The Truncated Icosahedron Star (TI Star): a doubling cycle and a golden cycle grown from the truncated icosahedron's own pyramids

The truncated icosahedron (TI) is the midpoint cell of DICTO's path between the White and the Red Queen (#76) and a station of his golden cycles. DICTO proposed filling its core with pentagonal and hexagonal pyramids, and then starting a new golden cycle from TI with those pyramids turned outwards, a spiked core. Cut from its centre, TI is exactly 12 pentagonal and 20 hexagonal pyramids; the hexagonal one carries on every base edge half the icosahedron's dihedral angle, the angle of the Cleo-Icos, because the hexagons of TI lie in the face planes of an icosahedron three times its edge. Turned outwards, the pyramids give the TI Star. Main theorem. Put the tip of each spike at s times the centre of its face. For s = 2 (each spike the mirror image of the pyramid inside) and for s = φ (each spike 1/φ of its height), TI, the TI Star, the hull of the 32 tips and TI s times larger form one turn of an endless cycle, each layer filled exactly by tetrahedra of three kinds over the corners and the two kinds of edge: for s = 2 a doubling cycle, the TI Star twice the volume of TI as DICTO's Cleo star is twice the icosahedron's; for s = φ a golden cycle, the TI Star φ times TI and one turn φ³ in volume. The hull of the tips is a pentakis dodecahedron, not the Catalan one; inside it each turn passes through two pairs of Platonic shells, the dodecahedron and icosahedron of TI's own face planes and those of the spike tips, each pierced by spikes, with a new core between them, the icosahedron truncated at 1/φ³ of each edge (eleven solids in all, complete in a stated sense; DICTO's growth rule carries each outer pair to the next inner pair). About one sphere the middle of every turn is its own polar, so the polar cycle shares four solids with the cycle and adds three, fourteen per turn. The hull of the tips, which DICTO named the Star Halo, starts a cycle of its own, and spiking each hull in turn gives a star chain that returns to the shape of the truncated icosahedron after four steps: read the other way it is the orbit of TI under DICTO's growth rule of Universal PiTet (#38), a PiTet cycle of its own. The spike tips touch none of the twelve stations of DICTO's golden cycle (#54) at any golden scale, so the two cycles are new and separate. All volumes are exact; 333 pieces per turn are free of overlap. DICTO's names: TI Star (Double and Golden), Star Halo. The star chain is the orbit of TI under DICTO's growth rule of Universal PiTet (#38). Computation and proof with Claude (Anthropic). More at dictospheres.com.