What type of map can i project onto a sphere to eliminate distortion

[VegaMan]

This is not actual "homework." I am building a POV globe and I want to get as accurate of a projection as possible. The images I will upload into the globe will be a simple 2 dimensional image projected onto a 3 dimensional plane. I have researched and it appears that a Mercator type map is the best bet for this but I am unsure. If a Mercator map will still provide distortion, does anyone know what the "aspect ratio" of the Earth would be, so that I can manually distort the image or should I not even make the image rectangular?

This is basically what I'm going for except with color

 

[chemisttree]

I'm not sure you can entirely eliminate distortion but you can minimize it with a Goodes Homolosine Projection. You will note that the north and south poles are ripped up a bit but if you graft a mercator version of those two land masses onto the homolosine you can probably get a good result. I have no idea how to do it. Maybe these folks do.