Plotting a fix number of points evenly across surface area of a circle or sphere. I would like to design an LED sphere, but I am having some trouble deciding on the placement of LEDs evenly across its surface area.
I would like there to be 32 evenly spaced LEDs across the circumference of the sphere. That is the easy part. I am somewhat certain that there is no way to map LEDs on to the rest of the sphere without some approximations. 
I noticed that when I designed this image in illustrator
side view of sphere and measured the lenghts of the colored lines that there was no obvious pattern. 
These were the values I got when I divided the lengths of the lines by each other, starting with the yellow line divided by the light red line:
1.96155694377703,
1.451739343459089,
1.272780965237935,
1.175870069605568,
1.111136167174052,
1.061595183331078,
1.019595348096224

These numbers seem totally random to me. Is there any perfect LED count and sphere radius that would eliminate the need to make approximations in LED count along each line of the sphere? With the numbers I have right now, calculating LEDs along each ring results in a decimal; which is a problem, because you can't split LEDs into fractions... If there are no perfect numbers, then how would I get as close to perfect as possible?
 A: It depends somewhat on how many points you want to have on the surface of the sphere.
The most you can have with absolute regularity (in that no point can be distinguished from any other) is 20; that would be the vertices of a dodecahedron or the face centers of an icosahedron.  
You can get 32 fairly regular looking points by combining the 12 vertices of the icosahedron with the extensions to the sphere surface of each of its 20 faces.  
If you want more points, this turns out to be the second simplest (the dodecahedron faces themselves the simplest) case in a series of Buckminster Fuller domes:  Each is made by placing 12 points  the face centers of the 5-sided faces of a dodecahedron, but then filling the remaining gaps with circlets of 1, 2, 3, ... hexagonal faces.  So for example, the next in this series is the "Bucky ball$ with 60 points.
The math to determine the coordinates of each point is reasonably straighforward, but you can for the three examples discussed just look up the coordinates on wikipedia pages for those figures.
