Dendritic structures

Mr. Horibe made a number of dendritic fullerenes which are similar to the Kepler's stellated polyhedrons. By using the Euler theorem, it is quite straightforward to show that, in a cage-like fullerene without hole, N₅-N₇=12, where N₅ and N₇ are the number of pentagons and heptagons, respectively. In the typical fullerenes where the local Gaussian curvature is positive everywhere, we should have N₇=0. The total number of pentagons is 12. Another way to put it is to introduce the so-called topological charge, 1 for a pentagon and -1 for a heptagon. So the topological charge of a cage-like fullerene is 12.

Heptagons always generate a negative Gaussian curvature. For a cage-like fullerene, whenever we introduce an extra heptagon, we have to include a pentagon in order to satisfy the identity N₅-N₇=12.

One can replace a pentagon in a Goldberg icosahedron (icosahedral fullerene) by 6 pentagons (a hemisphere of C20) and 5 heptagons to get a spike. So the topological for this area is still 1 after replacement. Similarly, we can do the same replacement for all other 11 pentagons to get a dendritic structure.



Of course, we can use the same kind of trick to "grow" a spike (a carbon nanotube endcapped with the hemisphere of a C₂₀) along the normal direction of any pentagon on any kind of graphitic structure.

Incidentally, this structure without the C20 caps is just the inner part of the high-genus fullerenes we have done before.