Three Hinges

I think I have to start today’s post with an apology. The reason for this is that I’m going to present a monster of a 3D model. It all boils down to a saying by Einstein, which was later simplified to

“Everything Should Be Made as Simple as Possible, But Not Simpler”

The original saying was a bit more precise:

“It can scarcely be denied that the supreme goal of all theory is to make the irreducible basic elements as simple and as few as possible without having to surrender the adequate representation of a single datum of experience.”

And, as far as I understand, this is such a model of the reflective interactions of elementary particles of energy /kaus/dots):

This monster of a model presents the three hinges presents in the unit sphere of reflection with a hinge angle of zero. As mentioned in the Hinged Expansion of Spacetime post, the diameter of the unit sphere is √12r/cos θ, where r is the radius of a kau and θ the hinge angle. When θ = 0, the diameter simplifies to √12r.

However, the single-hinge model from my last post ignored the fact that there are three hinges in the unit sphere, the edge hinges being the centers of the neighboring unit spheres and the central hinge being shared with both neighboring unit spheres. What this means is that when one chooses any value for θ, this three-hinged-shape can open in only one possible way. Unfortunately, I won’t be able to this opening in equations in this post. That will require some head-scratching.

So, unless I’m unable to convert this shape into sensible equations, this is the core of the theory of everything. As simple as possible, but not simpler. But as the saying goes “Do not count your chickens before they hatch”. I can only feel fully at ease when I have working equations. Before that I don’t know whether my eggs are dead or not. At least they don’t stink…