The generalized Sierpiński Arrowhead Curve

Abstract

We define special Hamiltonian-paths and special permutations of the up-facing dark tiles on a checked triangular grid related to the generalized Sierpiński Gasket. Our definitions and observations make possible the generalization of the Sierpiński Arrowhead Curve for all orders. We produce these symmetric recursive curves in many ways by two kinds of asymmetric paths which are in a bijective relation and unambiguously transformable into each other in any order. These node-rewriting and edge-rewriting recursive curves keep their self-avoiding and simple properties after the transformation and their cardinality specifies a new integer sequence. We show a transformation table to change the curves into each other and we give another table to change them into Lindenmayer-system strings both by the absolute direction codes of their edges.

16 pages, 10 figures, 3 tables