3 ms·
For anyone wondering why you cannot tile a sphere with hexes: this is a matter of Euler characteristic. Suppose that there is a tiling of sphere with N hexes.
by xyzzyz 13y ago
For anyone wondering why you cannot tile a sphere with hexes: this is a matter of Euler characteristic.
Suppose that there is a tiling of sphere with N hexes. Each hex has 6 edges, but every edge is shared by two hexes, so there are 6N/2 = 3N edges. Similarly, there are 6N/3 = 2N vertices.Thus, the Euler characteristic would be 2N - 3N + N = 0. Now, the sphere has Euler characteristic 2, so we get a contradiction.
Incidentally, a torus has Euler characteristic 0, and indeed it turns out that you can tile torus with hexes.
- marshray 13y agoCarbon nanotubes are mostly hexagons. They certainly could make toruses, if they wanted to.
- omaranto 13y agoThere's no need to assume exactly 3 hexagons meet at each vertex: if you only assume at least 3 hexagons meet at each vertex, then 6N >=3V, and we get that the Euler characteristic is <=0, which is still a contradiction. Of course if you allow vertices with only two hexagons meeting at them you can make a sphere: for example, take two hexagons and glue them around the edges, then stuff the middle to get a hexagonal cushion.