Hexagon grid on sphere without pentagon

Question

I wanted to make a tile-based world on a sphere. I would like to use hexagon-shaped tiles, though as far as I can figure out you need to have pentagons in there too to make it fit.

So, my question is, is it possible to make a hexagon grid on a sphere, without using pentagons? The hexagons don't have to be regular hexagons, if they are all close to regular that's more than enough.

Any help here?

Thanks a lot!

Answer 1

Impossible. This is mathemetical theorem. Only imprecise ways exist I guess.

http://en.wikipedia.org/wiki/Platonic_solid

http://en.wikipedia.org/wiki/Semiregular_polyhedron

Covering Earth with Hexagonal Map Tiles

Answer 2

The author of this article was experiencing similar problems and wrote the aforementioned article on the subject, with some JavaScript source code.

Basically, the idea is, that since it is mathematically impossible to remove the Pentagons, the only valid solution is to modify and distort your hexagonal grid to make the occasional pentagon stand out less. To make odd tiles the normal occurrence. He did this by creating a voronoi diagram over the icosphere rather than a straight hexagonal grid, and to repeatedly run the Lloyd relaxation algorithm over the points until they were more or less in the Pentagon to Octagon range.