The only map I can think of from the sphere to the plane (and periodically tiles) uses genus 0 Riemann surfaces and quadratic differentials. And I know he didn't do that.
Nonetheless, there's a fascinating work by Anton Zorich called "Flat Surfaces" (it's a great discussion with lots of pictures))
https://arxiv.org/abs/math/0609392
The geodesics on a flat plane is the straight line -- while the geodesics on a sphere are the great arcs. So there has to be a fair amount of distortion.
I could imagine just drawing a map of the world on the surface of a cube instead of a sphere. That might come out pretty bad.
Nonetheless, there's a fascinating work by Anton Zorich called "Flat Surfaces" (it's a great discussion with lots of pictures)) https://arxiv.org/abs/math/0609392 The geodesics on a flat plane is the straight line -- while the geodesics on a sphere are the great arcs. So there has to be a fair amount of distortion.
I could imagine just drawing a map of the world on the surface of a cube instead of a sphere. That might come out pretty bad.