The Gains of DICTO's Growth Rule: the exact growth of every closed star chain

DICTO's growth rule T of Universal PiTet (#38) puts a face through every corner of a solid, square to the line from the centre: T(S) = {x : v·x ≤ v·v for every corner v}. Run again and again, it carries some solids back to their own shape, larger; the factor is the chain's gain. In The Truncated Icosahedron Star (#78) DICTO noticed that the four-step factors of his star chains, 4.944 and 11.542, come close to 5 and 12, and proposed that such factors might form a new series of ratios. We find every gain exactly. T is polar reciprocation of the inverted corners, and on a solid with all its corners on one sphere it is plain polar reciprocation in that sphere. The main theorem follows: when a solid and its image both have their corners on spheres, of radii R and R′, the chain closes in two steps with gain (R′/R)²; for the Platonic solids this is (R/r)², circumradius over inradius squared: 9, 3 and 15 − 6√5. The star chain of #78 closes with factor sp/g, where g is the gain of T and p the period: φ⁴/g = 4.9443 and 16/g = 11.5418 for the truncated icosahedron, which explains DICTO's near-misses exactly. We give the exact gain of every closed chain among the Platonic, Archimedean and Catalan solids and of ten of DICTO's solids; all but the two omnitruncates and their duals close, in two or four steps. The gains form series by symmetry; the primes in their norms are the ones the number field allows, so a prime such as 89 or 109 in the gain of the truncated icosahedron is typical, not special. The chain of DICTO's Ogdoecontahedron passes through four of his named solids and then spreads without closing, and the cube and the dodecahedron with their edges cut back close at some depths and spread at others: depth, not the number of kinds of face, decides. 1. Introduction Draw a plane through every corner of a solid, square to the line from the centre, and keep what lies inside all the planes. A new solid appears, with a face for every corner of the old one. Do it again, and again: some solids come back to their own shape, only larger. DICTO's idea: the gains of the closed chains as a new series of ratios. Computation and proof with Claude (Anthropic). More at dictospheres.com.