Curves of order n generated by folding morphisms are walks on the square grid, also coded by sequences (starting with D) of n-1 U's and D's, the Up and Down folds. These are also known as n-folds. In the square grid they uniquely correspond to folding morphisms, which are a special class of morphisms sigma on the alphabet {a,b,c,d}. (There is in particular the requirement that the first two letters of sigma(a) are ab.) Here the letters a,b,c, and d correspond to the four possible steps of the walk.
A curve C = C1 of order n generates curves Cj of order n^j by the process of iterated folding. Iterated folding corresponds to iterates of the folding morphism.
Grid-filling or plane-filling means that all the points in arbitrary large balls of gridpoints are eventually visited by the Cj.
Perfect means that four 90-degree rotated copies of the curves Cj started at the origin will pass exactly twice through all grid-points as j tends to infinity (except the origin itself).
It is a theorem that a(
A022544(n)) = 0 for all n.
