If I were given the diameter of the sphere and in order for the patches to cover the entire surface perfectly how to know how big each circular unit should be and how many of them are needed?
If the circles are to all be the same diameter, I don't think it is geometrically possible. This is not my specialty, so please someone show me I'm wrong.
Six circles fit exactly around one circle when all are in the same plane and have the same diameter. This works for any diameter circle.
However, when you constrain the circles to the surface of the sphere, they can no longer be in the same plane. From the pole, as the surrounding circles adhere to the sphere's surface AND maintain their tangency to the pole circle their centers must move down and closer to the pole axis. If their diameter does not change, they must overlap. Maintaining their tangency to each other is in conflict with the other constraints.
Draw the six-around-one circle figure on a flat sheet of paper (a plane). Now begin to fold that paper into a sphere... the paper must be deformed - along with the circles - to achieve the approximate the sphere shape.
Check out this post - I think this is what you need
Re: Draw Golfball from the Inside
I don't think there is a single right answer. Obviously the smaller you make the circles the more of them can fit around the sphere. If you make the circle equal radius to the sphere then you can fit one (the equator).
But maybe you could look into the Tammes Problem, I think this deals with the same thing.
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