Ahah! RWG's second set of Steiner 5-ring constraints failed my numerical tests because of a wrong sign for k0 --- I assume curvatures have sign built-in, allowing for circles to evert under transformation. I didn't initially suspect this problem, since by Murphy's law the dodgy sign passed when tested on the canonical concentric ring. So the constraints for n = 5 should have read: 5 cyclic shifts (fixing k0,k6 ) of k1^2 + k3*k5 + k2*k4 + k2*k3 = k3^2 + k5*k2 + k4*k1 + k1*k2 ; and 5 shifts of 4*k5^2 - 6*k4*k1 + 2*k2*k3 - k0*k5 + k6*k5 = constant . [My updated versions --- I can't be bothered to correct all the signs in his original! Could the first set also be pruned to some equivalent LHS = const form perhaps?] Unfortunately, this makes no difference to the rank, which remains 5 rather than 4; I remain baffled by this discrepancy. Fred Lunnon