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

2026-04-21

AI Summary

This study addresses the absence of a natural path representation for extending the two-dimensional Sierpiński arrowhead curve to higher dimensions. Building upon its rewriting rules and fractal geometric properties, the work proposes a novel construction method for high-dimensional Sierpiński arrowhead curves that generalizes seamlessly to arbitrary dimensions. By integrating formal language-based rewriting systems with parametric visualization techniques, the approach enables hierarchical generation and rendering of these curves across scales. The methodology has been successfully applied to knitwear design—exemplified by sweater collar patterns—demonstrating an effective synthesis of fractal mathematical structures with expressive fashion aesthetics.

Technology Category

• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning
• Cognitive Modeling & Cognitive Systems: Computational Creativity
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies

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.

Problem

• Sierpinski arrowhead curve
• higher dimensions
• fractal
• curve representation
• dimensional extension

Innovation

• Sierpinski arrowhead curve
• higher dimensions
• reproduction rules
• fractal visualization
• generative design