your trit on the complex plane — the flat rail + the lift + the turn i makes
the horizontal gold axis is real: −1 and +1, the two ends of the flat rail. the vertical purple axis is imaginary: i (and −i below). multiplying by i is a 90° turn — the cyan arm sweeps e^{iθ}, and the corner-hops trace 1 → i → −1 → −i → 1. {1, i} together reach every point on the plane. i isn\u2019t a pole on the rail; it\u2019s the direction perpendicular to it — the lift that makes turning possible.
flat & two-sided — the watch battery (±i are the faces)
the plane is FLAT (zero curvature) and TWO-SIDED — a front and a back, exactly like a coin cell\u2019s + and − faces. NOT one-sided; no Möbius twist. the two faces are +i and −i: the two square roots of −1, the two directions rotation can turn (CCW / CW). flipping the battery = complex conjugation (i→−i) = reversing the spin. cap the plane with ∞ and it rounds up into the Bloch sphere — curved, still two-sided.
the canonical 3-phase trit — cube roots (back to the start)
the natural three-on-a-circle: the cube roots of unity 1, ω, ω² = e^{2πik/3}, three points 120° apart, summing to zero. this is the session\u2019s 3-phase — Axiom One\u2019s trigonal 120°, the rotating field, the three sapphire legs. your {−1, i, +1} slices real-vs-imaginary; the canonical trit slices 3-fold. same instinct, two honest geometries.
toddler corner
ELI5: line up three special numbers: −1, then i, then +1. The −1 and +1 are the two ends of a flat see-saw — the ordinary number-line you know, left and right. But i doesn\u2019t live on that see-saw at all; it stands straight UP off the middle of it, pointing in a brand-new direction. And its superpower is turning: multiply anything by i and it spins a quarter-turn. Now your watch-battery question: is this thing flat, and does it have one side or two? It\u2019s FLAT — a perfectly flat sheet, no curve. And it has TWO sides, just like a coin battery has a plus-face and a minus-face — there\u2019s no sneaky one-sided trickery. The cool bit: the two faces are +i and −i, which are just "spin this way" and "spin the other way." Turn the battery over and you\u2019ve flipped clockwise into counter-clockwise. And if you glue the far-away edge of the flat sheet all into a single point, it puffs up into a ball — the same globe from last time. Flat coin with a + and − face → blow it up → round world. The i is the little stand-up arrow that makes all the spinning, and there are exactly two ways it can spin: the two faces of your battery.