[chad-autry]

Notice how the lines are stretched, Not every hexagon in the tiling represents the same area if mapped back to a sphere. The non-polar edges of the tiles pictures are larger. If you actually printed it out, and tried to wrap it around a globe, the middles would have to expand or the edges would have to shrink for the tiles to join.

Edit: Basically he hasn't actually tiled a sphere with regular hexagons (which is what the proof said was impossible. He has titled a flat projection of a sphere with regular hexagons, which would have to be morphed to irregular hexagons if tiling a sphere when the projection was reversed.

Not taking away from the tiling, which is quite interesting in itself.

[jacobolus]

It’s possible to make a map projection from the sphere to a flat surface either preserve areas, or preserve angles (and local shapes), but not both. This projection preserves angles, but it could be modified to preserve area instead, or some compromise between the two.