Dear Colleagues,
We know that the soccer ball has 60 rotational symmetries and this is computed by the relation |G|=|orb_{G}(i)||stab_{G}(i)|where orb_{G}(i) denotes of the orbit of i in G and stab_{G}(i) is the stabilizer of i in G. In this case we consider the set of pentagons and we find that |orb_{G}(i)|=12 and |stab_{G}(i)|=5 so |G|=60.
The question is what if we consider hexagons instead of pentagons and calculate the order of our group then we find that |G|=120!!!!
The problem is in the 60 degree rotational symmetry about the line through the centers of two opposite hexagonal faces but how?!! In fact I do not know and I need your help.
Regards;
Raed.