Extensions of Hitomezashi Patterns

TL;DR

This paper explores advanced structural and classification results for hitomezashi patterns, including optimization and the analysis of long-stitch variants with non-axis-aligned directions, expanding understanding of these traditional Japanese embroidery designs.

Contribution

It provides a complete classification of long-stitch hitomezashi patterns and introduces new results on their structural properties and variants with non-axis-aligned stitches.

Findings

Classification of long-stitch patterns with stitches > 1

Optimization results on pattern arrangements

Analysis of non-axis-aligned stitch variants

Abstract

Hitomezashi, a form of traditional Japanese embroidery, gives rise to intricate arrangements of axis-parallel unit-length stitches in the plane. Pete studied these patterns in the context of percolation theory, and the first two authors recently investigated additional structural properties of them. In this paper, we establish several optimization-style results on hitomezashi patterns and provide a complete classification of "long-stitch" hitomezashi patterns in which stitches have length greater than 1. We also study variants in which stitches can have directions not parallel to the coordinate axes.