Mapping a World With Much Less Warped Distances?

[Sylrae]

Hey Cartographers;

I've made a bunch of world maps for my home games in the past several years, though typically the big rectangular kind. I was thinking about the idea of a way to do a world map that isn't distorting all the distance at the poles so much.

I've seen the Icosahedron variety, which cuts down on distance distortion a little bit, but I was hoping for something that would distort it less. When I was looking around, I came across Goldberg's Polyhedrons

As a bit of a thought experiment this evening, I put together an unwrap of one (the unwrap isn't perfect, as the goldberg polyhedrons have hexagons which aren't regular, and in my unwrap, they are (which makes my result less round). However, the basic idea is illustrate here. All the edges connect to give you a somewhat spherical polyhedron. My Unwrap is here.

Does anyone have a better unwrap (perhaps one that is more accurate than mine, as I apparently screwed up in thinking the Goldberg Polyhedron hexagons were all the same size) layout to put a world map on? (I'm looking for something with like, 100+ faces). Ideally most or all of the faces would be the same shape & size, but the most important part be that distance is on the same scale for all of the unwrapped faces, such that it can be assembled into a sort of pseudo-globe.

Perhaps an approach for doing this with 3d software?

Bonus points if I can use a tool to convert a traditional (warped) projection into one designed to drastically reduce said warping.

[johnvanvliet]

there is NO "perfect" way to map a sphere to a flat plane
that said
there are a few that are not bad and take into account the poles
like this GREAT!!!! map of the planet Mars
( not mine )
Martian Cartography - Unmanned Spaceflight.com
106 meg PDF
https://docs.google.com/uc?export=do...GNLRDRNSWI2QzQ

Mercator_projection
but Polarstereographic for near the two poles