The generalized Sierpi\'{n}ski Arrowhead Curve

TL;DR

This paper generalizes the Sierpiński Arrowhead Curve to all orders using special Hamiltonian paths and permutations on a triangular grid, introducing recursive symmetric curves with unique transformations and a new integer sequence.

Contribution

It extends the Sierpiński Arrowhead Curve to all orders through novel recursive curves, transformations, and a new integer sequence, with detailed tables for conversions.

Findings

Defined special Hamiltonian-paths and permutations for the curves

Established bijective relations and transformation tables

Introduced a new integer sequence from the recursive curves

Abstract

We define special Hamiltonian-paths and special permutations of the up-facing dark tiles on a checked triangular grid related to the generalized Sierpi\'{n}ski Gasket. Our definitions and observations make possible the generalization of the Sierpi\'{n}ski 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.