# Dirac belt (belt trick)

A cube tied to six fixed anchor pads by elastic ribbons (two in the first half), posed at any moment of a full
720 degree untangle. Turn the cube once (360 degrees) and the ribbons end up twisted in a way that no moving
of the ribbons can undo while the cube and pads stay put. Turn it twice (720 degrees) and they can be untangled
completely. This is the topology behind spin-1/2 particles: the rotation group SO(3) is double-covered by the unit
quaternions SU(2), and only a 720 degree turn is a loop that shrinks to nothing.

## How the shape is made

Every point p is turned rigidly about the centre by the unit quaternion Q(h(|p|), t):

- `h(r)` is 1 inside the ball of radius R0 = 1.76 a around the cube (a = half the cube size; the cube's corner is
  at 1.732 a), 0 outside R1 = anchor distance - 0.3 a, and smooth (C3) and monotone in between. Its slope is
  constant in log r over the middle 70 %, which keeps the ribbons' stretch as low as this construction allows.
- `Q(s, t)` follows the timeline below. The two untangling loops are Andrew Hanson's 360 and 720 degree quaternion
  paths (A. J. Hanson, *Visualizing Quaternions*, Morgan Kaufmann 2006).

Because each sphere |p| = r turns as one rigid piece, the map is smooth, one-to-one and volume-preserving at every
moment. Ribbons that are apart at rest therefore never touch each other, themselves, the cube or the pads, and
each ribbon keeps exactly its rest volume. The cube (inside R0) turns rigidly with Q(1, t), and the pads (outside
R1) never move.

Each ribbon is a closed solid: its top and bottom surfaces are B-splines fitted (to 0.001 mm at the default size)
through the exact mapped points, 241 along the length by 13 across, and its sides and ends are ruled between them.

## Timeline (`time`, seconds; the same as the video, frame = time x 60)

| Time | What happens |
|---|---|
| 0-3 | Rest |
| 3-9 | The cube turns 360 degrees about z; the ribbons twist |
| 9-15 | Hanson's 360 loop: the twist moves around but cannot be removed |
| 15-21 | The cube turns a second 360 degrees (720 in total) |
| 21-28 | Hanson's 720 untangle: back to rest, with the cube never turning back |
| 28-29.5 | Four more ribbons and pads grow out (nothing turns) |
| 29.5-37.5 | Six-ribbon finale: with all six ribbons attached the cube turns another 720 degrees, and the ribbons come back untangled |

Every phase starts and ends at rest or at a whole-turn pose, so the last moment matches the first exactly.

## Parameters

| Parameter | Meaning |
|---|---|
| `time` | Moment in the motion, 0 to 37.5 s |
| `ribbons` | `as in the video` (2, then 6 from 28 s), or a fixed `1`, `2` or `6`. The map moves all of space, so any count is valid at any moment |
| `cube_size` | Edge of the cube, mm |
| `ribbon_width` | mm, at most 0.5 x cube size |
| `ribbon_thickness` | mm, at most 0.02 x cube size (as in the video) and a quarter of the width |
| `ribbon_length` | From cube face to anchor pad, mm, at least 2.5 x cube size |

The defaults reproduce the video's proportions exactly (cube 40 mm, so one video unit is 80 mm).

## Layout and bodies

The cube is centred on the origin, ribbon `pz` points up (+z), and the cube's +x face carries a red inlay so its
turn is visible. Bodies: `cube`, `mark` (the inlay, flush with the face), `pad_<axis>` and `ribbon_<axis>` for each
ribbon present (`pz`, `nz`, `px`, `nx`, `py`, `ny`). Each ribbon end is glued to its cube face and to its pad face
(they touch; they don't overlap).

## Notes

- The ribbons stretch by up to about 9 times along their length in the 720 untangle, like elastic bands, which is
  why they are TPU 60A. Real belts bend rather than stretch; the topology is the same.
- Checked with CadQuery 2.8 / OCCT at the defaults every 0.5 s of the timeline (76 moments), plus six ribbons at
  8 moments through the turns and the untangle, plus the smallest allowed proportions (cube 20, width 10,
  thickness 0.4, length 50 mm) with six ribbons at 5 moments: every body a valid closed solid, no overlap between any
  two bodies (beyond the glued contacts), no ribbon self-intersection, ribbon volume equal to the rest volume within
  0.03 %, and fitted surfaces within 0.03 mm of the exact mapped points. The closest two ribbons come at the
  defaults is 4.2 mm (six ribbons, 22.5 s).
- STEVE's quick mass and volume figures for the ribbons can read about 3 % high: its default volume integration is
  coarse on these thin B-spline solids. The exact ribbon volume is width x thickness x length at every moment.