I'm studying the shapes of voting systems

Writing this out loud makes it a little more real. Yes, this is what I'm doing. Methods of putting together larger numbers of ballots and coming out with a winner have a shape, and this, somehow, is useful. I'm working on finding shapes. Right now, one of them is looking somewhat hexagonal for some numbers, although it can compress into a line or, when we allow more deviant behavior, morph into a triangle. Then, it also can be unfolded into a solid.

When I put it like that, it sounds like I've gone off the deep end. What will I do next, find the scent of an ethical code? The sound of ideological distributions? Maybe I should turn my problem around - design shapes, and find voting systems that produce them. I wonder what degree of complexity I would need to make a voting d20.