response 88 of 132:

Aug 30 06:15 UTC 1997

I don't think you can tile a sphere with any number of regular hexagons. 
You can tile a plane with them, though.

You need a few pentagons in order to get it to curl up into a sphere. 

If you look at a geodesic dome, a la Buckminster Fuller, you will see 
that it is made up of triangles, with 5 or 6 coming together at each 
point. The "dual" of this structure is made by using the center of each 
triangle as a node and connecting each new node to the adjacent ones. 
The dual of a geodesic dome is made up of hexagons and pentagons that 
tile a sphere. You can have many more hexagons if you like, but you need 
to have exactly 12 pentagons to complete the sphere. If you leave the 
hexagons out altogether, you have a dodecahedron.

The hexagons don't contribute any curling, which is why you can tile a 
plane with them. if they are regular, each one forms an inside angle of 
120 degrees at each vertex. Since there must be no more nor less than 3 
coming together at each point, they always sum to 360 degrees. Hence, no 
curling.

Curiously, although you can't tile a sphere with hexagons, you can tile 
a torus with them. To see this, section the torus and unwind it to form 
a cylinder, then cut the cylinder the long way and unwind it to form a 
rectangle. Tile the rectange as you would a plane, then wind it and 
stitch it back up the way you cut it.