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I like your electric fields idea regarding the applications of the twelve sections of the sphere but since I last posted on that thread I have run into a major roadblock to all the applications I was considering. The subdivisions do NOT appear to be of equal area as I had figured earlier in the thread, using a spherical calculator. I made some incorrect assumptions about angles and unfortunately it appears the subdivisions near the center of each diamond are of greater area then nearer the corners. So most applications are thwarted until I come up with a way to have each designated subsection span an equal area.
There are still some applications that will work if exactness is not required. If you quarter a section each of the quarters is of equal area, so if you want to divide the sphere into 48 equal sections for your application the scheme is still good.
One such idea is to for instance mount 48 cameras on a tower or aircraft each pointing in the direction of the center of a quartered section of each diamond. Then the output of the cameras could be sent in a one to one way to 48 screens positioned in the same pattern around a viewer. Thus placing the viewer at the top of the tower or in the craft.
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So the 48 cameras are some sort of virtual reality?
Application: What if you divide the Earth into sections to map satellites?
What about navigation, especially in Space?
What about video game 3D world positioning?
The changing shape and propagation of the radio wave?
When you draw you start with geometry primitives. Why not create a CAD script?
It is hard for me to think of something mathematical that doesn’t have an application no matter how small.
I know my application suggestions are not descriptive and it would take much work to complete them, but I envision a 3D model manipulated by change, shape, and division of spheres. Instead of editing polygons you’d edit spheres. If you tried 3D printing you know the inside of the model is structured like a bridge trusses. Those supports could be your spheres divisions.
I do need to learn some CAD and Rendering programs.
There are many applications I think for the general Sperical Rhombic Dodecahedral structure.
And interesting aspect is it filled with symmetries and dual figures. And it is the basis for a dense packing scheme that puts 12 spheres around a center sphere. This pattern is a cuboctrahedral pattern that when built out, putting always 12 spheres around each sphere in the same pattern it looks more and more like a cuboctahedron as you build out, and contains 3 intersectinng square planes and 4 intersecting hexagonal planes.
The math is already there, the switch is to think of a cube with the corners cut off to the center of the edges. The center of each edge is the center of a 1/12th section of the sphere.
I am working on a scheme to each sphere in the scheme. The scheme fills space and is completely scalable and very symmetric, having direct analogs to the cube, the sphere, the tetrahedron and the cuboctahedron.
I like your ideas. Please use the system freely. It is a mathematical system so I can not claim any ownership. What I will claim is the discovery of the way to break the sphere into TAR radians using the four axis of the cube that extend through the corners, assigning a color to each 360 group of great circles and using the intersection of the two colors in each of the diamonds to name each degree sized area. I recently found out that all the degree sized area are not equal in area, so there is as of yet not a clean mathematical way to use the system.
Where I see some possibility however is in labeling the spheres in the dense packing situation. There is a relationship between the spheres in the dense packing system and the TAR radian system but I have not discovered it yet.