I named my last post “Perfect Pétanque”. The name might have been slightly premature, as implied that I had finally figured out all of the equations describing the unit sphere of reflection. Or not just implied: I actually said so in the post.
So, wasn’t the statement correct? Well, it was, and it wasn’t. Looking at just one incident of reflection, not in connection with any other event, the model could have been perfectly valid. However, as soon as I took the shape and duplicated it to actually show reflection, I realized that the shape was slightly different. I was able to make a decent estimate of the corrected shape, and it probably isn’t that hard to write the equations for the new shape. But before I do that, here is the new model with two pétanques:
There’s quite a lot going on with this model, so you’ll have to look hard to imbibe everything. I guess the most important thing to note is that the model clearly shows two angles of reflection at the intersection of the neighboring unit spheres of reflection. The yellow flyby path ends in reflection that an angle of reflection identical to its flyby angle. The same is true for the shallower blue flyby paths.
This is at the same time very logical, but it also reveals that my intuition is almost always only partially correct. I probably should convert this two-pétanque model into accurate equations before trying to tackle the four-pétanque model. But I’ll almost certainly try to see whether I can figure the four-pétanque model out without the use of equations, because I’m that curious.
The core of the Theory of Everything is almost certainly nearly ready. I’d be surprised if this ended up being a dead end after all of these discoveries.
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