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.

翻译：谢尔宾斯基三角形和谢尔宾斯基箭尾曲线均定义于二维空间，可用于建模同一分形结构。尽管三角构造存在向任意维度的自然推广，但曲线表示的类似推广尚付阙如。本文通过分析二维谢尔宾斯基箭尾曲线的性质，基于生成规则建立其向任意维度的推广框架。基于该框架，我们进一步提出跨层级呈现此类曲线的可视化方法。鉴于几何图案在艺术领域（尤其是时装设计）中具有悠久的应用传统，本文最后以毛衣育克部位为例，展示了该可视化方法在针织服装中的具体应用。