Three tori, three X's, twelve gates — the lone rider of № 28, tripled and crossed into a closed core on the cube's own symmetry frame.
Take three rings and set them on three axes — vertical, horizontal, and one corner-to-corner diagonal. Each ring, seen down its own axis, crosses itself into an X: two strands, four ends. Stack the three and the count closes exactly.
This is not three arbitrary rings. The diagonal meets each face axis at arccos(1/√3) = 54.7356° — the magic angle (tetrahedral / NMR), the body diagonal of a cube. The vertical and horizontal are orthogonal (V·H = 0). So the three axes are the cube's symmetry frame, and the twelve gates fall exactly on the cube's twelve edges. Three mutually-set rings through one centre is the classic orthogonal link — the atom glyph, the gyroscope gimbal.
| gate | axis | strand · end | direction | cube edge group |
|---|---|---|---|---|
| G00–G01 | VERTICAL (0,0,1) | strand A/B · near ends | → forward | 4 edges ∥ z |
| G02–G03 | VERTICAL | strand A/B · far ends | ← back | 4 edges ∥ z |
| G04–G05 | HORIZONTAL (1,0,0) | strand A/B · near ends | → forward | 4 edges ∥ x |
| G06–G07 | HORIZONTAL | strand A/B · far ends | ← back | 4 edges ∥ x |
| G08–G09 | DIAGONAL (1,1,1)/√3 | strand A/B · near ends | → forward | 4 edges ∥ y |
| G10–G11 | DIAGONAL | strand A/B · far ends | ← back | 4 edges ∥ y |
REAL: the count closes, the magic angle is exact, and the twelve gates are the cube's twelve edges — the same twelve that give the cuboctahedron its vertices and the octave its tones. DOESN'T: three equal orthogonal circles through one centre generically intersect rather than link; true Borromean rings (cut one, the other two fall free) cannot be built from three flat round circles at all — it takes a gentle deformation (Freedman–Skora, 1987). The "X" is a projection: down-axis a ring crosses itself; side-on it is an ellipse — the gate is real, its X-shape is viewpoint. And № 28's catch rides along, now tripled: the core measures its own phase on three axes and still cannot inspect its own maker. Twelve gates — one hand on the lathe.