Figure 3.

Random growth is unlikely to produce a closed fullerene cage. (A) The order for spiral addition of faces for cage 36-15 is 55656665556555566565, where “5” specifies a pentagon and “6” specifies a hexagon. This list, which comes from a “recipe” for making 36-15 (Fowler and Manolopoulos (24)) that consists of the positions of the 12 pentagons (1,2,3,4,8,9,10,12,13,14,15,18,20). Lists for 36-15 could start on any of the 20 faces and go backward as well as forward, in this case without any replicates, a total of 40 lists. The 36-15 cage shown by this Schlegel diagram is the same as the one shown by the Schlegel diagram in Fig. 2 A but centered on a different face. (B) An example of a random order for spiral addition of faces with 12 pentagons and 8 hexagons is 66555656566655555556. After addition of the 20th face, the two circled vertices are not three-connected, and no further addition of faces could produce a closed fullerene.