Preface to the question: I came up with a method of transferring a 2d projection map of the earth onto a globe and made a webapp to help me do that here:
http://codepen.io/vez/full/YVWLRm/ (press alt and move the mouse to rotate the globe/translate the map. Click to place a point on the globe.)
My idea was to take a globe that can be rotated and allow the user to place points on it at the point with the least depth on the globe (always in the middle). When you rotate the sphere you also move a map behind it such that the whole width of the map will be translated given a full horizontal rotation of the sphere, and the height of the map translated given a full vertical rotation. The idea was to move the map so that the borders of land are in the middle circle and place points along the borders in order to capture the globe from the map.
This is where I run into my problem. I imagined that I could find a good 2d map projection so that when I transferred it onto the globe using this method it would be an accurate globe.
https://en.wikipedia.org/wiki/List_of_map_projections
I've heard that no 2d map is accurate, but surely one is the "inverse" of the method I am using to create this globe? So my question is: Using the method described, which type of 2d projection map of the earth, will result in an accurate globe?
As ALB pointed out it is theoretically impossible to map the plane to the sphere (There is no homeomorphism between the plane and the sphere).
in practice however, it is entirely possibly because there is a homeomorphism between the plane and the sphere with a single point removed. In computing this makes no difference because nothing is continuous anyways.
So the answer: The method I listed above is the approximate inverse of the Equirectangular Projection of the globe.