Generalisations and randomisation of the plane Koch curve

The Koch curve evolves from a base equilateral triangle by the trisection of each side and the replication of the original triangle on the mid-section, the process being repeated ad infinitum by the addition of sets of successively smaller triangles. The process is generalised to replace the trisectioning by (2k+1)-sectioning. It is shown that a square is the only other regular polygon on which the (2k+1)-sectioning procedure can be implemented. The Koch curves thus generated are strictly self-similar, their fractal dimensions being similarity dimensions and enclose simply connected areas. Randomisation of the generating procedure is also discussed.