Golden Cuts: the DICTO-Kepler Golden Truncations and the DICTO-GTT-Solids

Ten Greater and Lesser Golden Cut solids; at the octahedron, Icosaxes and Unborn Axes, which fill space and are sections of the 24-cell. A truncated triangle is a hexagon with all six angles 120° and sides alternating between two lengths; the regular hexagon is the case of equal sides. DICTO observed that this face runs through his recent solids. By exact computation over Q(√5) we show that among the 69 solids of the DICTO Solids dataset every hexagonal face with 3-fold symmetry is a truncated triangle: regular in nine solids, and non-regular, with its two sides in a ratio from Q(√5), in eight single solids and one assembly; the three single solids whose ratio is exactly φ, Icosaxes, Unborn Axes and White Knight, are the DICTO-GTT-Solids. We then prove that cutting every corner of a Platonic solid equally so that each original face becomes a truncated polygon with sides in the golden ratio φ gives exactly two solids for each Platonic solid, ten in all, the DICTO-Kepler golden truncations, which we name the Greater and Lesser Golden Cut solids. At the octahedron they are the two semi-uniform truncated octahedra of Koca et al., DICTO's Icosaxes and Unborn Axes, the pair that #48 found filling space. Both golden pairs are sections of regular four-dimensional solids at the same fractions of the way from the centre to a cell: in the 24-cell (octahedral pair) and in the 120-cell (icosahedral pair), at 1/φ² and 1/φ, either side of the Archimedean section at 1/2. The observation, the solids and the family names are DICTO's. Computation, proofs, exact checks and drafting with Claude (Anthropic); an independent review by Fable (Anthropic) checked every theorem and number.