Different preoccupation, keeping me awake:
A simplified world map, where the principal locations are the vertices of a Platonic solid.
If, in a conquest/influence game, maybe like Civ, your influence or travel times are based on how many edges you must cross to reach the target vertex...
This could be expanded to a bracket-style system, where once you've conquered your d6 (example), you move to a larger world (d6=8 or more vertices) seeded with opponents who come from d6 or larger base worlds, with each person's vertex/city built as a conglomerate of their previous world conquered...
I'm just trying to remember if d20's or d12's would have the most vertices, but I think it's the d12's... Cause you can embed a d12's vertices in the faces of a d20 and vice versa..? That's certainly true with a d6 and d8, which have 8 and 6 vertices respectively...
Now, I think I'm just rambling - hoping this writing out the thought will clear the way for sleep... 
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