[Math Lair] Maps

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No doubt you've seen a map of the Earth. Perhaps you've wondered about how to map the surface of a sphere (which the Earth approximates) onto a flat piece of paper. You could try an experiment: Take an orange, peel it in any way you like, and try to get all of the pieces to lie flat on a table or other flat surface. You'll probably find that, no matter how hard you try, you can't get the pieces to lie flat.

Similarly, it is impossible to make an exact scale map of a sphere, or any portion of a spherical surface, on a flat piece of paper. This was actually proven by Euler in 1751. No matter how we draw the map, there is bound to be some distortion of some type or another.