Yes, but as has been mentioned by others, that is impossible. So either one can use pentagons and hexagons (as I inadvertently did above ;-) or just pentagons.

My goal was trying to get a hex grid, with roughly "ignorable" distortion. I think the most straight forward way would be to use four "pseudo" (irregular) hexagons, and wrap them into a cube (ie: four "hexagons" made from a square and two triangles with 45 and 90 degree angels). You would then have a "round" map that could be subdivided. It would be far from perfect, but very straightforward -- and if mapped onto a planet "north-south" (aka pointy-side up) -- I think it would yield a similar distortion to what we are used to from various map projections:

        /\   /\   /\   /\  <- distorted sides
       /__\ /__\ /__\ /__\        ________       _______
      |  A |  B |  C |  D |      /       /\     |\  c  /|
      | __ | __ | __ | __ |     /   N   /bb\    |d \ /  |
       \  / \  / \  / \  /     /_______/bbbb\   |  /'\b |
        \/   \/   \/   \/      \aaaaaaa\bbbb/   |/__a__\|
                                \aaaaaaa\b /    top down view
                                 \_______\/    ("North pole")

When wrapped up into a cube, the south and north poles would lie at the intersection of the four hexagons. Unless I'm tricking myself, this has the interesting property that the distance from A to B and D is 1, while from A to C is 2 -- whichever way you "circumnavigate".

Being hexagons, they could be further subdivided into hexagons (as for most purposes this grid would be too coarse).

I'm not convinced it would be very useful as a "proper" map projection -- but might work for a game.