All Five Golden Pairs Are Slices of Four-Dimensional Solids: Golden Cuts beyond Plato

Slice a four-dimensional cube, the tesseract, straight across its longest diagonal and every slice is a tetrahedron with its corners cut off. At two golden heights the cuts are exactly the golden cuts of #88, one either side of the classical truncated tetrahedron at half height. The golden dodecahedra appear the same way in the 120-cell, so four of the five golden pairs of the regular solids are slices of regular four-dimensional solids. The cubes are the exception, and we prove it: no slice of a regular four-dimensional solid is a cube with its corners cut. They live one step away, in the rectified tesseract, the four-dimensional cube cut back to the middles of its edges, whose slices are cubes cut at every depth, both golden ones included. Beyond the regular solids golden cuts are rare: a solid with two kinds of face never has one; two solids come close, with every face golden or regular; two rhombic solids have their own golden pair; and the four star polyhedra of Kepler and Poinsot have two each. In #88 we showed that each Platonic solid has exactly two golden cuts, the Greater and Lesser Golden Cut solids, and that the icosahedral and octahedral pairs are sections of the 120-cell and the 24-cell. DICTO proposed carrying the golden cuts beyond the Platonic solids. Our main theorem is four-dimensional: the sections of the tesseract perpendicular to a long diagonal, between its two middle layers of corners, are exactly the tetrahedra cut at x = (1 − h)/(2 − h), so that the Greater Golden Cut Tetrahedron, the Archimedean truncated tetrahedron and the Lesser Golden Cut Tetrahedron are the sections at heights 1/φ², 1/2 and 1/φ; the golden tetrahedra are also sections of the 120-cell, and the Greater and Lesser Golden Cut Dodecahedra are its cell-first sections at 1/φ² and 1/φ of the way between two layers of corners. Hence the golden pairs of the dodecahedron, icosahedron, tetrahedron and octahedron are all sections of regular four-dimensional solids. The golden cubes are not: no hyperplane section of any of the six regular four-dimensional solids is a cube cut at its corners, at any depth (Theorem A). They are one step away: the sections x1 = h of the rectified tesseract with corners (±1, ±1, ±1, 0) are exactly the cubes cut at x = h/2, so the Greater and Lesser Golden Cut Cubes are its sections at h = 2x_G and 2x_L, as the golden octahedra are sections of the rectified 16-cell, the 24-cell (Theorem B); nine uniform polytopes of the tesseract and 24-cell families have golden cube sections. DICTO proposed that cut cubes and small octahedra fill space; they do at every depth, and this filling is a section of the rectified tesseractic honeycomb. Beyond Plato: the twelve golden cut depths of the regular triangle, square, pentagon, hexagon, octagon and decagon are all different, so no Archimedean solid has a cut making every face golden; only the icosidodecahedron and the snub dodecahedron have cuts making every face golden or regular, two each, at 1/(φ√5) and 1/3, a new family, the DICTO Golden-Regular Cuts (for the icosidodecahedron, Zefiro's 2011 regular-decagon and regular-hexagon stages, here read as golden), and DICTO added the two lesser ones, whose truncated triangles have ratio exactly φ, to the DICTO-GTT-Solids; among the Catalan solids only the rhombic dodecahedron and the rhombic triacontahedron have golden cuts, exactly two each, in closed form, which join the DICTO-Kepler golden truncations; and each Kepler–Poinsot star polyhedron has two, none of which is a section of the ten regular star polytopes in the icosahedral direction (floating-point search). All results are exact in the number fields of the solids, except where stated. The idea of carrying the golden cuts of #88 beyond the Platonic solids, the choice of this research, the golden cuts themselves, the names of the new solids and of the families, the two new members of the DICTO-GTT-Solids, the proposition that a cut cube should have a place where it takes the octahedron's place, the filling by cut cubes and octahedra, and the star-polytope search are DICTO's. Computation, proofs, exact checks and drafting with Claude (Anthropic).