Stitching Arrowhead Curves: Extending the Sierpinski Arrowhead Curve to Higher Dimensions
Eric Zimmermann, Stefan Bruckner

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.
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