Alright, making some progress. What I need now is the negative geometry of polyhedra. The negative geometry is Hyperbolic geometry. I have 1 great resolution so far.
Eucl = Ellipt unioned Hyperb.
It is summarized in one picture )|(
Where the elliptic ) is counterbalanced by the hyperbolic ( yielding the Euclidean straightline |
So, what is the Hyperbolic of circle, oval, ellipse, 3-gon, 4-gon, etc? Here we have some clue as the parabola and hyperbola appear to be "holes in space" so that the elliptical geometry figures can fit inside the hole.
So, what type of figure emerges when we insert a dodecahedron inside a cube, fill in the empty space. Then remove the dodecahedron and examine the figure remaining. We can call it a "hole". Is it a fancy type of parabola and hyperbola in 3rd D.
To answer some of these questions, I have fallen back to 2nd D. And trying to describe the hole inside a square where a circle fits. Now there are 4 points of contact of circle with square, and how do we describe those 4 crescent shaped figures, are they parabolas or hyperbolas triangle shaped with a curved hypotenuse.
Now, if we take all the n-gons, 3-gon, 4-gon, etc etc and placed inside circle we get many of these crescent shaped figures, 3 for the 3-gon, 4 for the 4-gon, interrupted only by vertices of the polygon. Now I think these are called Digons, crescent shaped figures.
So, now, in the Plane, in 2nd D the goal is to match up a positive curvature figure with a negative curvature and end up with a straightline
The curvature figures canceling one another yielding the straight line |
The goal in 3rd D, is for a positive curvature figure regular polyhedra, to combine with a negative and end up as a cube.
Getting tougher and tougher by the day, but, I have until January to achieve.