Ways to partition a sphere? first of all, sorry for the lack of terminology/ignorance on the subject, I just joined this website.

I need a sphere or sphere-like 3D shape, whose surface is partitioned into another geometric primitive, in some kind of grid. I would prefer these partitions to be hexagons, tiled into each other. The bigger the sphere surface is, the more hexagons are supposed to be present.

After some research, I found the truncated icosahedron, which looks quite similar to what I want, except it has some pentagons in there, which kills the premise I need to satisfy:

If i have an object in any given partition, I need to be able to travel that object all around the sphere, always passing through the center of the next partition, and it has to arrive the initial location in a straight line. The traveling direction is arbitrary but always is the middle of one of the edges of the geometric shape of the partition.

I need to be able, in a visualization sense, to have a whole line of the elementary geometric shapes be moved at once, as if it was a huge circular rubik puzzle.

EDIT:
http://en.wikipedia.org/wiki/Truncated_order-7_triangular_tiling
This might be what I am looking for to some extend.. ?

I know I am probably not explaining myself perfectly, but if anyone could help out it would be appreciated.
There is also the option to use other shapapes like
triangles with 72° angles, five meeting at a vertex
triangles with 90° angles, four meeting at a vertex
triangles with 120° angles, three meeting at a vertex
quadrilaterals with 120° angles, three meeting at a vertex
pentagons with 120° angles, three meeting at a vertex
It is impossible to partite a sphere only with hexagons but, for the purpose of a game, you may partite only its visible part. This kind of partition is done here. And pay attention: this reference is a joke.