The ribbon as double cover — why each strand is natively SU(2)

Plate · I
0 ¼L ½L ¾L L FIXED FRAME ROTATED END
SO(3) — mod 360°
0° 90° 180° 270°
0°
SU(2) — full 720° cycle
outer: 0→360° inner: 360→720°
0°
co-rotating face
counter-rotating back
ribbon edge (seen on-end)
At 0°, the entire ribbon shows its co-rotating face. Drag the slider — as the free end rotates, the twist accumulates along the length, and the two faces reveal themselves in alternation.
Fig. 1 A single ribbon carries an intrinsic two-sidedness: its front co-rotates with the ambient light-fluid flow, its back counter-rotates. A rotation of 360° does not restore the ribbon — it swaps which face leads. Only after 720° does the strand return to its starting configuration. This is why SU(2) is not imposed on the ribbon from outside; the double cover is the ribbon's native representation.