The INDYNA register derives the golden ratio twice from lack of space: in three dimensions from the counter-rotation of the two tetrahedra of the stella octangula, which halts at the icosahedron at the angle t* = arccos(φ²/√8) = 22.2388° (register L279), and in four dimensions from the tilt of the 96 face centres of the 24-cell, which produces the 600-cell at a* = arccos(φ√(3/8)) = 7.7612° (L283). We prove that these two angles are one: a* = 30° − t* exactly. Two further identities follow. The cell centres of the 600-cell on the unit 3-sphere lie at radius cos t*, so that the circumradius of every tetrahedral cell is sin t* = √(3/8)/φ = 0.378467, the lack-of-space radius of the four-dimensional model (L262). And when the dually inscribed icosahedron and dodecahedron invert about a three-fold axis, the two narrowest openings of the turn lie at ±60° and the side openings at exactly 60° ± t*; the narrowest opening has the closed form cos δ = ρ(6 − √5)/3 with ρ = r/R of the icosahedron. Three results of the register — the three-dimensional gold, the four-dimensional gold, the lack-of-space radius — and the geometry of the inversion are therefore governed by a single angle.
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Arber Gishto (2026) studied this question.