To see the half-twist in , begin with the point formula_55 in formula_39. This corresponds to a unique point of , namely formula_57. The path stops at formula_58, where it gives the point formula_59. Except for and , every point in the path lies on a different line through the origin. Therefore formula_60 travels once around the center circle of . However, while and lie in the same line of the ruling, they are on opposite sides of the origin. This change in sign is the algebraic manifestation of the half-twist.