves look like:

 If you look closely, half of the curves can be drawn as discs teetering on a cylinder, with the cylinder pointing at the center of the sphere. Each of these curves transition into a curve which can be drawn as a circle a around the center of the sphere, i.e. with a radius equal to that of the sphere. The third curve is otherwise identical to the first one, but with an opposite direction of rotation. For example, the curve rotates clockwise around the blue disc, but counterclockwise around the green one.

So, where does this insight lead me to? I don’t know yet. What I do know is that this model is a simplification that doesn’t take into account the need for a secondary curvature that leads to the formation of the saint Hannes knot. But at least it should be the first step to formulate the large-scale folding pattern of reflection. But honestly, answering this question just leads to more questions.

Perhaps I might be close to the final solution, but once more I realize how much there is to understand. This was a serious case of unknown unknowns. I think there’s going to be more of those coming around in the near future.