Upon having selected a likely bird motif tile, the geometric lines can be made more 'refined' by the addition of arcs. The aim behind this is to give the bird a more natural, life-like appearance, rather than the more severe geometric lines that obviously are of a somewhat stylised motif.
A similar principle to this can be applied to tessellations that are more ‘developed' in terms of their angularity, such as with a Greek cross tessellation. Here the line can be said to be of two segments (i). Again, a s-curve process can be applied, for example (j), of which such a line can be said to be arbitrary, as other combinations of the arc can be introduced, of equal validity. Consequently, the arbitrary line may thus not necessarily be the ‘best'. Therefore, examples of this type have to be analysed in an systematic manner to ensure that the best is indeed found. This is done in a series of steps, of which the first is noting that the line can be replaced with either a concave of convex arc, or alternatively left in situ, thus giving three possible arrangements. Similarly, the other line can have the same three possibilities as described, duly adapted for that particular purpose. From this therefore, simple calculation show that for each possibility there is 3 x 3 = 9 arrangements, i.e. nine distinct lines of every combination. Consequently, the nine arrangements can thus be shown as (nine) tiles as according to the Greek cross symmetry arrangement. Consequently, having thus investigated ‘all possibilities' the best example can thus be selected as a definitive motif. Of course, in the strictest sense all are equally valid, and could thus be shown as nine distinct tessellations. However, as they can be said to be (minor) variations of each other, to consider these as unique would be essentially an overstatement.