Two examples of hexagonal rings: (a) with 134 hexagonal sites (red), completed with 21 pentagonal sites (blue) on the inner border and with an outside border containing 34 heptagonal sites (green) and 21 hexagonal sites (red); (b) with 422 hexagonal sites (red), completed with 34 pentagonal sites (blue) on the inner border and with an outer border containing 55 heptagonal sites (green) and 34 hexagonal sites (red). The inner border has blue points inside, with zigzag steps between blue points containing one or two red points. This is labelled l or s in (a). The outer border has red points on the outside with one or two green points between two red points. The inner and outer sequences of l and s are the same, starting at the first `blue point' or the first `green point'. Three families of parastichies are clearly visible: they are characterized by three Fibonacci numbers (fu-1,fu,fu+1) indicating the increase of indices jumping from a point to a neighbouring point following a parastichy. These numbers are indicated on the figure [they correspond to u = 9 for (a) and u = 10 for (b)]. Notice that if we orient the border following parastichies defined by the smaller increase given by fu-1, the borders in (a) and (b) are in reverse orientation: this is associated with the parity of u.