To make it less distorted, you will have to make more cuts, seriously increasing the risk that the map does not accurately show the distance between points P and Q because the shortest path between them on the globe goes through a cut on your map.

In the limit, you could get something like http://en.wikipedia.org/wiki/Goode_homolosine_projection, but with infinitely many lobes. And that is a _could_: you can add branches like in Dymaxion wherever you feel like it.