Globe

[SSBLoveU]

The engineers loved the cube and the Great Orthogonality Theorem. The irreducible representation of a cube is just like a STAR TREK episode when they ponder what is going on with the rotating cube in space and radiation. The geodesic spheres just add another bit of thrills. Things like write the formula to show only sixty carbons are needed to make the 32 faces, and what point group does a buckyball belong too are great.

The molecule C60 was named after Buckminster Fuller, who was an architect and designed buildings around geodesic spheres, hence Buckminsterfullerene.

The answer is I = icosahedral = 6 five-fold axes => I(h)

The buckyball is the only molecule of a single atom to form a hollow spheroid, and it spins at over one hundred million times per second. According to John R.D. Copley, physicist at the National Institute of Standards and Technology, "there are 174 ways that [the buckyball] can vibrate." http://hubpages.com/hub/The-Buckyball

[wodan11]

Star Trek

STAR TREK is the correct use of the trademark.

sorry

[Hauldren Collider]

While my question was more like a joke, why not? I do not see too much difference between hexagon and triangle. Plus, one can create a globe out of triangles.

Adjacency of 3 means that cities would need to cover more than one tile, and in order to make it a regular shape while retaining a planar graph you would end up with....hexagons.

Also this:

You can't however create a globe out of uniformly tiled triangles (there aren't too many differences between triangles and hexagons, since a hexagon can be regarded as 6 triangles together). To map triangles into a globe, you need to have some 'missing'. Normally at the vertex of each triangle you have 6 triangles meeting (and those 6 form a nice hexagon centered on that vertex). You'll find that to have a globe, you need (for example) 12 places where there is a vertex that only has 5 triangles meeting. Which just happens to form a nice pentagon centered on that vertex.

So if you work with triangles to form a globe, you can then group them together into hexagons, and lo and behold, you find you have 12 pentagons lurking amongst them.

[MxM]

You can't however create a globe out of uniformly tiled triangles (there aren't too many differences between triangles and hexagons, since a hexagon can be regarded as 6 triangles together). To map triangles into a globe, you need to have some 'missing'. Normally at the vertex of each triangle you have 6 triangles meeting (and those 6 form a nice hexagon centered on that vertex). You'll find that to have a globe, you need (for example) 12 places where there is a vertex that only has 5 triangles meeting. Which just happens to form a nice pentagon centered on that vertex.

So if you work with triangles to form a globe, you can then group them together into hexagons, and lo and behold, you find you have 12 pentagons lurking amongst them.

Sure, but it is all triangles, identical ones. The problem with hexagon is that you simply can not do a sphere with them, you have to have pentagons, but with triangles you do!

And it is not "missing" triangles, it is just right amount to cover a sphere.

[MxM]

Adjacency of 3 means that cities would need to cover more than one tile, and in order to make it a regular shape while retaining a planar graph you would end up with....hexagons.

Not necessarily. You can have more shapes with triangles, including hexagons. Why is it worse than just hexagons?

[vulture]

Sure, but it is all triangles, identical ones. The problem with hexagon is that you simply can not do a sphere with them, you have to have pentagons, but with triangles you do!

And it is not "missing" triangles, it is just right amount to cover a sphere.

The triangles may themselves all be the same sizes and shapes, but they aren't identical in game terms. They have different numbers of neighbours, different numbers of triangles with 'n' moves, and so on. (not to mention the ugly issue of corner movement - 3 side moves and 9 corner moves giving 12 total possible movement directions with 3 different distances involved).

Since you can also tile a sphere with squares quite happily with the same issue of not all having the same number of neighbours (some squares are surrounded by 7 tiles instead of 8), they'd seem like a better choice than triangles if you wanted to go down that route since the corner issues are less severe.

I still prefer hexes for the lack of corner movements at all. In functional terms, a square with 7 neighbours rather than 8 and a pentagon with 5 neighbours rather than a hex with 6 are interchangeable.

[MxM]

Or you can have triangles and no corner movement, just 3 directions. Thus, they will be identical.