The conformation of nine-membered rings

Each of the 16 symmetrical conformations of puck- ered nine-membered rings is characterized by three amplitude-phase pairs which can be mapped onto the surface of a helical tube that closes into a torus. A projection along the tubular axis, assuming con- stant radii yields a composite map on which all symmetrical forms can be located unequivocally. The symmetrical forms and representative examples from the literature are analyzed quantitatively in terms of the six primitive forms defined by the three symmetry classes of displacement from a regular planar nonagon. Introduction The conformation of an N-membered cyclic com- pound can be described quantitatively as a linear combination of N- 3 primitive forms (Boeyens & Evans, 1989). For small ring systems these are typi- cally the same as the well-known low-energy sym- metrical forms, but for large N some of them could present chemically unlikely arrangements (Evans & Boeyens, 1989). As N increases the large number of contributing forms obscures the interpretation of the coefficients in the linear combinations, and for very large rings it is therefore preferable to resort to