Loops and Regions in Hitomezashi Patterns

TL;DR

This paper explores the mathematical properties of hitomezashi patterns, revealing new topological and combinatorial invariants of loops and providing an asymptotic estimate for the number of regions in random patterns.

Contribution

It establishes new congruence properties of loops in hitomezashi patterns and analyzes the expected number of regions in random patterns, advancing the mathematical understanding of these designs.

Findings

Loops have length ≡ 4 mod 8 and area ≡ 1 mod 4.

Every loop has odd width and height.

Expected number of regions in random patterns is asymptotically ((π^2 - 9)/12) * m * n.

Abstract

Hitomezashi patterns, which originate from traditional Japanese embroidery, are intricate arrangements of unit-length line segments called stitches. The stitches connect to form hitomezashi strands and hitomezashi loops, which divide the plane into regions. We investigate the deeper mathematical properties of these patterns, which also feature prominently in the study of corner percolation. It was previously known that every loop in a hitomezashi pattern has odd width and odd height. We additionally prove that such a loop has length congruent to $4$ modulo $8$ and area congruent to $1$ modulo $4$. Although these results are simple to state, their proofs require us to understand the delicate topological and combinatorial properties of slicing operations that can be applied to hitomezashi patterns. We also show that the expected number of regions in a random $m\times n$ hitomezashi pattern (chosen according to a natural random model) is asymptotically $\left(\frac{\pi^2-9}{12}+o(1)\right)mn$.