In my last post “Reflection Double Hihat” I presented a geometry of reflection that was at least supposed to be mathematically correct for reflection of elementary particles of energy (kaus/dots) where the reflection resulted in a helical path around a linear central path. In the post I promised that the shape could be applied to describe physically relevant reflection as well.
Well, in this post I attempted to do this. To begin with I have to confess that I had screwed up my color coding of the reflection shape. While I clearly was describing linear paths between reflections, I used yellow cylinders to describe these, like in this Half a Double Helix post. In today’s post I’ve gone back to the old color coding.
So, how accurate was my prediction that the shape would allow me to write the equations for reflection? As tends to be the case in my posts, the answer is “accurateish”. When I went back to the Blender image and tried to derive rules and even equations, I realized that I had already taken a misstep. In my post “The Unit Cell of Reflection” and even before I had clearly stated that between reflections the kaus travel from one circular plane to another, where the circular planes are connected to each other from a single point. This means that any reflection needs to be with connected hihats. Or more specifically, the reflection model has to be built around the unit cell of reflection with two hinged circular planes of reflection.
That isn’t to say that the simplified reflection model was useless. Rather, this simplification can be applied to the unit cell of reflection as well. If I don’t know the perfect way to draw a reflected unit cell, I can draw the unit cell with zero reflection, but all the required elements to derive the location of kaus at the point of reflection, as well as at the points of grazing. Here’s the model:
If you look closely, you see several important elements. Firstly, you see two transparent blue spheres depicting the centerpoints of outer (blue) reflecting dots and the radius (2r) of their connection to the inner (blue) reflecting dots. With zero reflection you can draw a single blue torus (with a radius of √3r) between the two spheres, depicting two merged circular planes of reflection. At the base of this blue torus I’ve drawn a red cylindrical hinge, depicting the only point in the shape that remains unchanged upon reflection. This means that the location of all the other shapes (especially the kaus) must be in relation to this spot.
After this, things turn a bit complicated, but I’ll try to keep the concept simple. The first thing to determine in the reflected shape is the shifted centers of the large blue spheres. After determining these two centers, I hopefully will be able to determine the location of the other kaus. I was about to say something a bit smarter, but realized that the only way for me to know whether my musings make any mathematical sense would be to formulate equations based on this shape, but I must admit that my mathematical skills aren’t that good. I can derive the equations eventually (hopefully), but speed isn’t my forte.
Also, if you look closer still, you can see that I’ve reverted back to the shape from Half a Double Helix post. This goes to show how complex the geometry is. I still feel confident that I’m edging closer to the finishing line, but now that I’ve found which equations I should formulate, I’m not at all sure whether this is an easy or a hard task. It seems each breakthrough is smaller than the previous and requires hours of staring at 3D models and trying small tweaks without knowing whether they improve or deteriorate the model.
If my model proves correct and not just a pipe dream, I can just imagine future historians of science and mathematics wondering how hard all of this was for me, when they’ve learned it in high school with clear instructions. Well almost no discovery is easy when you invent them. At least not the mathematical ones. The easy discoveries have been made.
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