In my last post I showed that the unit sphere of reflection is smaller than I thought. This means that the sphere that I previously considered a unit sphere requires renaming. As this sphere describes the hinging of two connected unit spheres, I’m renaming it a hinge sphere.
In today’s post I combine the unit and hinge spheres into a whole and show how they relate to each other. This time I change the order of the letters of the nodes presented in the Known Knowns -post, remove the letter not belonging to the smaller unit sphere of reflection and introduce a new node for its center. This is what the model looks like:
If you look at the letter, you can see that the center of the hinge sphere, A, is located at a different place than the center of the unit sphere, B. This means that when the hinge sphere is expanding, when the hinge angle is expanding, the location of the center of the unit sphere is changing as well. But the unit sphere is expanding as well. What cannot be shown here is what happens to the unit sphere when the hinge angle is expanded. You see, when the hinge angle is zero, the grazing (yellow) and reflection (blue) nodes are located on the surface of a unit sphere with the same radius of I think √(3/8) r, or ~0.612 r (I could be mistaken with this calculation, though). However, when the hinge angle increases above zero, the distance between the yellow nodes remains 2r, whereas the distance between the blue nodes increases above 2r. This means that there are two unit spheres of reflection centered at B: a smaller yellow one for the yellow nodes and a slightly larger blue one (only illustrated with blue toruses) for the blue nodes.
Whether what I say in this post is true depends on whether I can convert these ideas into equations. I’m quite convinced that I can, but before they’re ready, there might still be surprises ahead.
Comentários