The Hexagon Section: The Five 24-Cells as a 5×5 Matrix, the Observer Inside the Golden World, the Inclination of the Golden Decagon to the Rational Hexagon (cos α* = ρ), and the Twenty-Four Nodes on the Cut

Abstract

Using the static Hessian H of the 600-cell flag double (2640 nodes) at its home, we compute what the fivefold stage of Note 46 looks like in numbers. (i) Reduced to the 120 vertices and decomposed into the five 24-cells, the breathing stiffness is a 5×5 matrix with a single diagonal value a = −1.6590895 and a single off-diagonal value b = +0.7429724: one positive eigenvalue a + 4b = +1.3128 (all five in phase, the standing wave) and four degenerate negative ones a − b = −2.4021 (the counter-poles). (ii) Eliminating the golden nodes, the observer’s effective breathing stiffness is −5.534, identical for the rational and for every golden 24-cell (the symmetry prediction of Note 46), and negative, whereas each body alone has exactly one positive direction, its breathing: embedded among its four counter-poles the observer’s breathing changes sign. Locked in, the observer’s vertex radius is 0.8% (24-cell) or 2.0% (tesseract) smaller than alone. (iii) Through every vertex of the 600-cell pass four rational hexagons (the Hurwitz cycles, planes spanned by the real axis and the four axes of the stella octangula), six golden hexagons, six decagons and fifteen squares. The decagons through a vertex meet its rational hexagon only in the vertex and its antipode — the axis of inversion — and three of them are inclined to the hexagon by exactly α* = arctan(2/φ²) = 37.3774°, with cos α* = ρ = √((5 + 2√5)/15), the scale ratio of the icosahedron; the other three by arccos(1/√(9 + 12φ)) = 79.1877°. (iv) The hexagon plane contains exactly 24 of the 2640 nodes: six vertices at 60°k, six face centres at 30° + 60°k, and twelve cell centres at 60°k ± t*, t* = 22.2388° — every vertex sees its cell centres under the angle t*, and a* = 30° − t* is the arc between a face centre and its neighbouring cell centre. On this cut the softest static motion is the displacement along the ring (dipole), followed by the alternating-vertex pattern (the flip). No small number arises statically; all ratios are of order one.