Stitching Arrowhead Curves: Extending the Sierpinski Arrowhead Curve to Higher Dimensions
This work provides a theoretical extension of a known fractal curve to higher dimensions, which is incremental for mathematics but offers a novel visualization method for artistic applications.
The authors extend the Sierpinski arrowhead curve to arbitrary dimensions using reproduction rules, enabling visualization across levels and demonstrating an application in knitwear design.
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.