In my last post I showed a tentative way to get bikaus, that can fold into origamis with a specific radius to also fold into sphere. If you haven’t followed all of my posts, bikaus are the paths that two ends of molecules move between reflection, when the molecules are simplified into a vector. I haven’t told you this yet, but this model requires molecules to stretch along its axis. Molecular vibrations are a commonly known phenomenon. I’m not sure whether I need to invoke new vibrations to explain reflections. If I had to guess, probably not.
Anyhow, for a fleeting instance I thought that would need to scrap my unit sphere of reflection. But it doesn’t seem like I have to, after all. However, I probably need to ditch the old hinge. You see, the old hinge model seems to be incompatible with the bikau quadrilateral. I think the best way to explain is to show a 3D model of a folding bikau quadrilateral with a unit sphere of reflection:
The green diagonal line is the axis of folding. When the angle of folding is zero, the long blue and yellow lines, depicting the paths of motion form a rectangle with the short blue and yellow lines, depicting vectors connecting the two sides of a molecule at two instances of reflection. When the folding angle is zero, all the points of the shape are located within a single circle along the equator of the sphere. However, when folded, the points not along the diagonal rotate along circular paths around the unit sphere of reflection. These points can be located along a circle offset from the equator and with a smaller radius.
While this model of reflection doesn’t rule out the possibility of the circular planes of reflection touching each other, my intuition says that they won’t. I can’t exactly explain why I have a strong feeling about this, but it relates to what I found out in my previous posts.
While in the above model I have only one bikau quadrilateral within the sphere, I’m sure that there should be two of them in the final model, just like I’ve thought for a while now.
I could still be wrong about this. It wouldn’t be the last time. On the other hand, most of the time when I’ve been wrong about details, the theory has still been steadily progressing towards an ever more logical conclusion. Perhaps I can one day say that the theory is ready…
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