Mathematical specification of hitomezashi designs

TL;DR

The paper formalizes hitomezashi designs by encoding vertical and horizontal stitch lines with two binary strings $v$ and $w$, and studies duality between the front and reverse patterns. It develops a robust framework for binary-encoded hitomezashi, proves self-duality conditions, and provides encodings for a suite of traditional patterns, highlighting which are self-dual and how duality manifests across orientations. The authors then connect hitomezashi loops to Fibonacci snowflakes and introduce Pell persimmon polyomino patterns via Pell words, culminating in the Persimmon-Snowflake Conjecture that the largest loop in a Pell persimmon of order $n$ matches the Fibonacci snowflake of order $n$. These results bridge textile art with combinatorial and number-theoretic structures, enabling new design approaches and inviting further mathematical exploration of duality, loop geometry, and lattice tilings in sashiko-inspired patterns.

Abstract

Two mathematical aspects of the centuries-old Japanese sashiko stitching form hitomezashi are discussed: the encoding of designs using words from a binary alphabet, and duality. Traditional hitomezashi designs are analysed using these two ideas. Self-dual hitomezashi designs related to Fibonacci snowflakes, which we term Pell persimmon polyomino patterns, are proposed. Both these designs and the binary words used to generate them appear to be new to their respective literatures.

Figures (21)

• Figure 1: A mathematically-designed stitch pattern for hitomezashi. This piece has been worked on white Aida cloth using blue cotton thread. The traditional roles of blue and white --- indigo-dyed cloth and white thread --- have been exchanged.
• Figure 2: Notice how different sequences of vertical stitches on the left and on the right interact with common lines of horizontal stitches to give different patterns. This piece has been worked with cotton thread on blue hessian (burlap).
• Figure 3: Hitomezashi as a stitch form can incorporate crossing stitches and diagonal stitches. This design features both.
• Figure 4: Starting to decorate a small square coaster with hitomezashi. The first ten vertical lines of running stitch have been completed.
• Figure 5: The two sides of a coaster decorated with hitomezashi. On the reverse of a pattern of offset crosses jūjizashi, a regular pattern of small squares and stepped lines (kuchi and yamagata) is formed. This is the dual pattern.