# 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.

• You can't partition a sphere into hexagons. Euler's theorem (the one about $v-e+f$) forbids it. – Gerry Myerson Dec 28 '13 at 2:44
• what if its not a sphere, but something similar? – Grimshaw Dec 28 '13 at 2:46
• I guarantee you, you can't tile a sphere with hexagons. It's not open for debate: it's a theorem, and I've told you how to start looking for a proof. – Gerry Myerson Dec 28 '13 at 2:58
• You can do it with hexagons and pentagons like a soccer ball. – Dylan Yott Dec 28 '13 at 3:29
• The article on spherical tilings (ways to cover the sphere with different shapes) on wikipedia might be useful to you. en.wikipedia.org/wiki/Spherical_polyhedron – Sak Dec 28 '13 at 4:09