Stitching Arrowhead Curves: Extending the Sierpinski Arrowhead Curve to Higher Dimensions

TL;DR

This paper extends the Sierpinski arrowhead curve from two dimensions to higher dimensions using reproduction rules, enabling new visualizations and applications in arts like knitwear.

Contribution

It introduces a novel method to generalize the arrowhead curve to arbitrary dimensions, bridging fractal geometry and artistic visualization.

Findings

Formulated an extension of the arrowhead curve to higher dimensions.

Developed a visualization approach for these higher-dimensional curves.

Applied the visualization in artistic design, specifically in knitwear.

Abstract

The Sierpinski triangle and the Sierpinski arrowhead curve are both defined in dimension 2 and can be used to model the same fractal. While a natural extension of the triangular construction to arbitrary dimensions exists, an analogous extension of the curve representation does not. In this article, we analyze the properties of the two-dimensional Sierpinski arrowhead curve to formulate an extension to arbitrary dimensions based on reproduction rules. Building on this formulation, we demonstrate a way to visualize such curves in a comparative manner across levels. Finally, as geometric patterns have a long history in the arts, and especially in fashion, we exemplify this visualization approach in knitwear, specifically in the yoke of a sweater.