The DICTO Kepler–Poinsot–UP Cycle

Universal PiTet grows from the Cleo star by the tightest solid round the last, and its shells come out thin. DICTO specified a taller version ("think stellation, think Gaudí"): keep every station a true star and fill only between stars. With the icosahedron I of edge 1, the great stellated dodecahedron (spikes to 3× the face distance) has a dodecahedral hull of edge φ, the small stellated dodecahedron on that hull (spires to √5× the face distance) has an icosahedral hull of edge φ³, and the cycle repeats for ever, ×φ³ in length and ×φ⁹ in volume each turn. The spires stand φ² and φ times their face radius and the hull steps ×φ then ×φ², balancing within each turn; the stars are exactly 3× and √5× their cores in volume. Every shell is filled exactly by the Cleo star's own pattern: the spikes, inward pyramids from the hull's faces and valley tetrahedra along the core's edges. In the first shell these are DICTO's CLEO, cap and AXE. The two shells meet in golden reciprocity, PiTet's own principle of opposites: the stars' points stand φ² and φ times their face radius, and the inward pyramids of the other shell are exactly their reciprocals, 1/φ² and 1/φ. All exact in ℚ(√5). Companion to Universal PiTet (UP), DICTO DOI 10.5281/zenodo.23463305.