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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.

Grimshaw
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    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 02:44
  • what if its not a sphere, but something similar? – Grimshaw Dec 28 '13 at 02:46
  • and, is there any other base shape to partition the sphere? I need a even grid of these shapes , which I then wrap around the sphere-like shape. – Grimshaw Dec 28 '13 at 02:47
  • http://s3.goodfon.com/wallpaper/previews-middle/578145.jpg This image seems to show exactly what I was picturing for my shape, a perfect tiling of hexagons. However I don't know if the back of the sphere in the image is correct or has imperfections..This is what I am trying to find out basically. – Grimshaw Dec 28 '13 at 02:55
  • I think you can partition a torus into hexagons. For a sphere, only triangles, squares, and pentagons. – Gerry Myerson Dec 28 '13 at 02:56
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    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 02:58
  • I believe you, I am just looking for geometrical possibilities for what I need. This is meant for a computer game, I figured how to implement my game, but I would like the pieces on the surface of my sphere to move through meaningful tiles. I guess I will have to fake the effect of hexagon tiles, thanks! – Grimshaw Dec 28 '13 at 03:21
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    You can do it with hexagons and pentagons like a soccer ball. – Dylan Yott Dec 28 '13 at 03:29
  • Following on @DylanYott's point, you can do it with many hexagons but will have to have exactly $12$ pentagons. The image you link to has lots of hexagons and the pentagons get lost. The first comment explains why-note that $v=3f$ and $e=3f-n/2$ where $n$ is the number of pentagons. – Ross Millikan Dec 28 '13 at 04:08
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    The article on spherical tilings (ways to cover the sphere with different shapes) on wikipedia might be useful to you. http://en.wikipedia.org/wiki/Spherical_polyhedron – Sak Dec 28 '13 at 04:09
  • For game development purposes, there is a known approach to start from a certain hexagon/pentagon partition then partition it further by subdividing these tiles into smaller hexagons/pentagons recursively. – mbaitoff Dec 28 '13 at 05:14
  • Also look at geodesic spheres, e.g. in this question. – MvG Dec 28 '13 at 16:32
  • This question concentrates on the dual, with mostly hexagons instead of triangles. – MvG Dec 28 '13 at 16:39

2 Answers2

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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

Promenycles
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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.

Luís Gonçalves
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