Now here is a subject that a reader can spend an entire lifetime on, absorbed, learning and trying to figure out. I am talking of the Conic Sections combined with Regular Polyhedra. If one has an open mind, they can learn something new about them, almost every day. Looking at them from a unconcerned view-- they might think, well, there are only 5 regular polyhedra and 5 conic sections, so be done and finished with them in 6 months. But, however, they are deeply complex and complicated.
What I am attempting to do, is unify them, much in the same manner as unifying Elliptic Geometry and Hyperbolic Geometry to be Euclidean Geometry, as symmetry breaking.
There are few clues that they can be unified. One clue is that there are 5 in each. circle, oval, ellipse, parabola, hyperbola;; tetrah, cube, octah, icos, dodec. Another clue is that icos is dual to dodec and cube is dual to octah, and ellipse is dual to circle and parabola dual to hyperbola.
Last night I was playing around with them of my dice models and noticed something special. I think it is called a dihedral angle for one edge and two planes.
This angle though is not dihedral.
If we look at the regular polyhedra, there are 3 angles of use-- 90 degrees, mostly 60 degrees and then 108 degrees in pentagons.
Now, what throws the angles off, is the oddball 108. But, upon examining that 108 degrees in the icosahedron, I realized that the 5 sides of the pentagon in the icosahedron are occupied by 5 equilateral triangles of 60 degrees.
In the icosahedron it is a dihedral angle of 138.18... degrees, but the pentagon angle is 108 degrees.
So, the question here is, on a octahedron, removing 1/2 of the figure to be a square pyramid. What is the angle of the square base to the edge? Is it 60 degrees, the same as the 60 degrees in the face angles. Can the face angle differ from the base to edge angle.