the core, its Bloch vector, and the twisting ribbon
the sapphire core is the spinor. the orange arrow is its Bloch vector (the observable spin direction) — it precesses and returns after 360°. the purple ribbon ties the core to the fixed outer ring (the world): as you turn the core it twists. at 360° one full twist — stuck. at 720° two twists — which slide off (the belt trick). the ribbon shows the hidden −1. drag to rotate the view.
the spinor sign — the hidden −1
top: the spinor phase clock — R(θ) turns by the HALF-angle θ/2, so at θ=360° it points the opposite way (−1) and only at 720° is it back. middle: the sign of the state (cos(θ/2)): +1 → 0 → −1 (at 360°) → 0 → +1 (at 720°). bottom: the two returns — Bloch vector home every 360°, spinor home only every 720°.
toddler corner
ELI5: hold a dinner plate flat on your palm and keep it flat while you spin it around. After one full turn (360°) your arm is all twisted up — you can\u2019t undo the twist without letting go. But keep going the SAME way for one more full turn (720° total), sweeping your hand up over your shoulder, and — magic — your arm comes untwisted, no cheating. Your arm just proved something deep: in the world of turning, ONE full spin is not "back to start," but TWO full spins is. Tiny particles like electrons live by exactly this rule — spin one is called a "spinor." Our sapphire is one. Turn it once around and it secretly flips to its own negative (like a photo turned into its shadow); turn it twice and it\u2019s truly itself again. We tied a ribbon from the gem to the wall so you can SEE the twist that a plain number would hide: one twist is stuck, two twists slip free. That hidden flip — the minus sign after one turn — is the whole secret of what a spinor is, now wrapped around the core.