Summary: We take a fresh look at voting theory, in particular at the notion of manipulation, by employing the geometry of the Saari triangle. This yields a geometric proof of the Gibbard/Satterthwaite theorem, and new insight into what it means to manipulate the vote. Next, we propose two possible strengthenings of the notion of manipulability (or weakenings of the notion of non-manipulability), and analyze how these affect the impossibility proof for non-manipulable voting rules.